Article CFCONT_1, MML version 4.99.1005
:: CFCONT_1:funcnot 1 => CFCONT_1:func 1
definition
let a1 be Function-like Relation of COMPLEX,COMPLEX;
let a2 be Function-like quasi_total Relation of NAT,COMPLEX;
assume rng a2 c= dom a1;
func A1 * A2 -> Function-like quasi_total Relation of NAT,COMPLEX equals
a2 * a1;
end;
:: CFCONT_1:def 1
theorem
for b1 being Function-like Relation of COMPLEX,COMPLEX
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
st rng b2 c= dom b1
holds b1 * b2 = b2 * b1;
:: CFCONT_1:prednot 1 => CFCONT_1:pred 1
definition
let a1 be Function-like Relation of COMPLEX,COMPLEX;
let a2 be Element of COMPLEX;
pred A1 is_continuous_in A2 means
a2 in dom a1 &
(for b1 being Function-like quasi_total Relation of NAT,COMPLEX
st rng b1 c= dom a1 & b1 is convergent & lim b1 = a2
holds a1 * b1 is convergent & a1 /. a2 = lim (a1 * b1));
end;
:: CFCONT_1:dfs 2
definiens
let a1 be Function-like Relation of COMPLEX,COMPLEX;
let a2 be Element of COMPLEX;
To prove
a1 is_continuous_in a2
it is sufficient to prove
thus a2 in dom a1 &
(for b1 being Function-like quasi_total Relation of NAT,COMPLEX
st rng b1 c= dom a1 & b1 is convergent & lim b1 = a2
holds a1 * b1 is convergent & a1 /. a2 = lim (a1 * b1));
:: CFCONT_1:def 2
theorem
for b1 being Function-like Relation of COMPLEX,COMPLEX
for b2 being Element of COMPLEX holds
b1 is_continuous_in b2
iff
b2 in dom b1 &
(for b3 being Function-like quasi_total Relation of NAT,COMPLEX
st rng b3 c= dom b1 & b3 is convergent & lim b3 = b2
holds b1 * b3 is convergent & b1 /. b2 = lim (b1 * b3));
:: CFCONT_1:th 2
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,COMPLEX holds
b1 = b2 - b3
iff
for b4 being Element of NAT holds
b1 . b4 = (b2 . b4) - (b3 . b4);
:: CFCONT_1:th 3
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,COMPLEX holds
rng (b2 ^\ b1) c= rng b2;
:: CFCONT_1:th 4
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
for b3 being Function-like Relation of COMPLEX,COMPLEX
st rng b2 c= dom b3
holds b2 . b1 in dom b3;
:: CFCONT_1:th 5
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,COMPLEX holds
b1 in rng b2
iff
ex b3 being Element of NAT st
b1 = b2 . b3;
:: CFCONT_1:th 6
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,COMPLEX holds
b2 . b1 in rng b2;
:: CFCONT_1:th 7
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
st b1 is subsequence of b2
holds rng b1 c= rng b2;
:: CFCONT_1:th 8
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
st b1 is subsequence of b2 & b2 is non-zero
holds b1 is non-zero;
:: CFCONT_1:th 9
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
for b3 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL holds
(b1 + b2) * b3 = (b1 * b3) + (b2 * b3) &
(b1 - b2) * b3 = (b1 * b3) - (b2 * b3) &
(b1 (#) b2) * b3 = (b1 * b3) (#) (b2 * b3);
:: CFCONT_1:th 10
theorem
for b1 being Element of COMPLEX
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
for b3 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL holds
(b1 (#) b2) * b3 = b1 (#) (b2 * b3);
:: CFCONT_1:th 11
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL holds
(- b1) * b2 = - (b1 * b2) &
|.b1.| * b2 = |.b1 * b2.|;
:: CFCONT_1:th 12
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL holds
(b1 * b2) " = b1 " * b2;
:: CFCONT_1:th 13
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
for b3 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL holds
(b1 /" b2) * b3 = (b1 * b3) /" (b2 * b3);
:: CFCONT_1:th 14
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Element of bool COMPLEX
st for b3 being Element of NAT holds
b1 . b3 in b2
holds rng b1 c= b2;
:: CFCONT_1:th 16
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
for b3 being Function-like Relation of COMPLEX,COMPLEX
st rng b2 c= dom b3
holds (b3 * b2) . b1 = b3 /. (b2 . b1);
:: CFCONT_1:th 17
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
for b3 being Function-like Relation of COMPLEX,COMPLEX
st rng b2 c= dom b3
holds (b3 * b2) ^\ b1 = b3 * (b2 ^\ b1);
:: CFCONT_1:th 18
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2, b3 being Function-like Relation of COMPLEX,COMPLEX
st rng b1 c= (dom b2) /\ dom b3
holds (b2 + b3) * b1 = (b2 * b1) + (b3 * b1) &
(b2 - b3) * b1 = (b2 * b1) - (b3 * b1) &
(b2 (#) b3) * b1 = (b2 * b1) (#) (b3 * b1);
:: CFCONT_1:th 19
theorem
for b1 being Element of COMPLEX
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
for b3 being Function-like Relation of COMPLEX,COMPLEX
st rng b2 c= dom b3
holds (b1 (#) b3) * b2 = b1 (#) (b3 * b2);
:: CFCONT_1:th 20
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Function-like Relation of COMPLEX,COMPLEX
st rng b1 c= dom b2
holds - (b2 * b1) = (- b2) * b1;
:: CFCONT_1:th 21
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Function-like Relation of COMPLEX,COMPLEX
st rng b1 c= dom (b2 ^)
holds b2 * b1 is non-zero;
:: CFCONT_1:th 22
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Function-like Relation of COMPLEX,COMPLEX
st rng b1 c= dom (b2 ^)
holds b2 ^ * b1 = (b2 * b1) ";
:: CFCONT_1:th 23
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Function-like Relation of COMPLEX,COMPLEX
for b3 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL
st rng b1 c= dom b2
holds Re ((b2 * b1) * b3) = Re (b2 * (b1 * b3));
:: CFCONT_1:th 24
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Function-like Relation of COMPLEX,COMPLEX
for b3 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL
st rng b1 c= dom b2
holds Im ((b2 * b1) * b3) = Im (b2 * (b1 * b3));
:: CFCONT_1:th 25
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Function-like Relation of COMPLEX,COMPLEX
for b3 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL
st rng b1 c= dom b2
holds (b2 * b1) * b3 = b2 * (b1 * b3);
:: CFCONT_1:th 26
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
for b3 being Function-like Relation of COMPLEX,COMPLEX
st rng b1 c= dom b3 & b2 is subsequence of b1
holds b3 * b2 is subsequence of b3 * b1;
:: CFCONT_1:th 27
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
for b3 being Function-like Relation of COMPLEX,COMPLEX
st b3 is total(COMPLEX, COMPLEX)
holds (b3 * b2) . b1 = b3 /. (b2 . b1);
:: CFCONT_1:th 28
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
for b3 being Function-like Relation of COMPLEX,COMPLEX
st b3 is total(COMPLEX, COMPLEX)
holds b3 * (b2 ^\ b1) = (b3 * b2) ^\ b1;
:: CFCONT_1:th 29
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2, b3 being Function-like Relation of COMPLEX,COMPLEX
st b2 is total(COMPLEX, COMPLEX) & b3 is total(COMPLEX, COMPLEX)
holds (b2 + b3) * b1 = (b2 * b1) + (b3 * b1) &
(b2 - b3) * b1 = (b2 * b1) - (b3 * b1) &
(b2 (#) b3) * b1 = (b2 * b1) (#) (b3 * b1);
:: CFCONT_1:th 30
theorem
for b1 being Element of COMPLEX
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
for b3 being Function-like Relation of COMPLEX,COMPLEX
st b3 is total(COMPLEX, COMPLEX)
holds (b1 (#) b3) * b2 = b1 (#) (b3 * b2);
:: CFCONT_1:th 31
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
for b3 being Function-like Relation of COMPLEX,COMPLEX
st rng b2 c= dom (b3 | b1)
holds (b3 | b1) * b2 = b3 * b2;
:: CFCONT_1:th 32
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
for b3 being Element of bool COMPLEX
for b4 being Function-like Relation of COMPLEX,COMPLEX
st rng b2 c= dom (b4 | b1) &
(rng b2 c= dom (b4 | b3) or b1 c= b3)
holds (b4 | b1) * b2 = (b4 | b3) * b2;
:: CFCONT_1:th 33
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
for b3 being Function-like Relation of COMPLEX,COMPLEX
st rng b2 c= dom (b3 | b1) & b3 " {0} = {}
holds (b3 ^ | b1) * b2 = ((b3 | b1) * b2) ";
:: CFCONT_1:attrnot 1 => FUNCT_1:attr 3
definition
let a1 be Relation-like Function-like set;
attr a1 is constant means
ex b1 being Element of COMPLEX st
for b2 being Element of NAT holds
a1 . b2 = b1;
end;
:: CFCONT_1:dfs 3
definiens
let a1 be Function-like quasi_total Relation of NAT,COMPLEX;
To prove
a1 is constant
it is sufficient to prove
thus ex b1 being Element of COMPLEX st
for b2 being Element of NAT holds
a1 . b2 = b1;
:: CFCONT_1:def 4
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
b1 is constant
iff
ex b2 being Element of COMPLEX st
for b3 being Element of NAT holds
b1 . b3 = b2;
:: CFCONT_1:th 34
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
b1 is constant
iff
ex b2 being Element of COMPLEX st
rng b1 = {b2};
:: CFCONT_1:th 35
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
b1 is constant
iff
for b2 being Element of NAT holds
b1 . b2 = b1 . (b2 + 1);
:: CFCONT_1:th 36
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
b1 is constant
iff
for b2, b3 being Element of NAT holds
b1 . b2 = b1 . (b2 + b3);
:: CFCONT_1:th 37
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
b1 is constant
iff
for b2, b3 being Element of NAT holds
b1 . b2 = b1 . b3;
:: CFCONT_1:th 38
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,COMPLEX holds
b2 ^\ b1 is subsequence of b2;
:: CFCONT_1:th 39
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
st b1 is subsequence of b2 & b2 is convergent
holds b1 is convergent;
:: CFCONT_1:th 40
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
st b1 is subsequence of b2 & b2 is convergent
holds lim b1 = lim b2;
:: CFCONT_1:th 41
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
st b1 is convergent &
(ex b3 being Element of NAT st
for b4 being Element of NAT
st b3 <= b4
holds b2 . b4 = b1 . b4)
holds b2 is convergent;
:: CFCONT_1:th 42
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
st b1 is convergent &
(ex b3 being Element of NAT st
for b4 being Element of NAT
st b3 <= b4
holds b2 . b4 = b1 . b4)
holds lim b1 = lim b2;
:: CFCONT_1:th 43
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
st b2 is convergent
holds b2 ^\ b1 is convergent & lim (b2 ^\ b1) = lim b2;
:: CFCONT_1:th 44
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
st b1 is convergent &
(ex b3 being Element of NAT st
b1 = b2 ^\ b3)
holds b2 is convergent;
:: CFCONT_1:th 45
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
st b1 is convergent &
(ex b3 being Element of NAT st
b1 = b2 ^\ b3)
holds lim b2 = lim b1;
:: CFCONT_1:th 46
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
st b1 is convergent & lim b1 <> 0
holds ex b2 being Element of NAT st
b1 ^\ b2 is non-zero;
:: CFCONT_1:th 47
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
st b1 is convergent & lim b1 <> 0
holds ex b2 being Function-like quasi_total Relation of NAT,COMPLEX st
b2 is subsequence of b1 & b2 is non-zero;
:: CFCONT_1:th 48
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
st b1 is constant
holds b1 is convergent;
:: CFCONT_1:th 49
theorem
for b1 being Element of COMPLEX
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
st (b2 is constant & b1 in rng b2 or b2 is constant &
(ex b3 being Element of NAT st
b2 . b3 = b1))
holds lim b2 = b1;
:: CFCONT_1:th 50
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
st b1 is constant
for b2 being Element of NAT holds
lim b1 = b1 . b2;
:: CFCONT_1:th 51
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
st b1 is convergent & lim b1 <> 0
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
st b2 is subsequence of b1 & b2 is non-zero
holds lim (b2 ") = (lim b1) ";
:: CFCONT_1:th 52
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
st b1 is constant & b2 is convergent
holds lim (b1 + b2) = (b1 . 0) + lim b2 &
lim (b1 - b2) = (b1 . 0) - lim b2 &
lim (b2 - b1) = (lim b2) - (b1 . 0) &
lim (b1 (#) b2) = (b1 . 0) * lim b2;
:: CFCONT_1:sch 1
scheme CFCONT_1:sch 1
ex b1 being Function-like quasi_total Relation of NAT,COMPLEX st
for b2 being Element of NAT holds
P1[b2, b1 . b2]
provided
for b1 being Element of NAT holds
ex b2 being Element of COMPLEX st
P1[b1, b2];
:: CFCONT_1:th 53
theorem
for b1 being Element of COMPLEX
for b2 being Function-like Relation of COMPLEX,COMPLEX holds
b2 is_continuous_in b1
iff
b1 in dom b2 &
(for b3 being Function-like quasi_total Relation of NAT,COMPLEX
st rng b3 c= dom b2 &
b3 is convergent &
lim b3 = b1 &
(for b4 being Element of NAT holds
b3 . b4 <> b1)
holds b2 * b3 is convergent & b2 /. b1 = lim (b2 * b3));
:: CFCONT_1:th 54
theorem
for b1 being Element of COMPLEX
for b2 being Function-like Relation of COMPLEX,COMPLEX holds
b2 is_continuous_in b1
iff
b1 in dom b2 &
(for b3 being Element of REAL
st 0 < b3
holds ex b4 being Element of REAL st
0 < b4 &
(for b5 being Element of COMPLEX
st b5 in dom b2 & |.b5 - b1.| < b4
holds |.(b2 /. b5) - (b2 /. b1).| < b3));
:: CFCONT_1:th 55
theorem
for b1 being Element of COMPLEX
for b2, b3 being Function-like Relation of COMPLEX,COMPLEX
st b2 is_continuous_in b1 & b3 is_continuous_in b1
holds b2 + b3 is_continuous_in b1 & b2 - b3 is_continuous_in b1 & b2 (#) b3 is_continuous_in b1;
:: CFCONT_1:th 56
theorem
for b1, b2 being Element of COMPLEX
for b3 being Function-like Relation of COMPLEX,COMPLEX
st b3 is_continuous_in b1
holds b2 (#) b3 is_continuous_in b1;
:: CFCONT_1:th 57
theorem
for b1 being Element of COMPLEX
for b2 being Function-like Relation of COMPLEX,COMPLEX
st b2 is_continuous_in b1
holds - b2 is_continuous_in b1;
:: CFCONT_1:th 58
theorem
for b1 being Element of COMPLEX
for b2 being Function-like Relation of COMPLEX,COMPLEX
st b2 is_continuous_in b1 & b2 /. b1 <> 0
holds b2 ^ is_continuous_in b1;
:: CFCONT_1:th 59
theorem
for b1 being Element of COMPLEX
for b2, b3 being Function-like Relation of COMPLEX,COMPLEX
st b2 is_continuous_in b1 & b2 /. b1 <> 0 & b3 is_continuous_in b1
holds b3 / b2 is_continuous_in b1;
:: CFCONT_1:prednot 2 => CFCONT_1:pred 2
definition
let a1 be Function-like Relation of COMPLEX,COMPLEX;
let a2 be set;
pred A1 is_continuous_on A2 means
a2 c= dom a1 &
(for b1 being Element of COMPLEX
st b1 in a2
holds a1 | a2 is_continuous_in b1);
end;
:: CFCONT_1:dfs 4
definiens
let a1 be Function-like Relation of COMPLEX,COMPLEX;
let a2 be set;
To prove
a1 is_continuous_on a2
it is sufficient to prove
thus a2 c= dom a1 &
(for b1 being Element of COMPLEX
st b1 in a2
holds a1 | a2 is_continuous_in b1);
:: CFCONT_1:def 5
theorem
for b1 being Function-like Relation of COMPLEX,COMPLEX
for b2 being set holds
b1 is_continuous_on b2
iff
b2 c= dom b1 &
(for b3 being Element of COMPLEX
st b3 in b2
holds b1 | b2 is_continuous_in b3);
:: CFCONT_1:th 60
theorem
for b1 being set
for b2 being Function-like Relation of COMPLEX,COMPLEX holds
b2 is_continuous_on b1
iff
b1 c= dom b2 &
(for b3 being Function-like quasi_total Relation of NAT,COMPLEX
st rng b3 c= b1 & b3 is convergent & lim b3 in b1
holds b2 * b3 is convergent & b2 /. lim b3 = lim (b2 * b3));
:: CFCONT_1:th 61
theorem
for b1 being set
for b2 being Function-like Relation of COMPLEX,COMPLEX holds
b2 is_continuous_on b1
iff
b1 c= dom b2 &
(for b3 being Element of COMPLEX
for b4 being Element of REAL
st b3 in b1 & 0 < b4
holds ex b5 being Element of REAL st
0 < b5 &
(for b6 being Element of COMPLEX
st b6 in b1 & |.b6 - b3.| < b5
holds |.(b2 /. b6) - (b2 /. b3).| < b4));
:: CFCONT_1:th 62
theorem
for b1 being set
for b2 being Function-like Relation of COMPLEX,COMPLEX holds
b2 is_continuous_on b1
iff
b2 | b1 is_continuous_on b1;
:: CFCONT_1:th 63
theorem
for b1, b2 being set
for b3 being Function-like Relation of COMPLEX,COMPLEX
st b3 is_continuous_on b1 & b2 c= b1
holds b3 is_continuous_on b2;
:: CFCONT_1:th 64
theorem
for b1 being Element of COMPLEX
for b2 being Function-like Relation of COMPLEX,COMPLEX
st b1 in dom b2
holds b2 is_continuous_on {b1};
:: CFCONT_1:th 65
theorem
for b1 being set
for b2, b3 being Function-like Relation of COMPLEX,COMPLEX
st b2 is_continuous_on b1 & b3 is_continuous_on b1
holds b2 + b3 is_continuous_on b1 & b2 - b3 is_continuous_on b1 & b2 (#) b3 is_continuous_on b1;
:: CFCONT_1:th 66
theorem
for b1, b2 being set
for b3, b4 being Function-like Relation of COMPLEX,COMPLEX
st b3 is_continuous_on b1 & b4 is_continuous_on b2
holds b3 + b4 is_continuous_on b1 /\ b2 & b3 - b4 is_continuous_on b1 /\ b2 & b3 (#) b4 is_continuous_on b1 /\ b2;
:: CFCONT_1:th 67
theorem
for b1 being Element of COMPLEX
for b2 being set
for b3 being Function-like Relation of COMPLEX,COMPLEX
st b3 is_continuous_on b2
holds b1 (#) b3 is_continuous_on b2;
:: CFCONT_1:th 68
theorem
for b1 being set
for b2 being Function-like Relation of COMPLEX,COMPLEX
st b2 is_continuous_on b1
holds - b2 is_continuous_on b1;
:: CFCONT_1:th 69
theorem
for b1 being set
for b2 being Function-like Relation of COMPLEX,COMPLEX
st b2 is_continuous_on b1 & b2 " {0} = {}
holds b2 ^ is_continuous_on b1;
:: CFCONT_1:th 70
theorem
for b1 being set
for b2 being Function-like Relation of COMPLEX,COMPLEX
st b2 is_continuous_on b1 &
(b2 | b1) " {0} = {}
holds b2 ^ is_continuous_on b1;
:: CFCONT_1:th 71
theorem
for b1 being set
for b2, b3 being Function-like Relation of COMPLEX,COMPLEX
st b2 is_continuous_on b1 & b2 " {0} = {} & b3 is_continuous_on b1
holds b3 / b2 is_continuous_on b1;
:: CFCONT_1:th 72
theorem
for b1 being Function-like Relation of COMPLEX,COMPLEX
st b1 is total(COMPLEX, COMPLEX) &
(for b2, b3 being Element of COMPLEX holds
b1 /. (b2 + b3) = (b1 /. b2) + (b1 /. b3)) &
(ex b2 being Element of COMPLEX st
b1 is_continuous_in b2)
holds b1 is_continuous_on COMPLEX;
:: CFCONT_1:attrnot 2 => CFCONT_1:attr 1
definition
let a1 be set;
attr a1 is compact means
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
st rng b1 c= a1
holds ex b2 being Function-like quasi_total Relation of NAT,COMPLEX st
b2 is subsequence of b1 & b2 is convergent & lim b2 in a1;
end;
:: CFCONT_1:dfs 5
definiens
let a1 be set;
To prove
a1 is compact
it is sufficient to prove
thus for b1 being Function-like quasi_total Relation of NAT,COMPLEX
st rng b1 c= a1
holds ex b2 being Function-like quasi_total Relation of NAT,COMPLEX st
b2 is subsequence of b1 & b2 is convergent & lim b2 in a1;
:: CFCONT_1:def 6
theorem
for b1 being set holds
b1 is compact
iff
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
st rng b2 c= b1
holds ex b3 being Function-like quasi_total Relation of NAT,COMPLEX st
b3 is subsequence of b2 & b3 is convergent & lim b3 in b1;
:: CFCONT_1:th 73
theorem
for b1 being Function-like Relation of COMPLEX,COMPLEX
st dom b1 is compact & b1 is_continuous_on dom b1
holds rng b1 is compact;
:: CFCONT_1:th 74
theorem
for b1 being Element of bool COMPLEX
for b2 being Function-like Relation of COMPLEX,COMPLEX
st b1 c= dom b2 & b1 is compact & b2 is_continuous_on b1
holds b2 .: b1 is compact;