Article NCFCONT1, MML version 4.99.1005
:: NCFCONT1:th 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 - b3 = b2 + - b3;
:: NCFCONT1:th 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
- b2 = (- 1r) * b2;
:: NCFCONT1:funcnot 1 => NCFCONT1:func 1
definition
let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a3 be Function-like Relation of the carrier of a1,the carrier of a2;
func ||.A3.|| -> Function-like Relation of the carrier of a1,REAL means
dom it = dom a3 &
(for b1 being Element of the carrier of a1
st b1 in dom it
holds it . b1 = ||.a3 /. b1.||);
end;
:: NCFCONT1:def 2
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Function-like Relation of the carrier of b1,REAL holds
b4 = ||.b3.||
iff
dom b4 = dom b3 &
(for b5 being Element of the carrier of b1
st b5 in dom b4
holds b4 . b5 = ||.b3 /. b5.||);
:: NCFCONT1:funcnot 2 => NCFCONT1:func 2
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
let a3 be Function-like Relation of the carrier of a1,the carrier of a2;
func ||.A3.|| -> Function-like Relation of the carrier of a1,REAL means
dom it = dom a3 &
(for b1 being Element of the carrier of a1
st b1 in dom it
holds it . b1 = ||.a3 /. b1.||);
end;
:: NCFCONT1:def 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Function-like Relation of the carrier of b1,REAL holds
b4 = ||.b3.||
iff
dom b4 = dom b3 &
(for b5 being Element of the carrier of b1
st b5 in dom b4
holds b4 . b5 = ||.b3 /. b5.||);
:: NCFCONT1:funcnot 3 => NCFCONT1:func 3
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
let a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a3 be Function-like Relation of the carrier of a1,the carrier of a2;
func ||.A3.|| -> Function-like Relation of the carrier of a1,REAL means
dom it = dom a3 &
(for b1 being Element of the carrier of a1
st b1 in dom it
holds it . b1 = ||.a3 /. b1.||);
end;
:: NCFCONT1:def 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Function-like Relation of the carrier of b1,REAL holds
b4 = ||.b3.||
iff
dom b4 = dom b3 &
(for b5 being Element of the carrier of b1
st b5 in dom b4
holds b4 . b5 = ||.b3 /. b5.||);
:: NCFCONT1:modenot 1 => NCFCONT1:mode 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be Element of the carrier of a1;
mode Neighbourhood of A2 -> Element of bool the carrier of a1 means
ex b1 being Element of REAL st
0 < b1 &
{b2 where b2 is Element of the carrier of a1: ||.b2 - a2.|| < b1} c= it;
end;
:: NCFCONT1:dfs 4
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be Element of the carrier of a1;
let a3 be Element of bool the carrier of a1;
To prove
a3 is Neighbourhood of a2
it is sufficient to prove
thus ex b1 being Element of REAL st
0 < b1 &
{b2 where b2 is Element of the carrier of a1: ||.b2 - a2.|| < b1} c= a3;
:: NCFCONT1:def 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1 holds
b3 is Neighbourhood of b2
iff
ex b4 being Element of REAL st
0 < b4 &
{b5 where b5 is Element of the carrier of b1: ||.b5 - b2.|| < b4} c= b3;
:: NCFCONT1:th 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Element of the carrier of b1
for b3 being Element of REAL
st 0 < b3
holds {b4 where b4 is Element of the carrier of b1: ||.b4 - b2.|| < b3} is Neighbourhood of b2;
:: NCFCONT1:th 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Element of the carrier of b1
for b3 being Neighbourhood of b2 holds
b2 in b3;
:: NCFCONT1:attrnot 1 => NCFCONT1:attr 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be Element of bool the carrier of a1;
attr a2 is compact means
for b1 being Function-like quasi_total Relation of NAT,the carrier of a1
st rng b1 c= a2
holds ex b2 being Function-like quasi_total Relation of NAT,the carrier of a1 st
b2 is subsequence of b1 & b2 is convergent(a1) & lim b2 in a2;
end;
:: NCFCONT1:dfs 5
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be Element of bool the carrier of a1;
To prove
a2 is compact
it is sufficient to prove
thus for b1 being Function-like quasi_total Relation of NAT,the carrier of a1
st rng b1 c= a2
holds ex b2 being Function-like quasi_total Relation of NAT,the carrier of a1 st
b2 is subsequence of b1 & b2 is convergent(a1) & lim b2 in a2;
:: NCFCONT1:def 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Element of bool the carrier of b1 holds
b2 is compact(b1)
iff
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st rng b3 c= b2
holds ex b4 being Function-like quasi_total Relation of NAT,the carrier of b1 st
b4 is subsequence of b3 & b4 is convergent(b1) & lim b4 in b2;
:: NCFCONT1:attrnot 2 => NCFCONT1:attr 2
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be Element of bool the carrier of a1;
attr a2 is closed means
for b1 being Function-like quasi_total Relation of NAT,the carrier of a1
st rng b1 c= a2 & b1 is convergent(a1)
holds lim b1 in a2;
end;
:: NCFCONT1:dfs 6
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be Element of bool the carrier of a1;
To prove
a2 is closed
it is sufficient to prove
thus for b1 being Function-like quasi_total Relation of NAT,the carrier of a1
st rng b1 c= a2 & b1 is convergent(a1)
holds lim b1 in a2;
:: NCFCONT1:def 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Element of bool the carrier of b1 holds
b2 is closed(b1)
iff
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st rng b3 c= b2 & b3 is convergent(b1)
holds lim b3 in b2;
:: NCFCONT1:attrnot 3 => NCFCONT1:attr 3
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be Element of bool the carrier of a1;
attr a2 is open means
a2 ` is closed(a1);
end;
:: NCFCONT1:dfs 7
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be Element of bool the carrier of a1;
To prove
a2 is open
it is sufficient to prove
thus a2 ` is closed(a1);
:: NCFCONT1:def 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Element of bool the carrier of b1 holds
b2 is open(b1)
iff
b2 ` is closed(b1);
:: NCFCONT1:funcnot 4 => NCFCONT1:func 4
definition
let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a3 be Function-like Relation of the carrier of a1,the carrier of a2;
let a4 be Function-like quasi_total Relation of NAT,the carrier of a1;
assume rng a4 c= dom a3;
func A3 * A4 -> Function-like quasi_total Relation of NAT,the carrier of a2 equals
a4 * a3;
end;
:: NCFCONT1:def 9
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
st rng b4 c= dom b3
holds b3 * b4 = b4 * b3;
:: NCFCONT1:funcnot 5 => NCFCONT1:func 5
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
let a3 be Function-like Relation of the carrier of a1,the carrier of a2;
let a4 be Function-like quasi_total Relation of NAT,the carrier of a1;
assume rng a4 c= dom a3;
func A3 * A4 -> Function-like quasi_total Relation of NAT,the carrier of a2 equals
a4 * a3;
end;
:: NCFCONT1:def 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
st rng b4 c= dom b3
holds b3 * b4 = b4 * b3;
:: NCFCONT1:funcnot 6 => NCFCONT1:func 6
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
let a3 be Function-like Relation of the carrier of a2,the carrier of a1;
let a4 be Function-like quasi_total Relation of NAT,the carrier of a2;
assume rng a4 c= dom a3;
func A3 * A4 -> Function-like quasi_total Relation of NAT,the carrier of a1 equals
a4 * a3;
end;
:: NCFCONT1:def 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b2,the carrier of b1
for b4 being Function-like quasi_total Relation of NAT,the carrier of b2
st rng b4 c= dom b3
holds b3 * b4 = b4 * b3;
:: NCFCONT1:funcnot 7 => NCFCONT1:func 7
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be Function-like Relation of the carrier of a1,COMPLEX;
let a3 be Function-like quasi_total Relation of NAT,the carrier of a1;
assume rng a3 c= dom a2;
func A2 * A3 -> Function-like quasi_total Relation of NAT,COMPLEX equals
a3 * a2;
end;
:: NCFCONT1:def 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like Relation of the carrier of b1,COMPLEX
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st rng b3 c= dom b2
holds b2 * b3 = b3 * b2;
:: NCFCONT1:funcnot 8 => NCFCONT1:func 8
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
let a2 be Function-like Relation of the carrier of a1,COMPLEX;
let a3 be Function-like quasi_total Relation of NAT,the carrier of a1;
assume rng a3 c= dom a2;
func A2 * A3 -> Function-like quasi_total Relation of NAT,COMPLEX equals
a3 * a2;
end;
:: NCFCONT1:def 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like Relation of the carrier of b1,COMPLEX
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st rng b3 c= dom b2
holds b2 * b3 = b3 * b2;
:: NCFCONT1:funcnot 9 => NCFCONT1:func 9
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be Function-like Relation of the carrier of a1,REAL;
let a3 be Function-like quasi_total Relation of NAT,the carrier of a1;
assume rng a3 c= dom a2;
func A2 * A3 -> Function-like quasi_total Relation of NAT,REAL equals
a3 * a2;
end;
:: NCFCONT1:def 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like Relation of the carrier of b1,REAL
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st rng b3 c= dom b2
holds b2 * b3 = b3 * b2;
:: NCFCONT1:prednot 1 => NCFCONT1:pred 1
definition
let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a3 be Function-like Relation of the carrier of a1,the carrier of a2;
let a4 be Element of the carrier of a1;
pred A3 is_continuous_in A4 means
a4 in dom a3 &
(for b1 being Function-like quasi_total Relation of NAT,the carrier of a1
st rng b1 c= dom a3 & b1 is convergent(a1) & lim b1 = a4
holds a3 * b1 is convergent(a2) & a3 /. a4 = lim (a3 * b1));
end;
:: NCFCONT1:dfs 14
definiens
let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a3 be Function-like Relation of the carrier of a1,the carrier of a2;
let a4 be Element of the carrier of a1;
To prove
a3 is_continuous_in a4
it is sufficient to prove
thus a4 in dom a3 &
(for b1 being Function-like quasi_total Relation of NAT,the carrier of a1
st rng b1 c= dom a3 & b1 is convergent(a1) & lim b1 = a4
holds a3 * b1 is convergent(a2) & a3 /. a4 = lim (a3 * b1));
:: NCFCONT1:def 15
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Element of the carrier of b1 holds
b3 is_continuous_in b4
iff
b4 in dom b3 &
(for b5 being Function-like quasi_total Relation of NAT,the carrier of b1
st rng b5 c= dom b3 & b5 is convergent(b1) & lim b5 = b4
holds b3 * b5 is convergent(b2) & b3 /. b4 = lim (b3 * b5));
:: NCFCONT1:prednot 2 => NCFCONT1:pred 2
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
let a3 be Function-like Relation of the carrier of a1,the carrier of a2;
let a4 be Element of the carrier of a1;
pred A3 is_continuous_in A4 means
a4 in dom a3 &
(for b1 being Function-like quasi_total Relation of NAT,the carrier of a1
st rng b1 c= dom a3 & b1 is convergent(a1) & lim b1 = a4
holds a3 * b1 is convergent(a2) & a3 /. a4 = lim (a3 * b1));
end;
:: NCFCONT1:dfs 15
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
let a3 be Function-like Relation of the carrier of a1,the carrier of a2;
let a4 be Element of the carrier of a1;
To prove
a3 is_continuous_in a4
it is sufficient to prove
thus a4 in dom a3 &
(for b1 being Function-like quasi_total Relation of NAT,the carrier of a1
st rng b1 c= dom a3 & b1 is convergent(a1) & lim b1 = a4
holds a3 * b1 is convergent(a2) & a3 /. a4 = lim (a3 * b1));
:: NCFCONT1:def 16
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Element of the carrier of b1 holds
b3 is_continuous_in b4
iff
b4 in dom b3 &
(for b5 being Function-like quasi_total Relation of NAT,the carrier of b1
st rng b5 c= dom b3 & b5 is convergent(b1) & lim b5 = b4
holds b3 * b5 is convergent(b2) & b3 /. b4 = lim (b3 * b5));
:: NCFCONT1:prednot 3 => NCFCONT1:pred 3
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
let a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a3 be Function-like Relation of the carrier of a1,the carrier of a2;
let a4 be Element of the carrier of a1;
pred A3 is_continuous_in A4 means
a4 in dom a3 &
(for b1 being Function-like quasi_total Relation of NAT,the carrier of a1
st rng b1 c= dom a3 & b1 is convergent(a1) & lim b1 = a4
holds a3 * b1 is convergent(a2) & a3 /. a4 = lim (a3 * b1));
end;
:: NCFCONT1:dfs 16
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
let a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a3 be Function-like Relation of the carrier of a1,the carrier of a2;
let a4 be Element of the carrier of a1;
To prove
a3 is_continuous_in a4
it is sufficient to prove
thus a4 in dom a3 &
(for b1 being Function-like quasi_total Relation of NAT,the carrier of a1
st rng b1 c= dom a3 & b1 is convergent(a1) & lim b1 = a4
holds a3 * b1 is convergent(a2) & a3 /. a4 = lim (a3 * b1));
:: NCFCONT1:def 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Element of the carrier of b1 holds
b3 is_continuous_in b4
iff
b4 in dom b3 &
(for b5 being Function-like quasi_total Relation of NAT,the carrier of b1
st rng b5 c= dom b3 & b5 is convergent(b1) & lim b5 = b4
holds b3 * b5 is convergent(b2) & b3 /. b4 = lim (b3 * b5));
:: NCFCONT1:prednot 4 => NCFCONT1:pred 4
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be Function-like Relation of the carrier of a1,COMPLEX;
let a3 be Element of the carrier of a1;
pred A2 is_continuous_in A3 means
a3 in dom a2 &
(for b1 being Function-like quasi_total Relation of NAT,the carrier of a1
st rng b1 c= dom a2 & b1 is convergent(a1) & lim b1 = a3
holds a2 * b1 is convergent & a2 /. a3 = lim (a2 * b1));
end;
:: NCFCONT1:dfs 17
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be Function-like Relation of the carrier of a1,COMPLEX;
let a3 be Element of the carrier of a1;
To prove
a2 is_continuous_in a3
it is sufficient to prove
thus a3 in dom a2 &
(for b1 being Function-like quasi_total Relation of NAT,the carrier of a1
st rng b1 c= dom a2 & b1 is convergent(a1) & lim b1 = a3
holds a2 * b1 is convergent & a2 /. a3 = lim (a2 * b1));
:: NCFCONT1:def 18
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like Relation of the carrier of b1,COMPLEX
for b3 being Element of the carrier of b1 holds
b2 is_continuous_in b3
iff
b3 in dom b2 &
(for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
st rng b4 c= dom b2 & b4 is convergent(b1) & lim b4 = b3
holds b2 * b4 is convergent & b2 /. b3 = lim (b2 * b4));
:: NCFCONT1:prednot 5 => NCFCONT1:pred 5
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be Function-like Relation of the carrier of a1,REAL;
let a3 be Element of the carrier of a1;
pred A2 is_continuous_in A3 means
a3 in dom a2 &
(for b1 being Function-like quasi_total Relation of NAT,the carrier of a1
st rng b1 c= dom a2 & b1 is convergent(a1) & lim b1 = a3
holds a2 * b1 is convergent & a2 /. a3 = lim (a2 * b1));
end;
:: NCFCONT1:dfs 18
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be Function-like Relation of the carrier of a1,REAL;
let a3 be Element of the carrier of a1;
To prove
a2 is_continuous_in a3
it is sufficient to prove
thus a3 in dom a2 &
(for b1 being Function-like quasi_total Relation of NAT,the carrier of a1
st rng b1 c= dom a2 & b1 is convergent(a1) & lim b1 = a3
holds a2 * b1 is convergent & a2 /. a3 = lim (a2 * b1));
:: NCFCONT1:def 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like Relation of the carrier of b1,REAL
for b3 being Element of the carrier of b1 holds
b2 is_continuous_in b3
iff
b3 in dom b2 &
(for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
st rng b4 c= dom b2 & b4 is convergent(b1) & lim b4 = b3
holds b2 * b4 is convergent & b2 /. b3 = lim (b2 * b4));
:: NCFCONT1:prednot 6 => NCFCONT1:pred 6
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
let a2 be Function-like Relation of the carrier of a1,COMPLEX;
let a3 be Element of the carrier of a1;
pred A2 is_continuous_in A3 means
a3 in dom a2 &
(for b1 being Function-like quasi_total Relation of NAT,the carrier of a1
st rng b1 c= dom a2 & b1 is convergent(a1) & lim b1 = a3
holds a2 * b1 is convergent & a2 /. a3 = lim (a2 * b1));
end;
:: NCFCONT1:dfs 19
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
let a2 be Function-like Relation of the carrier of a1,COMPLEX;
let a3 be Element of the carrier of a1;
To prove
a2 is_continuous_in a3
it is sufficient to prove
thus a3 in dom a2 &
(for b1 being Function-like quasi_total Relation of NAT,the carrier of a1
st rng b1 c= dom a2 & b1 is convergent(a1) & lim b1 = a3
holds a2 * b1 is convergent & a2 /. a3 = lim (a2 * b1));
:: NCFCONT1:def 20
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like Relation of the carrier of b1,COMPLEX
for b3 being Element of the carrier of b1 holds
b2 is_continuous_in b3
iff
b3 in dom b2 &
(for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
st rng b4 c= dom b2 & b4 is convergent(b1) & lim b4 = b3
holds b2 * b4 is convergent & b2 /. b3 = lim (b2 * b4));
:: NCFCONT1:th 5
theorem
for b1 being Element of NAT
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b4 being Function-like quasi_total Relation of NAT,the carrier of b2
for b5 being Function-like Relation of the carrier of b2,the carrier of b3
st rng b4 c= dom b5
holds b4 . b1 in dom b5;
:: NCFCONT1:th 6
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like quasi_total Relation of NAT,the carrier of b2
for b5 being Function-like Relation of the carrier of b2,the carrier of b3
st rng b4 c= dom b5
holds b4 . b1 in dom b5;
:: NCFCONT1:th 7
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like quasi_total Relation of NAT,the carrier of b3
for b5 being Function-like Relation of the carrier of b3,the carrier of b2
st rng b4 c= dom b5
holds b4 . b1 in dom b5;
:: NCFCONT1:th 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being set holds
b3 in rng b2
iff
ex b4 being Element of NAT st
b3 = b2 . b4;
:: NCFCONT1:th 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b3 is subsequence of b2
holds rng b3 c= rng b2;
:: NCFCONT1:th 10
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
st rng b4 c= dom b3
for b5 being Element of NAT holds
(b3 * b4) . b5 = b3 /. (b4 . b5);
:: NCFCONT1:th 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
st rng b4 c= dom b3
for b5 being Element of NAT holds
(b3 * b4) . b5 = b3 /. (b4 . b5);
:: NCFCONT1:th 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b2,the carrier of b1
for b4 being Function-like quasi_total Relation of NAT,the carrier of b2
st rng b4 c= dom b3
for b5 being Element of NAT holds
(b3 * b4) . b5 = b3 /. (b4 . b5);
:: NCFCONT1:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like Relation of the carrier of b1,COMPLEX
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st rng b3 c= dom b2
for b4 being Element of NAT holds
(b2 * b3) . b4 = b2 /. (b3 . b4);
:: NCFCONT1:th 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like Relation of the carrier of b1,REAL
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st rng b3 c= dom b2
for b4 being Element of NAT holds
(b2 * b3) . b4 = b2 /. (b3 . b4);
:: NCFCONT1:th 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like Relation of the carrier of b1,COMPLEX
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st rng b3 c= dom b2
for b4 being Element of NAT holds
(b2 * b3) . b4 = b2 /. (b3 . b4);
:: NCFCONT1:th 16
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
for b5 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL
st rng b4 c= dom b3
holds (b3 * b4) * b5 = b3 * (b4 * b5);
:: NCFCONT1:th 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
for b5 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL
st rng b4 c= dom b3
holds (b3 * b4) * b5 = b3 * (b4 * b5);
:: NCFCONT1:th 18
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b2,the carrier of b1
for b4 being Function-like quasi_total Relation of NAT,the carrier of b2
for b5 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL
st rng b4 c= dom b3
holds (b3 * b4) * b5 = b3 * (b4 * b5);
:: NCFCONT1:th 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like Relation of the carrier of b1,COMPLEX
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
for b4 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL
st rng b3 c= dom b2
holds (b2 * b3) * b4 = b2 * (b3 * b4);
:: NCFCONT1:th 20
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like Relation of the carrier of b1,REAL
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
for b4 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL
st rng b3 c= dom b2
holds (b2 * b3) * b4 = b2 * (b3 * b4);
:: NCFCONT1:th 21
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like Relation of the carrier of b1,COMPLEX
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
for b4 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL
st rng b3 c= dom b2
holds (b2 * b3) * b4 = b2 * (b3 * b4);
:: NCFCONT1:th 22
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4, b5 being Function-like quasi_total Relation of NAT,the carrier of b1
st rng b4 c= dom b3 & b5 is subsequence of b4
holds b3 * b5 is subsequence of b3 * b4;
:: NCFCONT1:th 23
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4, b5 being Function-like quasi_total Relation of NAT,the carrier of b1
st rng b4 c= dom b3 & b5 is subsequence of b4
holds b3 * b5 is subsequence of b3 * b4;
:: NCFCONT1:th 24
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b2,the carrier of b1
for b4, b5 being Function-like quasi_total Relation of NAT,the carrier of b2
st rng b4 c= dom b3 & b5 is subsequence of b4
holds b3 * b5 is subsequence of b3 * b4;
:: NCFCONT1:th 25
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Element of NAT
for b3 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL holds
(b1 * b3) . b2 = b1 . (b3 . b2);
:: NCFCONT1:th 26
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like Relation of the carrier of b1,COMPLEX
for b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b1
st rng b3 c= dom b2 & b4 is subsequence of b3
holds b2 * b4 is subsequence of b2 * b3;
:: NCFCONT1:th 27
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like Relation of the carrier of b1,REAL
for b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b1
st rng b3 c= dom b2 & b4 is subsequence of b3
holds b2 * b4 is subsequence of b2 * b3;
:: NCFCONT1:th 28
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like Relation of the carrier of b1,COMPLEX
for b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b1
st rng b3 c= dom b2 & b4 is subsequence of b3
holds b2 * b4 is subsequence of b2 * b3;
:: NCFCONT1:th 29
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Element of the carrier of b1 holds
b3 is_continuous_in b4
iff
b4 in dom b3 &
(for b5 being Element of REAL
st 0 < b5
holds ex b6 being Element of REAL st
0 < b6 &
(for b7 being Element of the carrier of b1
st b7 in dom b3 & ||.b7 - b4.|| < b6
holds ||.(b3 /. b7) - (b3 /. b4).|| < b5));
:: NCFCONT1:th 30
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Element of the carrier of b1 holds
b3 is_continuous_in b4
iff
b4 in dom b3 &
(for b5 being Element of REAL
st 0 < b5
holds ex b6 being Element of REAL st
0 < b6 &
(for b7 being Element of the carrier of b1
st b7 in dom b3 & ||.b7 - b4.|| < b6
holds ||.(b3 /. b7) - (b3 /. b4).|| < b5));
:: NCFCONT1:th 31
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b2,the carrier of b1
for b4 being Element of the carrier of b2 holds
b3 is_continuous_in b4
iff
b4 in dom b3 &
(for b5 being Element of REAL
st 0 < b5
holds ex b6 being Element of REAL st
0 < b6 &
(for b7 being Element of the carrier of b2
st b7 in dom b3 & ||.b7 - b4.|| < b6
holds ||.(b3 /. b7) - (b3 /. b4).|| < b5));
:: NCFCONT1:th 32
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like Relation of the carrier of b1,REAL
for b3 being Element of the carrier of b1 holds
b2 is_continuous_in b3
iff
b3 in dom b2 &
(for b4 being Element of REAL
st 0 < b4
holds ex b5 being Element of REAL st
0 < b5 &
(for b6 being Element of the carrier of b1
st b6 in dom b2 & ||.b6 - b3.|| < b5
holds abs ((b2 /. b6) - (b2 /. b3)) < b4));
:: NCFCONT1:th 33
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like Relation of the carrier of b1,COMPLEX
for b3 being Element of the carrier of b1 holds
b2 is_continuous_in b3
iff
b3 in dom b2 &
(for b4 being Element of REAL
st 0 < b4
holds ex b5 being Element of REAL st
0 < b5 &
(for b6 being Element of the carrier of b1
st b6 in dom b2 & ||.b6 - b3.|| < b5
holds |.(b2 /. b6) - (b2 /. b3).| < b4));
:: NCFCONT1:th 34
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like Relation of the carrier of b1,COMPLEX
for b3 being Element of the carrier of b1 holds
b2 is_continuous_in b3
iff
b3 in dom b2 &
(for b4 being Element of REAL
st 0 < b4
holds ex b5 being Element of REAL st
0 < b5 &
(for b6 being Element of the carrier of b1
st b6 in dom b2 & ||.b6 - b3.|| < b5
holds |.(b2 /. b6) - (b2 /. b3).| < b4));
:: NCFCONT1:th 35
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Element of the carrier of b1 holds
b3 is_continuous_in b4
iff
b4 in dom b3 &
(for b5 being Neighbourhood of b3 /. b4 holds
ex b6 being Neighbourhood of b4 st
for b7 being Element of the carrier of b1
st b7 in dom b3 & b7 in b6
holds b3 /. b7 in b5);
:: NCFCONT1:th 36
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Element of the carrier of b1 holds
b3 is_continuous_in b4
iff
b4 in dom b3 &
(for b5 being Neighbourhood of b3 /. b4 holds
ex b6 being Neighbourhood of b4 st
for b7 being Element of the carrier of b1
st b7 in dom b3 & b7 in b6
holds b3 /. b7 in b5);
:: NCFCONT1:th 37
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b2,the carrier of b1
for b4 being Element of the carrier of b2 holds
b3 is_continuous_in b4
iff
b4 in dom b3 &
(for b5 being Neighbourhood of b3 /. b4 holds
ex b6 being Neighbourhood of b4 st
for b7 being Element of the carrier of b2
st b7 in dom b3 & b7 in b6
holds b3 /. b7 in b5);
:: NCFCONT1:th 38
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Element of the carrier of b1 holds
b3 is_continuous_in b4
iff
b4 in dom b3 &
(for b5 being Neighbourhood of b3 /. b4 holds
ex b6 being Neighbourhood of b4 st
b3 .: b6 c= b5);
:: NCFCONT1:th 39
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Element of the carrier of b1 holds
b3 is_continuous_in b4
iff
b4 in dom b3 &
(for b5 being Neighbourhood of b3 /. b4 holds
ex b6 being Neighbourhood of b4 st
b3 .: b6 c= b5);
:: NCFCONT1:th 40
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b2,the carrier of b1
for b4 being Element of the carrier of b2 holds
b3 is_continuous_in b4
iff
b4 in dom b3 &
(for b5 being Neighbourhood of b3 /. b4 holds
ex b6 being Neighbourhood of b4 st
b3 .: b6 c= b5);
:: NCFCONT1:th 41
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Element of the carrier of b1
st b4 in dom b3 &
(ex b5 being Neighbourhood of b4 st
(dom b3) /\ b5 = {b4})
holds b3 is_continuous_in b4;
:: NCFCONT1:th 42
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Element of the carrier of b1
st b4 in dom b3 &
(ex b5 being Neighbourhood of b4 st
(dom b3) /\ b5 = {b4})
holds b3 is_continuous_in b4;
:: NCFCONT1:th 43
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b2,the carrier of b1
for b4 being Element of the carrier of b2
st b4 in dom b3 &
(ex b5 being Neighbourhood of b4 st
(dom b3) /\ b5 = {b4})
holds b3 is_continuous_in b4;
:: NCFCONT1:th 44
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3, b4 being Function-like Relation of the carrier of b1,the carrier of b2
for b5 being Function-like quasi_total Relation of NAT,the carrier of b1
st rng b5 c= (dom b3) /\ dom b4
holds (b3 + b4) * b5 = (b3 * b5) + (b4 * b5) &
(b3 - b4) * b5 = (b3 * b5) - (b4 * b5);
:: NCFCONT1:th 45
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4 being Function-like Relation of the carrier of b1,the carrier of b2
for b5 being Function-like quasi_total Relation of NAT,the carrier of b1
st rng b5 c= (dom b3) /\ dom b4
holds (b3 + b4) * b5 = (b3 * b5) + (b4 * b5) &
(b3 - b4) * b5 = (b3 * b5) - (b4 * b5);
:: NCFCONT1:th 46
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4 being Function-like Relation of the carrier of b2,the carrier of b1
for b5 being Function-like quasi_total Relation of NAT,the carrier of b2
st rng b5 c= (dom b3) /\ dom b4
holds (b3 + b4) * b5 = (b3 * b5) + (b4 * b5) &
(b3 - b4) * b5 = (b3 * b5) - (b4 * b5);
:: NCFCONT1:th 47
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
for b5 being Element of COMPLEX
st rng b4 c= dom b3
holds (b5 (#) b3) * b4 = b5 * (b3 * b4);
:: NCFCONT1:th 48
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
for b5 being Element of REAL
st rng b4 c= dom b3
holds (b5 (#) b3) * b4 = b5 * (b3 * b4);
:: NCFCONT1:th 49
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b2,the carrier of b1
for b4 being Function-like quasi_total Relation of NAT,the carrier of b2
for b5 being Element of COMPLEX
st rng b4 c= dom b3
holds (b5 (#) b3) * b4 = b5 * (b3 * b4);
:: NCFCONT1:th 50
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
st rng b4 c= dom b3
holds ||.b3 * b4.|| = ||.b3.|| * b4 &
- (b3 * b4) = (- b3) * b4;
:: NCFCONT1:th 51
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
st rng b4 c= dom b3
holds ||.b3 * b4.|| = ||.b3.|| * b4 &
- (b3 * b4) = (- b3) * b4;
:: NCFCONT1:th 52
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b2,the carrier of b1
for b4 being Function-like quasi_total Relation of NAT,the carrier of b2
st rng b4 c= dom b3
holds ||.b3 * b4.|| = ||.b3.|| * b4 &
- (b3 * b4) = (- b3) * b4;
:: NCFCONT1:th 53
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3, b4 being Function-like Relation of the carrier of b1,the carrier of b2
for b5 being Element of the carrier of b1
st b3 is_continuous_in b5 & b4 is_continuous_in b5
holds b3 + b4 is_continuous_in b5 & b3 - b4 is_continuous_in b5;
:: NCFCONT1:th 54
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4 being Function-like Relation of the carrier of b1,the carrier of b2
for b5 being Element of the carrier of b1
st b3 is_continuous_in b5 & b4 is_continuous_in b5
holds b3 + b4 is_continuous_in b5 & b3 - b4 is_continuous_in b5;
:: NCFCONT1:th 55
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4 being Function-like Relation of the carrier of b2,the carrier of b1
for b5 being Element of the carrier of b2
st b3 is_continuous_in b5 & b4 is_continuous_in b5
holds b3 + b4 is_continuous_in b5 & b3 - b4 is_continuous_in b5;
:: NCFCONT1:th 56
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Element of the carrier of b1
for b5 being Element of COMPLEX
st b3 is_continuous_in b4
holds b5 (#) b3 is_continuous_in b4;
:: NCFCONT1:th 57
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Element of the carrier of b1
for b5 being Element of REAL
st b3 is_continuous_in b4
holds b5 (#) b3 is_continuous_in b4;
:: NCFCONT1:th 58
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b2,the carrier of b1
for b4 being Element of the carrier of b2
for b5 being Element of COMPLEX
st b3 is_continuous_in b4
holds b5 (#) b3 is_continuous_in b4;
:: NCFCONT1:th 59
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Element of the carrier of b1
st b3 is_continuous_in b4
holds ||.b3.|| is_continuous_in b4 & - b3 is_continuous_in b4;
:: NCFCONT1:th 60
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Element of the carrier of b1
st b3 is_continuous_in b4
holds ||.b3.|| is_continuous_in b4 & - b3 is_continuous_in b4;
:: NCFCONT1:th 61
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b2,the carrier of b1
for b4 being Element of the carrier of b2
st b3 is_continuous_in b4
holds ||.b3.|| is_continuous_in b4 & - b3 is_continuous_in b4;
:: NCFCONT1:prednot 7 => NCFCONT1:pred 7
definition
let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a3 be Function-like Relation of the carrier of a1,the carrier of a2;
let a4 be set;
pred A3 is_continuous_on A4 means
a4 c= dom a3 &
(for b1 being Element of the carrier of a1
st b1 in a4
holds a3 | a4 is_continuous_in b1);
end;
:: NCFCONT1:dfs 20
definiens
let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a3 be Function-like Relation of the carrier of a1,the carrier of a2;
let a4 be set;
To prove
a3 is_continuous_on a4
it is sufficient to prove
thus a4 c= dom a3 &
(for b1 being Element of the carrier of a1
st b1 in a4
holds a3 | a4 is_continuous_in b1);
:: NCFCONT1:def 21
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being set holds
b3 is_continuous_on b4
iff
b4 c= dom b3 &
(for b5 being Element of the carrier of b1
st b5 in b4
holds b3 | b4 is_continuous_in b5);
:: NCFCONT1:prednot 8 => NCFCONT1:pred 8
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
let a3 be Function-like Relation of the carrier of a1,the carrier of a2;
let a4 be set;
pred A3 is_continuous_on A4 means
a4 c= dom a3 &
(for b1 being Element of the carrier of a1
st b1 in a4
holds a3 | a4 is_continuous_in b1);
end;
:: NCFCONT1:dfs 21
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
let a3 be Function-like Relation of the carrier of a1,the carrier of a2;
let a4 be set;
To prove
a3 is_continuous_on a4
it is sufficient to prove
thus a4 c= dom a3 &
(for b1 being Element of the carrier of a1
st b1 in a4
holds a3 | a4 is_continuous_in b1);
:: NCFCONT1:def 22
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being set holds
b3 is_continuous_on b4
iff
b4 c= dom b3 &
(for b5 being Element of the carrier of b1
st b5 in b4
holds b3 | b4 is_continuous_in b5);
:: NCFCONT1:prednot 9 => NCFCONT1:pred 9
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
let a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a3 be Function-like Relation of the carrier of a1,the carrier of a2;
let a4 be set;
pred A3 is_continuous_on A4 means
a4 c= dom a3 &
(for b1 being Element of the carrier of a1
st b1 in a4
holds a3 | a4 is_continuous_in b1);
end;
:: NCFCONT1:dfs 22
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
let a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a3 be Function-like Relation of the carrier of a1,the carrier of a2;
let a4 be set;
To prove
a3 is_continuous_on a4
it is sufficient to prove
thus a4 c= dom a3 &
(for b1 being Element of the carrier of a1
st b1 in a4
holds a3 | a4 is_continuous_in b1);
:: NCFCONT1:def 23
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being set holds
b3 is_continuous_on b4
iff
b4 c= dom b3 &
(for b5 being Element of the carrier of b1
st b5 in b4
holds b3 | b4 is_continuous_in b5);
:: NCFCONT1:prednot 10 => NCFCONT1:pred 10
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be Function-like Relation of the carrier of a1,COMPLEX;
let a3 be set;
pred A2 is_continuous_on A3 means
a3 c= dom a2 &
(for b1 being Element of the carrier of a1
st b1 in a3
holds a2 | a3 is_continuous_in b1);
end;
:: NCFCONT1:dfs 23
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be Function-like Relation of the carrier of a1,COMPLEX;
let a3 be set;
To prove
a2 is_continuous_on a3
it is sufficient to prove
thus a3 c= dom a2 &
(for b1 being Element of the carrier of a1
st b1 in a3
holds a2 | a3 is_continuous_in b1);
:: NCFCONT1:def 24
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like Relation of the carrier of b1,COMPLEX
for b3 being set holds
b2 is_continuous_on b3
iff
b3 c= dom b2 &
(for b4 being Element of the carrier of b1
st b4 in b3
holds b2 | b3 is_continuous_in b4);
:: NCFCONT1:prednot 11 => NCFCONT1:pred 11
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be Function-like Relation of the carrier of a1,REAL;
let a3 be set;
pred A2 is_continuous_on A3 means
a3 c= dom a2 &
(for b1 being Element of the carrier of a1
st b1 in a3
holds a2 | a3 is_continuous_in b1);
end;
:: NCFCONT1:dfs 24
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be Function-like Relation of the carrier of a1,REAL;
let a3 be set;
To prove
a2 is_continuous_on a3
it is sufficient to prove
thus a3 c= dom a2 &
(for b1 being Element of the carrier of a1
st b1 in a3
holds a2 | a3 is_continuous_in b1);
:: NCFCONT1:def 25
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like Relation of the carrier of b1,REAL
for b3 being set holds
b2 is_continuous_on b3
iff
b3 c= dom b2 &
(for b4 being Element of the carrier of b1
st b4 in b3
holds b2 | b3 is_continuous_in b4);
:: NCFCONT1:prednot 12 => NCFCONT1:pred 12
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
let a2 be Function-like Relation of the carrier of a1,COMPLEX;
let a3 be set;
pred A2 is_continuous_on A3 means
a3 c= dom a2 &
(for b1 being Element of the carrier of a1
st b1 in a3
holds a2 | a3 is_continuous_in b1);
end;
:: NCFCONT1:dfs 25
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
let a2 be Function-like Relation of the carrier of a1,COMPLEX;
let a3 be set;
To prove
a2 is_continuous_on a3
it is sufficient to prove
thus a3 c= dom a2 &
(for b1 being Element of the carrier of a1
st b1 in a3
holds a2 | a3 is_continuous_in b1);
:: NCFCONT1:def 26
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like Relation of the carrier of b1,COMPLEX
for b3 being set holds
b2 is_continuous_on b3
iff
b3 c= dom b2 &
(for b4 being Element of the carrier of b1
st b4 in b3
holds b2 | b3 is_continuous_in b4);
:: NCFCONT1:th 62
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being set
for b4 being Function-like Relation of the carrier of b1,the carrier of b2 holds
b4 is_continuous_on b3
iff
b3 c= dom b4 &
(for b5 being Function-like quasi_total Relation of NAT,the carrier of b1
st rng b5 c= b3 & b5 is convergent(b1) & lim b5 in b3
holds b4 * b5 is convergent(b2) & b4 /. lim b5 = lim (b4 * b5));
:: NCFCONT1:th 63
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being set
for b4 being Function-like Relation of the carrier of b1,the carrier of b2 holds
b4 is_continuous_on b3
iff
b3 c= dom b4 &
(for b5 being Function-like quasi_total Relation of NAT,the carrier of b1
st rng b5 c= b3 & b5 is convergent(b1) & lim b5 in b3
holds b4 * b5 is convergent(b2) & b4 /. lim b5 = lim (b4 * b5));
:: NCFCONT1:th 64
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being set
for b4 being Function-like Relation of the carrier of b2,the carrier of b1 holds
b4 is_continuous_on b3
iff
b3 c= dom b4 &
(for b5 being Function-like quasi_total Relation of NAT,the carrier of b2
st rng b5 c= b3 & b5 is convergent(b2) & lim b5 in b3
holds b4 * b5 is convergent(b1) & b4 /. lim b5 = lim (b4 * b5));
:: NCFCONT1:th 65
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being set
for b4 being Function-like Relation of the carrier of b1,the carrier of b2 holds
b4 is_continuous_on b3
iff
b3 c= dom b4 &
(for b5 being Element of the carrier of b1
for b6 being Element of REAL
st b5 in b3 & 0 < b6
holds ex b7 being Element of REAL st
0 < b7 &
(for b8 being Element of the carrier of b1
st b8 in b3 & ||.b8 - b5.|| < b7
holds ||.(b4 /. b8) - (b4 /. b5).|| < b6));
:: NCFCONT1:th 66
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being set
for b4 being Function-like Relation of the carrier of b1,the carrier of b2 holds
b4 is_continuous_on b3
iff
b3 c= dom b4 &
(for b5 being Element of the carrier of b1
for b6 being Element of REAL
st b5 in b3 & 0 < b6
holds ex b7 being Element of REAL st
0 < b7 &
(for b8 being Element of the carrier of b1
st b8 in b3 & ||.b8 - b5.|| < b7
holds ||.(b4 /. b8) - (b4 /. b5).|| < b6));
:: NCFCONT1:th 67
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being set
for b4 being Function-like Relation of the carrier of b2,the carrier of b1 holds
b4 is_continuous_on b3
iff
b3 c= dom b4 &
(for b5 being Element of the carrier of b2
for b6 being Element of REAL
st b5 in b3 & 0 < b6
holds ex b7 being Element of REAL st
0 < b7 &
(for b8 being Element of the carrier of b2
st b8 in b3 & ||.b8 - b5.|| < b7
holds ||.(b4 /. b8) - (b4 /. b5).|| < b6));
:: NCFCONT1:th 68
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being set
for b3 being Function-like Relation of the carrier of b1,COMPLEX holds
b3 is_continuous_on b2
iff
b2 c= dom b3 &
(for b4 being Element of the carrier of b1
for b5 being Element of REAL
st b4 in b2 & 0 < b5
holds ex b6 being Element of REAL st
0 < b6 &
(for b7 being Element of the carrier of b1
st b7 in b2 & ||.b7 - b4.|| < b6
holds |.(b3 /. b7) - (b3 /. b4).| < b5));
:: NCFCONT1:th 69
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being set
for b3 being Function-like Relation of the carrier of b1,REAL holds
b3 is_continuous_on b2
iff
b2 c= dom b3 &
(for b4 being Element of the carrier of b1
for b5 being Element of REAL
st b4 in b2 & 0 < b5
holds ex b6 being Element of REAL st
0 < b6 &
(for b7 being Element of the carrier of b1
st b7 in b2 & ||.b7 - b4.|| < b6
holds abs ((b3 /. b7) - (b3 /. b4)) < b5));
:: NCFCONT1:th 70
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being set
for b3 being Function-like Relation of the carrier of b1,COMPLEX holds
b3 is_continuous_on b2
iff
b2 c= dom b3 &
(for b4 being Element of the carrier of b1
for b5 being Element of REAL
st b4 in b2 & 0 < b5
holds ex b6 being Element of REAL st
0 < b6 &
(for b7 being Element of the carrier of b1
st b7 in b2 & ||.b7 - b4.|| < b6
holds |.(b3 /. b7) - (b3 /. b4).| < b5));
:: NCFCONT1:th 71
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being set
for b4 being Function-like Relation of the carrier of b1,the carrier of b2 holds
b4 is_continuous_on b3
iff
b4 | b3 is_continuous_on b3;
:: NCFCONT1:th 72
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being set
for b4 being Function-like Relation of the carrier of b1,the carrier of b2 holds
b4 is_continuous_on b3
iff
b4 | b3 is_continuous_on b3;
:: NCFCONT1:th 73
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being set
for b4 being Function-like Relation of the carrier of b2,the carrier of b1 holds
b4 is_continuous_on b3
iff
b4 | b3 is_continuous_on b3;
:: NCFCONT1:th 74
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being set
for b3 being Function-like Relation of the carrier of b1,COMPLEX holds
b3 is_continuous_on b2
iff
b3 | b2 is_continuous_on b2;
:: NCFCONT1:th 75
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being set
for b3 being Function-like Relation of the carrier of b1,REAL holds
b3 is_continuous_on b2
iff
b3 | b2 is_continuous_on b2;
:: NCFCONT1:th 76
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being set
for b3 being Function-like Relation of the carrier of b1,COMPLEX holds
b3 is_continuous_on b2
iff
b3 | b2 is_continuous_on b2;
:: NCFCONT1:th 77
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3, b4 being set
for b5 being Function-like Relation of the carrier of b1,the carrier of b2
st b5 is_continuous_on b3 & b4 c= b3
holds b5 is_continuous_on b4;
:: NCFCONT1:th 78
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4 being set
for b5 being Function-like Relation of the carrier of b1,the carrier of b2
st b5 is_continuous_on b3 & b4 c= b3
holds b5 is_continuous_on b4;
:: NCFCONT1:th 79
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4 being set
for b5 being Function-like Relation of the carrier of b2,the carrier of b1
st b5 is_continuous_on b3 & b4 c= b3
holds b5 is_continuous_on b4;
:: NCFCONT1:th 80
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Element of the carrier of b1
st b4 in dom b3
holds b3 is_continuous_on {b4};
:: NCFCONT1:th 81
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Element of the carrier of b1
st b4 in dom b3
holds b3 is_continuous_on {b4};
:: NCFCONT1:th 82
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b2,the carrier of b1
for b4 being Element of the carrier of b2
st b4 in dom b3
holds b3 is_continuous_on {b4};
:: NCFCONT1:th 83
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being set
for b4, b5 being Function-like Relation of the carrier of b1,the carrier of b2
st b4 is_continuous_on b3 & b5 is_continuous_on b3
holds b4 + b5 is_continuous_on b3 & b4 - b5 is_continuous_on b3;
:: NCFCONT1:th 84
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being set
for b4, b5 being Function-like Relation of the carrier of b1,the carrier of b2
st b4 is_continuous_on b3 & b5 is_continuous_on b3
holds b4 + b5 is_continuous_on b3 & b4 - b5 is_continuous_on b3;
:: NCFCONT1:th 85
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being set
for b4, b5 being Function-like Relation of the carrier of b2,the carrier of b1
st b4 is_continuous_on b3 & b5 is_continuous_on b3
holds b4 + b5 is_continuous_on b3 & b4 - b5 is_continuous_on b3;
:: NCFCONT1:th 86
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3, b4 being set
for b5, b6 being Function-like Relation of the carrier of b1,the carrier of b2
st b5 is_continuous_on b3 & b6 is_continuous_on b4
holds b5 + b6 is_continuous_on b3 /\ b4 & b5 - b6 is_continuous_on b3 /\ b4;
:: NCFCONT1:th 87
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4 being set
for b5, b6 being Function-like Relation of the carrier of b1,the carrier of b2
st b5 is_continuous_on b3 & b6 is_continuous_on b4
holds b5 + b6 is_continuous_on b3 /\ b4 & b5 - b6 is_continuous_on b3 /\ b4;
:: NCFCONT1:th 88
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4 being set
for b5, b6 being Function-like Relation of the carrier of b2,the carrier of b1
st b5 is_continuous_on b3 & b6 is_continuous_on b4
holds b5 + b6 is_continuous_on b3 /\ b4 & b5 - b6 is_continuous_on b3 /\ b4;
:: NCFCONT1:th 89
theorem
for b1 being Element of COMPLEX
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b4 being set
for b5 being Function-like Relation of the carrier of b2,the carrier of b3
st b5 is_continuous_on b4
holds b1 (#) b5 is_continuous_on b4;
:: NCFCONT1:th 90
theorem
for b1 being Element of REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being set
for b5 being Function-like Relation of the carrier of b2,the carrier of b3
st b5 is_continuous_on b4
holds b1 (#) b5 is_continuous_on b4;
:: NCFCONT1:th 91
theorem
for b1 being Element of COMPLEX
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being set
for b5 being Function-like Relation of the carrier of b3,the carrier of b2
st b5 is_continuous_on b4
holds b1 (#) b5 is_continuous_on b4;
:: NCFCONT1:th 92
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being set
for b4 being Function-like Relation of the carrier of b1,the carrier of b2
st b4 is_continuous_on b3
holds ||.b4.|| is_continuous_on b3 & - b4 is_continuous_on b3;
:: NCFCONT1:th 93
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being set
for b4 being Function-like Relation of the carrier of b1,the carrier of b2
st b4 is_continuous_on b3
holds ||.b4.|| is_continuous_on b3 & - b4 is_continuous_on b3;
:: NCFCONT1:th 94
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being set
for b4 being Function-like Relation of the carrier of b2,the carrier of b1
st b4 is_continuous_on b3
holds ||.b4.|| is_continuous_on b3 & - b4 is_continuous_on b3;
:: NCFCONT1:th 95
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
st b3 is total(the carrier of b1, the carrier of b2) &
(for b4, b5 being Element of the carrier of b1 holds
b3 /. (b4 + b5) = (b3 /. b4) + (b3 /. b5)) &
(ex b4 being Element of the carrier of b1 st
b3 is_continuous_in b4)
holds b3 is_continuous_on the carrier of b1;
:: NCFCONT1:th 96
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
st b3 is total(the carrier of b1, the carrier of b2) &
(for b4, b5 being Element of the carrier of b1 holds
b3 /. (b4 + b5) = (b3 /. b4) + (b3 /. b5)) &
(ex b4 being Element of the carrier of b1 st
b3 is_continuous_in b4)
holds b3 is_continuous_on the carrier of b1;
:: NCFCONT1:th 97
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b2,the carrier of b1
st b3 is total(the carrier of b2, the carrier of b1) &
(for b4, b5 being Element of the carrier of b2 holds
b3 /. (b4 + b5) = (b3 /. b4) + (b3 /. b5)) &
(ex b4 being Element of the carrier of b2 st
b3 is_continuous_in b4)
holds b3 is_continuous_on the carrier of b2;
:: NCFCONT1:th 98
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
st dom b3 is compact(b1) & b3 is_continuous_on dom b3
holds rng b3 is compact(b2);
:: NCFCONT1:th 99
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
st dom b3 is compact(b1) & b3 is_continuous_on dom b3
holds rng b3 is compact(b2);
:: NCFCONT1:th 100
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b2,the carrier of b1
st dom b3 is compact(b2) & b3 is_continuous_on dom b3
holds rng b3 is compact(b1);
:: NCFCONT1:th 101
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like Relation of the carrier of b1,COMPLEX
st dom b2 is compact(b1) & b2 is_continuous_on dom b2
holds rng b2 is compact;
:: NCFCONT1:th 102
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like Relation of the carrier of b1,REAL
st dom b2 is compact(b1) & b2 is_continuous_on dom b2
holds rng b2 is compact;
:: NCFCONT1:th 103
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like Relation of the carrier of b1,COMPLEX
st dom b2 is compact(b1) & b2 is_continuous_on dom b2
holds rng b2 is compact;
:: NCFCONT1:th 104
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Element of bool the carrier of b1
for b4 being Function-like Relation of the carrier of b1,the carrier of b2
st b3 c= dom b4 & b3 is compact(b1) & b4 is_continuous_on b3
holds b4 .: b3 is compact(b2);
:: NCFCONT1:th 105
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Element of bool the carrier of b1
for b4 being Function-like Relation of the carrier of b1,the carrier of b2
st b3 c= dom b4 & b3 is compact(b1) & b4 is_continuous_on b3
holds b4 .: b3 is compact(b2);
:: NCFCONT1:th 106
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Element of bool the carrier of b2
for b4 being Function-like Relation of the carrier of b2,the carrier of b1
st b3 c= dom b4 & b3 is compact(b2) & b4 is_continuous_on b3
holds b4 .: b3 is compact(b1);
:: NCFCONT1:th 107
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like Relation of the carrier of b1,REAL
st dom b2 <> {} & dom b2 is compact(b1) & b2 is_continuous_on dom b2
holds ex b3, b4 being Element of the carrier of b1 st
b3 in dom b2 & b4 in dom b2 & b2 /. b3 = upper_bound rng b2 & b2 /. b4 = lower_bound rng b2;
:: NCFCONT1:th 108
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
st dom b3 <> {} & dom b3 is compact(b1) & b3 is_continuous_on dom b3
holds ex b4, b5 being Element of the carrier of b1 st
b4 in dom b3 &
b5 in dom b3 &
||.b3.|| /. b4 = upper_bound rng ||.b3.|| &
||.b3.|| /. b5 = lower_bound rng ||.b3.||;
:: NCFCONT1:th 109
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
st dom b3 <> {} & dom b3 is compact(b1) & b3 is_continuous_on dom b3
holds ex b4, b5 being Element of the carrier of b1 st
b4 in dom b3 &
b5 in dom b3 &
||.b3.|| /. b4 = upper_bound rng ||.b3.|| &
||.b3.|| /. b5 = lower_bound rng ||.b3.||;
:: NCFCONT1:th 110
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b2,the carrier of b1
st dom b3 <> {} & dom b3 is compact(b2) & b3 is_continuous_on dom b3
holds ex b4, b5 being Element of the carrier of b2 st
b4 in dom b3 &
b5 in dom b3 &
||.b3.|| /. b4 = upper_bound rng ||.b3.|| &
||.b3.|| /. b5 = lower_bound rng ||.b3.||;
:: NCFCONT1:th 111
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being set
for b4 being Function-like Relation of the carrier of b1,the carrier of b2 holds
||.b4.|| | b3 = ||.b4 | b3.||;
:: NCFCONT1:th 112
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being set
for b4 being Function-like Relation of the carrier of b1,the carrier of b2 holds
||.b4.|| | b3 = ||.b4 | b3.||;
:: NCFCONT1:th 113
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being set
for b4 being Function-like Relation of the carrier of b2,the carrier of b1 holds
||.b4.|| | b3 = ||.b4 | b3.||;
:: NCFCONT1:th 114
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Element of bool the carrier of b1
st b4 <> {} & b4 c= dom b3 & b4 is compact(b1) & b3 is_continuous_on b4
holds ex b5, b6 being Element of the carrier of b1 st
b5 in b4 &
b6 in b4 &
||.b3.|| /. b5 = upper_bound (||.b3.|| .: b4) &
||.b3.|| /. b6 = lower_bound (||.b3.|| .: b4);
:: NCFCONT1:th 115
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Element of bool the carrier of b1
st b4 <> {} & b4 c= dom b3 & b4 is compact(b1) & b3 is_continuous_on b4
holds ex b5, b6 being Element of the carrier of b1 st
b5 in b4 &
b6 in b4 &
||.b3.|| /. b5 = upper_bound (||.b3.|| .: b4) &
||.b3.|| /. b6 = lower_bound (||.b3.|| .: b4);
:: NCFCONT1:th 116
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b2,the carrier of b1
for b4 being Element of bool the carrier of b2
st b4 <> {} & b4 c= dom b3 & b4 is compact(b2) & b3 is_continuous_on b4
holds ex b5, b6 being Element of the carrier of b2 st
b5 in b4 &
b6 in b4 &
||.b3.|| /. b5 = upper_bound (||.b3.|| .: b4) &
||.b3.|| /. b6 = lower_bound (||.b3.|| .: b4);
:: NCFCONT1:th 117
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like Relation of the carrier of b1,REAL
for b3 being Element of bool the carrier of b1
st b3 <> {} & b3 c= dom b2 & b3 is compact(b1) & b2 is_continuous_on b3
holds ex b4, b5 being Element of the carrier of b1 st
b4 in b3 & b5 in b3 & b2 /. b4 = upper_bound (b2 .: b3) & b2 /. b5 = lower_bound (b2 .: b3);
:: NCFCONT1:prednot 13 => NCFCONT1:pred 13
definition
let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a3 be set;
let a4 be Function-like Relation of the carrier of a1,the carrier of a2;
pred A4 is_Lipschitzian_on A3 means
a3 c= dom a4 &
(ex b1 being Element of REAL st
0 < b1 &
(for b2, b3 being Element of the carrier of a1
st b2 in a3 & b3 in a3
holds ||.(a4 /. b2) - (a4 /. b3).|| <= b1 * ||.b2 - b3.||));
end;
:: NCFCONT1:dfs 26
definiens
let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a3 be set;
let a4 be Function-like Relation of the carrier of a1,the carrier of a2;
To prove
a4 is_Lipschitzian_on a3
it is sufficient to prove
thus a3 c= dom a4 &
(ex b1 being Element of REAL st
0 < b1 &
(for b2, b3 being Element of the carrier of a1
st b2 in a3 & b3 in a3
holds ||.(a4 /. b2) - (a4 /. b3).|| <= b1 * ||.b2 - b3.||));
:: NCFCONT1:def 27
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being set
for b4 being Function-like Relation of the carrier of b1,the carrier of b2 holds
b4 is_Lipschitzian_on b3
iff
b3 c= dom b4 &
(ex b5 being Element of REAL st
0 < b5 &
(for b6, b7 being Element of the carrier of b1
st b6 in b3 & b7 in b3
holds ||.(b4 /. b6) - (b4 /. b7).|| <= b5 * ||.b6 - b7.||));
:: NCFCONT1:prednot 14 => NCFCONT1:pred 14
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
let a3 be set;
let a4 be Function-like Relation of the carrier of a1,the carrier of a2;
pred A4 is_Lipschitzian_on A3 means
a3 c= dom a4 &
(ex b1 being Element of REAL st
0 < b1 &
(for b2, b3 being Element of the carrier of a1
st b2 in a3 & b3 in a3
holds ||.(a4 /. b2) - (a4 /. b3).|| <= b1 * ||.b2 - b3.||));
end;
:: NCFCONT1:dfs 27
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
let a3 be set;
let a4 be Function-like Relation of the carrier of a1,the carrier of a2;
To prove
a4 is_Lipschitzian_on a3
it is sufficient to prove
thus a3 c= dom a4 &
(ex b1 being Element of REAL st
0 < b1 &
(for b2, b3 being Element of the carrier of a1
st b2 in a3 & b3 in a3
holds ||.(a4 /. b2) - (a4 /. b3).|| <= b1 * ||.b2 - b3.||));
:: NCFCONT1:def 28
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being set
for b4 being Function-like Relation of the carrier of b1,the carrier of b2 holds
b4 is_Lipschitzian_on b3
iff
b3 c= dom b4 &
(ex b5 being Element of REAL st
0 < b5 &
(for b6, b7 being Element of the carrier of b1
st b6 in b3 & b7 in b3
holds ||.(b4 /. b6) - (b4 /. b7).|| <= b5 * ||.b6 - b7.||));
:: NCFCONT1:prednot 15 => NCFCONT1:pred 15
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
let a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a3 be set;
let a4 be Function-like Relation of the carrier of a1,the carrier of a2;
pred A4 is_Lipschitzian_on A3 means
a3 c= dom a4 &
(ex b1 being Element of REAL st
0 < b1 &
(for b2, b3 being Element of the carrier of a1
st b2 in a3 & b3 in a3
holds ||.(a4 /. b2) - (a4 /. b3).|| <= b1 * ||.b2 - b3.||));
end;
:: NCFCONT1:dfs 28
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
let a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a3 be set;
let a4 be Function-like Relation of the carrier of a1,the carrier of a2;
To prove
a4 is_Lipschitzian_on a3
it is sufficient to prove
thus a3 c= dom a4 &
(ex b1 being Element of REAL st
0 < b1 &
(for b2, b3 being Element of the carrier of a1
st b2 in a3 & b3 in a3
holds ||.(a4 /. b2) - (a4 /. b3).|| <= b1 * ||.b2 - b3.||));
:: NCFCONT1:def 29
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being set
for b4 being Function-like Relation of the carrier of b1,the carrier of b2 holds
b4 is_Lipschitzian_on b3
iff
b3 c= dom b4 &
(ex b5 being Element of REAL st
0 < b5 &
(for b6, b7 being Element of the carrier of b1
st b6 in b3 & b7 in b3
holds ||.(b4 /. b6) - (b4 /. b7).|| <= b5 * ||.b6 - b7.||));
:: NCFCONT1:prednot 16 => NCFCONT1:pred 16
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be set;
let a3 be Function-like Relation of the carrier of a1,COMPLEX;
pred A3 is_Lipschitzian_on A2 means
a2 c= dom a3 &
(ex b1 being Element of REAL st
0 < b1 &
(for b2, b3 being Element of the carrier of a1
st b2 in a2 & b3 in a2
holds |.(a3 /. b2) - (a3 /. b3).| <= b1 * ||.b2 - b3.||));
end;
:: NCFCONT1:dfs 29
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be set;
let a3 be Function-like Relation of the carrier of a1,COMPLEX;
To prove
a3 is_Lipschitzian_on a2
it is sufficient to prove
thus a2 c= dom a3 &
(ex b1 being Element of REAL st
0 < b1 &
(for b2, b3 being Element of the carrier of a1
st b2 in a2 & b3 in a2
holds |.(a3 /. b2) - (a3 /. b3).| <= b1 * ||.b2 - b3.||));
:: NCFCONT1:def 30
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being set
for b3 being Function-like Relation of the carrier of b1,COMPLEX holds
b3 is_Lipschitzian_on b2
iff
b2 c= dom b3 &
(ex b4 being Element of REAL st
0 < b4 &
(for b5, b6 being Element of the carrier of b1
st b5 in b2 & b6 in b2
holds |.(b3 /. b5) - (b3 /. b6).| <= b4 * ||.b5 - b6.||));
:: NCFCONT1:prednot 17 => NCFCONT1:pred 17
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be set;
let a3 be Function-like Relation of the carrier of a1,REAL;
pred A3 is_Lipschitzian_on A2 means
a2 c= dom a3 &
(ex b1 being Element of REAL st
0 < b1 &
(for b2, b3 being Element of the carrier of a1
st b2 in a2 & b3 in a2
holds abs ((a3 /. b2) - (a3 /. b3)) <= b1 * ||.b2 - b3.||));
end;
:: NCFCONT1:dfs 30
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be set;
let a3 be Function-like Relation of the carrier of a1,REAL;
To prove
a3 is_Lipschitzian_on a2
it is sufficient to prove
thus a2 c= dom a3 &
(ex b1 being Element of REAL st
0 < b1 &
(for b2, b3 being Element of the carrier of a1
st b2 in a2 & b3 in a2
holds abs ((a3 /. b2) - (a3 /. b3)) <= b1 * ||.b2 - b3.||));
:: NCFCONT1:def 31
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being set
for b3 being Function-like Relation of the carrier of b1,REAL holds
b3 is_Lipschitzian_on b2
iff
b2 c= dom b3 &
(ex b4 being Element of REAL st
0 < b4 &
(for b5, b6 being Element of the carrier of b1
st b5 in b2 & b6 in b2
holds abs ((b3 /. b5) - (b3 /. b6)) <= b4 * ||.b5 - b6.||));
:: NCFCONT1:prednot 18 => NCFCONT1:pred 18
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
let a2 be set;
let a3 be Function-like Relation of the carrier of a1,COMPLEX;
pred A3 is_Lipschitzian_on A2 means
a2 c= dom a3 &
(ex b1 being Element of REAL st
0 < b1 &
(for b2, b3 being Element of the carrier of a1
st b2 in a2 & b3 in a2
holds |.(a3 /. b2) - (a3 /. b3).| <= b1 * ||.b2 - b3.||));
end;
:: NCFCONT1:dfs 31
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
let a2 be set;
let a3 be Function-like Relation of the carrier of a1,COMPLEX;
To prove
a3 is_Lipschitzian_on a2
it is sufficient to prove
thus a2 c= dom a3 &
(ex b1 being Element of REAL st
0 < b1 &
(for b2, b3 being Element of the carrier of a1
st b2 in a2 & b3 in a2
holds |.(a3 /. b2) - (a3 /. b3).| <= b1 * ||.b2 - b3.||));
:: NCFCONT1:def 32
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being set
for b3 being Function-like Relation of the carrier of b1,COMPLEX holds
b3 is_Lipschitzian_on b2
iff
b2 c= dom b3 &
(ex b4 being Element of REAL st
0 < b4 &
(for b5, b6 being Element of the carrier of b1
st b5 in b2 & b6 in b2
holds |.(b3 /. b5) - (b3 /. b6).| <= b4 * ||.b5 - b6.||));
:: NCFCONT1:th 118
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3, b4 being set
for b5 being Function-like Relation of the carrier of b1,the carrier of b2
st b5 is_Lipschitzian_on b3 & b4 c= b3
holds b5 is_Lipschitzian_on b4;
:: NCFCONT1:th 119
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4 being set
for b5 being Function-like Relation of the carrier of b1,the carrier of b2
st b5 is_Lipschitzian_on b3 & b4 c= b3
holds b5 is_Lipschitzian_on b4;
:: NCFCONT1:th 120
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4 being set
for b5 being Function-like Relation of the carrier of b2,the carrier of b1
st b5 is_Lipschitzian_on b3 & b4 c= b3
holds b5 is_Lipschitzian_on b4;
:: NCFCONT1:th 121
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3, b4 being set
for b5, b6 being Function-like Relation of the carrier of b1,the carrier of b2
st b5 is_Lipschitzian_on b3 & b6 is_Lipschitzian_on b4
holds b5 + b6 is_Lipschitzian_on b3 /\ b4;
:: NCFCONT1:th 122
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4 being set
for b5, b6 being Function-like Relation of the carrier of b1,the carrier of b2
st b5 is_Lipschitzian_on b3 & b6 is_Lipschitzian_on b4
holds b5 + b6 is_Lipschitzian_on b3 /\ b4;
:: NCFCONT1:th 123
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4 being set
for b5, b6 being Function-like Relation of the carrier of b2,the carrier of b1
st b5 is_Lipschitzian_on b3 & b6 is_Lipschitzian_on b4
holds b5 + b6 is_Lipschitzian_on b3 /\ b4;
:: NCFCONT1:th 124
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3, b4 being set
for b5, b6 being Function-like Relation of the carrier of b1,the carrier of b2
st b5 is_Lipschitzian_on b3 & b6 is_Lipschitzian_on b4
holds b5 - b6 is_Lipschitzian_on b3 /\ b4;
:: NCFCONT1:th 125
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4 being set
for b5, b6 being Function-like Relation of the carrier of b1,the carrier of b2
st b5 is_Lipschitzian_on b3 & b6 is_Lipschitzian_on b4
holds b5 - b6 is_Lipschitzian_on b3 /\ b4;
:: NCFCONT1:th 126
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4 being set
for b5, b6 being Function-like Relation of the carrier of b2,the carrier of b1
st b5 is_Lipschitzian_on b3 & b6 is_Lipschitzian_on b4
holds b5 - b6 is_Lipschitzian_on b3 /\ b4;
:: NCFCONT1:th 127
theorem
for b1 being Element of COMPLEX
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b4 being set
for b5 being Function-like Relation of the carrier of b2,the carrier of b3
st b5 is_Lipschitzian_on b4
holds b1 (#) b5 is_Lipschitzian_on b4;
:: NCFCONT1:th 128
theorem
for b1 being Element of REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being set
for b5 being Function-like Relation of the carrier of b2,the carrier of b3
st b5 is_Lipschitzian_on b4
holds b1 (#) b5 is_Lipschitzian_on b4;
:: NCFCONT1:th 129
theorem
for b1 being Element of COMPLEX
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being set
for b5 being Function-like Relation of the carrier of b3,the carrier of b2
st b5 is_Lipschitzian_on b4
holds b1 (#) b5 is_Lipschitzian_on b4;
:: NCFCONT1:th 130
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being set
for b4 being Function-like Relation of the carrier of b1,the carrier of b2
st b4 is_Lipschitzian_on b3
holds - b4 is_Lipschitzian_on b3 & ||.b4.|| is_Lipschitzian_on b3;
:: NCFCONT1:th 131
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being set
for b4 being Function-like Relation of the carrier of b1,the carrier of b2
st b4 is_Lipschitzian_on b3
holds - b4 is_Lipschitzian_on b3 & ||.b4.|| is_Lipschitzian_on b3;
:: NCFCONT1:th 132
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being set
for b4 being Function-like Relation of the carrier of b2,the carrier of b1
st b4 is_Lipschitzian_on b3
holds - b4 is_Lipschitzian_on b3 & ||.b4.|| is_Lipschitzian_on b3;
:: NCFCONT1:th 133
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being set
for b4 being Function-like Relation of the carrier of b1,the carrier of b2
st b3 c= dom b4 & b4 is_constant_on b3
holds b4 is_Lipschitzian_on b3;
:: NCFCONT1:th 134
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being set
for b4 being Function-like Relation of the carrier of b1,the carrier of b2
st b3 c= dom b4 & b4 is_constant_on b3
holds b4 is_Lipschitzian_on b3;
:: NCFCONT1:th 135
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being set
for b4 being Function-like Relation of the carrier of b2,the carrier of b1
st b3 c= dom b4 & b4 is_constant_on b3
holds b4 is_Lipschitzian_on b3;
:: NCFCONT1:th 136
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Element of bool the carrier of b1 holds
id b2 is_Lipschitzian_on b2;
:: NCFCONT1:th 137
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being set
for b4 being Function-like Relation of the carrier of b1,the carrier of b2
st b4 is_Lipschitzian_on b3
holds b4 is_continuous_on b3;
:: NCFCONT1:th 138
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being set
for b4 being Function-like Relation of the carrier of b1,the carrier of b2
st b4 is_Lipschitzian_on b3
holds b4 is_continuous_on b3;
:: NCFCONT1:th 139
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being set
for b4 being Function-like Relation of the carrier of b2,the carrier of b1
st b4 is_Lipschitzian_on b3
holds b4 is_continuous_on b3;
:: NCFCONT1:th 140
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being set
for b3 being Function-like Relation of the carrier of b1,COMPLEX
st b3 is_Lipschitzian_on b2
holds b3 is_continuous_on b2;
:: NCFCONT1:th 141
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being set
for b3 being Function-like Relation of the carrier of b1,REAL
st b3 is_Lipschitzian_on b2
holds b3 is_continuous_on b2;
:: NCFCONT1:th 142
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being set
for b3 being Function-like Relation of the carrier of b1,COMPLEX
st b3 is_Lipschitzian_on b2
holds b3 is_continuous_on b2;
:: NCFCONT1:th 143
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
st ex b4 being Element of the carrier of b2 st
rng b3 = {b4}
holds b3 is_continuous_on dom b3;
:: NCFCONT1:th 144
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
st ex b4 being Element of the carrier of b2 st
rng b3 = {b4}
holds b3 is_continuous_on dom b3;
:: NCFCONT1:th 145
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b2,the carrier of b1
st ex b4 being Element of the carrier of b1 st
rng b3 = {b4}
holds b3 is_continuous_on dom b3;
:: NCFCONT1:th 146
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being set
for b4 being Function-like Relation of the carrier of b1,the carrier of b2
st b3 c= dom b4 & b4 is_constant_on b3
holds b4 is_continuous_on b3;
:: NCFCONT1:th 147
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being set
for b4 being Function-like Relation of the carrier of b1,the carrier of b2
st b3 c= dom b4 & b4 is_constant_on b3
holds b4 is_continuous_on b3;
:: NCFCONT1:th 148
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being set
for b4 being Function-like Relation of the carrier of b2,the carrier of b1
st b3 c= dom b4 & b4 is_constant_on b3
holds b4 is_continuous_on b3;
:: NCFCONT1:th 149
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like Relation of the carrier of b1,the carrier of b1
st for b3 being Element of the carrier of b1
st b3 in dom b2
holds b2 /. b3 = b3
holds b2 is_continuous_on dom b2;
:: NCFCONT1:th 150
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like Relation of the carrier of b1,the carrier of b1
st b2 = id dom b2
holds b2 is_continuous_on dom b2;
:: NCFCONT1:th 151
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like Relation of the carrier of b1,the carrier of b1
for b3 being Element of bool the carrier of b1
st b3 c= dom b2 & b2 | b3 = id b3
holds b2 is_continuous_on b3;
:: NCFCONT1:th 152
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being set
for b3 being Function-like Relation of the carrier of b1,the carrier of b1
for b4 being Element of COMPLEX
for b5 being Element of the carrier of b1
st b2 c= dom b3 &
(for b6 being Element of the carrier of b1
st b6 in b2
holds b3 /. b6 = (b4 * b6) + b5)
holds b3 is_continuous_on b2;
:: NCFCONT1:th 153
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like Relation of the carrier of b1,REAL
st for b3 being Element of the carrier of b1
st b3 in dom b2
holds b2 /. b3 = ||.b3.||
holds b2 is_continuous_on dom b2;
:: NCFCONT1:th 154
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being set
for b3 being Function-like Relation of the carrier of b1,REAL
st b2 c= dom b3 &
(for b4 being Element of the carrier of b1
st b4 in b2
holds b3 /. b4 = ||.b4.||)
holds b3 is_continuous_on b2;