Article BHSP_3, MML version 4.99.1005
:: BHSP_3:attrnot 1 => BHSP_3:attr 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
attr a2 is Cauchy means
for b1 being Element of REAL
st 0 < b1
holds ex b2 being Element of NAT st
for b3, b4 being Element of NAT
st b2 <= b3 & b2 <= b4
holds dist(a2 . b3,a2 . b4) < b1;
end;
:: BHSP_3:dfs 1
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
To prove
a2 is Cauchy
it is sufficient to prove
thus for b1 being Element of REAL
st 0 < b1
holds ex b2 being Element of NAT st
for b3, b4 being Element of NAT
st b2 <= b3 & b2 <= b4
holds dist(a2 . b3,a2 . b4) < b1;
:: BHSP_3:def 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 is Cauchy(b1)
iff
for b3 being Element of REAL
st 0 < b3
holds ex b4 being Element of NAT st
for b5, b6 being Element of NAT
st b4 <= b5 & b4 <= b6
holds dist(b2 . b5,b2 . b6) < b3;
:: BHSP_3:prednot 1 => BHSP_3:attr 1
notation
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
synonym a2 is_Cauchy_sequence for Cauchy;
end;
:: BHSP_3:th 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is constant
holds b2 is Cauchy(b1);
:: BHSP_3:th 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 is Cauchy(b1)
iff
for b3 being Element of REAL
st 0 < b3
holds ex b4 being Element of NAT st
for b5, b6 being Element of NAT
st b4 <= b5 & b4 <= b6
holds ||.(b2 . b5) - (b2 . b6).|| < b3;
:: BHSP_3:th 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is Cauchy(b1) & b3 is Cauchy(b1)
holds b2 + b3 is Cauchy(b1);
:: BHSP_3:th 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is Cauchy(b1) & b3 is Cauchy(b1)
holds b2 - b3 is Cauchy(b1);
:: BHSP_3:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of REAL
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b3 is Cauchy(b1)
holds b2 * b3 is Cauchy(b1);
:: BHSP_3:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is Cauchy(b1)
holds - b2 is Cauchy(b1);
:: BHSP_3:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b3 is Cauchy(b1)
holds b3 + b2 is Cauchy(b1);
:: BHSP_3:th 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b3 is Cauchy(b1)
holds b3 - b2 is Cauchy(b1);
:: BHSP_3:th 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is convergent(b1)
holds b2 is Cauchy(b1);
:: BHSP_3:prednot 2 => BHSP_3:pred 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
let a2, a3 be Function-like quasi_total Relation of NAT,the carrier of a1;
pred A2 is_compared_to A3 means
for b1 being Element of REAL
st 0 < b1
holds ex b2 being Element of NAT st
for b3 being Element of NAT
st b2 <= b3
holds dist(a2 . b3,a3 . b3) < b1;
end;
:: BHSP_3:dfs 2
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
let a2, a3 be Function-like quasi_total Relation of NAT,the carrier of a1;
To prove
a2 is_compared_to a3
it is sufficient to prove
thus for b1 being Element of REAL
st 0 < b1
holds ex b2 being Element of NAT st
for b3 being Element of NAT
st b2 <= b3
holds dist(a2 . b3,a3 . b3) < b1;
:: BHSP_3:def 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 is_compared_to b3
iff
for b4 being Element of REAL
st 0 < b4
holds ex b5 being Element of NAT st
for b6 being Element of NAT
st b5 <= b6
holds dist(b2 . b6,b3 . b6) < b4;
:: BHSP_3:th 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 is_compared_to b2;
:: BHSP_3:th 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is_compared_to b3
holds b3 is_compared_to b2;
:: BHSP_3:prednot 3 => BHSP_3:pred 2
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
let a2, a3 be Function-like quasi_total Relation of NAT,the carrier of a1;
redefine pred a2 is_compared_to a3;
symmetry;
:: for a1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
:: for a2, a3 being Function-like quasi_total Relation of NAT,the carrier of a1
:: st a2 is_compared_to a3
:: holds a3 is_compared_to a2;
reflexivity;
:: for a1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
:: for a2 being Function-like quasi_total Relation of NAT,the carrier of a1 holds
:: a2 is_compared_to a2;
end;
:: BHSP_3:th 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is_compared_to b3 & b3 is_compared_to b4
holds b2 is_compared_to b4;
:: BHSP_3:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 is_compared_to b3
iff
for b4 being Element of REAL
st 0 < b4
holds ex b5 being Element of NAT st
for b6 being Element of NAT
st b5 <= b6
holds ||.(b2 . b6) - (b3 . b6).|| < b4;
:: BHSP_3:th 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st ex b4 being Element of NAT st
for b5 being Element of NAT
st b4 <= b5
holds b2 . b5 = b3 . b5
holds b2 is_compared_to b3;
:: BHSP_3:th 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is Cauchy(b1) & b2 is_compared_to b3
holds b3 is Cauchy(b1);
:: BHSP_3:th 16
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is convergent(b1) & b2 is_compared_to b3
holds b3 is convergent(b1);
:: BHSP_3:th 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of the carrier of b1
for b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b1
st b3 is convergent(b1) & lim b3 = b2 & b3 is_compared_to b4
holds b4 is convergent(b1) & lim b4 = b2;
:: BHSP_3:attrnot 2 => BHSP_3:attr 2
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
attr a2 is bounded means
ex b1 being Element of REAL st
0 < b1 &
(for b2 being Element of NAT holds
||.a2 . b2.|| <= b1);
end;
:: BHSP_3:dfs 3
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
To prove
a2 is bounded
it is sufficient to prove
thus ex b1 being Element of REAL st
0 < b1 &
(for b2 being Element of NAT holds
||.a2 . b2.|| <= b1);
:: BHSP_3:def 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 is bounded(b1)
iff
ex b3 being Element of REAL st
0 < b3 &
(for b4 being Element of NAT holds
||.b2 . b4.|| <= b3);
:: BHSP_3:th 18
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is bounded(b1) & b3 is bounded(b1)
holds b2 + b3 is bounded(b1);
:: BHSP_3:th 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is bounded(b1)
holds - b2 is bounded(b1);
:: BHSP_3:th 20
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is bounded(b1) & b3 is bounded(b1)
holds b2 - b3 is bounded(b1);
:: BHSP_3:th 21
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of REAL
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b3 is bounded(b1)
holds b2 * b3 is bounded(b1);
:: BHSP_3:th 22
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is constant
holds b2 is bounded(b1);
:: BHSP_3:th 23
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of NAT holds
ex b4 being Element of REAL st
0 < b4 &
(for b5 being Element of NAT
st b5 <= b3
holds ||.b2 . b5.|| < b4);
:: BHSP_3:th 24
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is convergent(b1)
holds b2 is bounded(b1);
:: BHSP_3:th 25
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is bounded(b1) & b2 is_compared_to b3
holds b3 is bounded(b1);
:: BHSP_3:funcnot 1 => BHSP_3:func 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
let a2 be Function-like quasi_total natural-valued increasing Relation of NAT,REAL;
let a3 be Function-like quasi_total Relation of NAT,the carrier of a1;
redefine func a3 * a2 -> Function-like quasi_total Relation of NAT,the carrier of a1;
end;
:: BHSP_3:modenot 1 => BHSP_3:mode 1
definition
let a1 be 1-sorted;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
mode subsequence of A2 -> Function-like quasi_total Relation of NAT,the carrier of a1 means
ex b1 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL st
it = b1 * a2;
end;
:: BHSP_3:dfs 4
definiens
let a1 be 1-sorted;
let a2, a3 be Function-like quasi_total Relation of NAT,the carrier of a1;
To prove
a3 is subsequence of a2
it is sufficient to prove
thus ex b1 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL st
a3 = b1 * a2;
:: BHSP_3:def 4
theorem
for b1 being 1-sorted
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b3 is subsequence of b2
iff
ex b4 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL st
b3 = b4 * b2;
:: BHSP_3:th 26
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL
for b4 being Element of NAT holds
(b2 * b3) . b4 = b2 . (b3 . b4);
:: BHSP_3:th 27
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 is subsequence of b2;
:: BHSP_3:th 28
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is subsequence of b3 & b3 is subsequence of b4
holds b2 is subsequence of b4;
:: BHSP_3:th 29
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is constant & b3 is subsequence of b2
holds b3 is constant;
:: BHSP_3:th 30
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is constant & b3 is subsequence of b2
holds b2 = b3;
:: BHSP_3:th 31
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is bounded(b1) & b3 is subsequence of b2
holds b3 is bounded(b1);
:: BHSP_3:th 32
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is convergent(b1) & b3 is subsequence of b2
holds b3 is convergent(b1);
:: BHSP_3:th 33
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is subsequence of b3 & b3 is convergent(b1)
holds lim b2 = lim b3;
:: BHSP_3:th 34
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is Cauchy(b1) & b3 is subsequence of b2
holds b3 is Cauchy(b1);
:: BHSP_3:funcnot 2 => BHSP_3:func 2
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
let a3 be Element of NAT;
func A2 ^\ A3 -> Function-like quasi_total Relation of NAT,the carrier of a1 means
for b1 being Element of NAT holds
it . b1 = a2 . (b1 + a3);
end;
:: BHSP_3:def 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of NAT
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b4 = b2 ^\ b3
iff
for b5 being Element of NAT holds
b4 . b5 = b2 . (b5 + b3);
:: BHSP_3:th 35
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 ^\ 0 = b2;
:: BHSP_3:th 36
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3, b4 being Element of NAT holds
(b2 ^\ b3) ^\ b4 = (b2 ^\ b4) ^\ b3;
:: BHSP_3:th 37
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3, b4 being Element of NAT holds
(b2 ^\ b3) ^\ b4 = b2 ^\ (b3 + b4);
:: BHSP_3:th 38
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
for b4 being Element of NAT holds
(b2 + b3) ^\ b4 = (b2 ^\ b4) + (b3 ^\ b4);
:: BHSP_3:th 39
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of NAT holds
(- b2) ^\ b3 = - (b2 ^\ b3);
:: BHSP_3:th 40
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
for b4 being Element of NAT holds
(b2 - b3) ^\ b4 = (b2 ^\ b4) - (b3 ^\ b4);
:: BHSP_3:th 41
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of REAL
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
for b4 being Element of NAT holds
(b2 * b3) ^\ b4 = b2 * (b3 ^\ b4);
:: BHSP_3:th 42
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL
for b4 being Element of NAT holds
(b2 * b3) ^\ b4 = b2 * (b3 ^\ b4);
:: BHSP_3:th 43
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of NAT holds
b2 ^\ b3 is subsequence of b2;
:: BHSP_3:th 44
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of NAT
st b2 is convergent(b1)
holds b2 ^\ b3 is convergent(b1) & lim (b2 ^\ b3) = lim b2;
:: BHSP_3:th 46
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is convergent(b1) &
(ex b4 being Element of NAT st
b2 = b3 ^\ b4)
holds b3 is convergent(b1);
:: BHSP_3:th 47
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is Cauchy(b1) &
(ex b4 being Element of NAT st
b2 = b3 ^\ b4)
holds b3 is Cauchy(b1);
:: BHSP_3:th 48
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of NAT
st b2 is Cauchy(b1)
holds b2 ^\ b3 is Cauchy(b1);
:: BHSP_3:th 49
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
for b4 being Element of NAT
st b2 is_compared_to b3
holds b2 ^\ b4 is_compared_to b3 ^\ b4;
:: BHSP_3:th 50
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of NAT
st b2 is bounded(b1)
holds b2 ^\ b3 is bounded(b1);
:: BHSP_3:th 51
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of NAT
st b2 is constant
holds b2 ^\ b3 is constant;
:: BHSP_3:attrnot 3 => BHSP_3:attr 3
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
attr a1 is complete means
for b1 being Function-like quasi_total Relation of NAT,the carrier of a1
st b1 is Cauchy(a1)
holds b1 is convergent(a1);
end;
:: BHSP_3:dfs 6
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
To prove
a1 is complete
it is sufficient to prove
thus for b1 being Function-like quasi_total Relation of NAT,the carrier of a1
st b1 is Cauchy(a1)
holds b1 is convergent(a1);
:: BHSP_3:def 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR holds
b1 is complete
iff
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is Cauchy(b1)
holds b2 is convergent(b1);
:: BHSP_3:prednot 4 => BHSP_3:attr 3
notation
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
synonym a1 is_complete_space for complete;
end;
:: BHSP_3:th 53
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b1 is complete & b2 is Cauchy(b1)
holds b2 is bounded(b1);
:: BHSP_3:attrnot 4 => BHSP_3:attr 4
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
attr a1 is Hilbert means
a1 is non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR &
a1 is complete;
end;
:: BHSP_3:dfs 7
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
To prove
a1 is Hilbert
it is sufficient to prove
thus a1 is non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR &
a1 is complete;
:: BHSP_3:def 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR holds
b1 is Hilbert
iff
b1 is non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR &
b1 is complete;
:: BHSP_3:prednot 5 => BHSP_3:attr 4
notation
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
synonym a1 is_Hilbert_space for Hilbert;
end;