Article BHSP_2, MML version 4.99.1005
:: BHSP_2:attrnot 1 => BHSP_2: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 convergent means
ex b1 being Element of the carrier of a1 st
for b2 being Element of REAL
st 0 < b2
holds ex b3 being Element of NAT st
for b4 being Element of NAT
st b3 <= b4
holds dist(a2 . b4,b1) < b2;
end;
:: BHSP_2: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 convergent
it is sufficient to prove
thus ex b1 being Element of the carrier of a1 st
for b2 being Element of REAL
st 0 < b2
holds ex b3 being Element of NAT st
for b4 being Element of NAT
st b3 <= b4
holds dist(a2 . b4,b1) < b2;
:: BHSP_2: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 convergent(b1)
iff
ex b3 being Element of the carrier of b1 st
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) < b4;
:: BHSP_2: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 convergent(b1);
:: BHSP_2:th 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
st b2 is convergent(b1) &
(ex b4 being Element of NAT st
for b5 being Element of NAT
st b4 <= b5
holds b3 . b5 = b2 . b5)
holds b3 is convergent(b1);
:: BHSP_2: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 convergent(b1) & b3 is convergent(b1)
holds b2 + b3 is convergent(b1);
:: BHSP_2: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 convergent(b1) & b3 is convergent(b1)
holds b2 - b3 is convergent(b1);
:: BHSP_2: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 convergent(b1)
holds b2 * b3 is convergent(b1);
:: BHSP_2: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 convergent(b1)
holds - b2 is convergent(b1);
:: BHSP_2: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 convergent(b1)
holds b3 + b2 is convergent(b1);
:: BHSP_2: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 convergent(b1)
holds b3 - b2 is convergent(b1);
:: BHSP_2: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 holds
b2 is convergent(b1)
iff
ex b3 being Element of the carrier of b1 st
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.|| < b4;
:: BHSP_2:funcnot 1 => BHSP_2: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 Relation of NAT,the carrier of a1;
assume a2 is convergent(a1);
func lim A2 -> Element of the carrier of a1 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,it) < b1;
end;
:: BHSP_2:def 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
st b2 is convergent(b1)
for b3 being Element of the carrier of b1 holds
b3 = lim b2
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) < b4;
:: BHSP_2:th 10
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 constant & b2 in proj2 b3
holds lim b3 = b2;
:: BHSP_2:th 11
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 constant &
(ex b4 being Element of NAT st
b3 . b4 = b2)
holds lim b3 = b2;
:: BHSP_2:th 12
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
for b5 being Element of NAT
st b4 <= b5
holds b3 . b5 = b2 . b5)
holds lim b2 = lim b3;
:: BHSP_2: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
st b2 is convergent(b1) & b3 is convergent(b1)
holds lim (b2 + b3) = (lim b2) + lim b3;
:: BHSP_2: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 b2 is convergent(b1) & b3 is convergent(b1)
holds lim (b2 - b3) = (lim b2) - lim b3;
:: BHSP_2:th 15
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 convergent(b1)
holds lim (b2 * b3) = b2 * lim b3;
:: BHSP_2:th 16
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 lim - b2 = - lim b2;
:: BHSP_2: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 being Function-like quasi_total Relation of NAT,the carrier of b1
st b3 is convergent(b1)
holds lim (b3 + b2) = (lim b3) + b2;
:: BHSP_2:th 18
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 convergent(b1)
holds lim (b3 - b2) = (lim b3) - b2;
:: BHSP_2:th 19
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 convergent(b1)
holds lim b3 = b2
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 ||.(b3 . b6) - b2.|| < b4;
:: BHSP_2:funcnot 2 => BHSP_2: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;
func ||.A2.|| -> Function-like quasi_total Relation of NAT,REAL means
for b1 being Element of NAT holds
it . b1 = ||.a2 . b1.||;
end;
:: BHSP_2: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
for b3 being Function-like quasi_total Relation of NAT,REAL holds
b3 = ||.b2.||
iff
for b4 being Element of NAT holds
b3 . b4 = ||.b2 . b4.||;
:: BHSP_2:th 20
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 convergent;
:: BHSP_2: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 the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b3 is convergent(b1) & lim b3 = b2
holds ||.b3.|| is convergent & lim ||.b3.|| = ||.b2.||;
:: BHSP_2:th 22
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 convergent(b1) & lim b3 = b2
holds ||.b3 - b2.|| is convergent & lim ||.b3 - b2.|| = 0;
:: BHSP_2:funcnot 3 => BHSP_2:func 3
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 the carrier of a1;
func dist(A2,A3) -> Function-like quasi_total Relation of NAT,REAL means
for b1 being Element of NAT holds
it . b1 = dist(a2 . b1,a3);
end;
:: BHSP_2:def 4
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 the carrier of b1
for b4 being Function-like quasi_total Relation of NAT,REAL holds
b4 = dist(b2,b3)
iff
for b5 being Element of NAT holds
b4 . b5 = dist(b2 . b5,b3);
:: BHSP_2:th 23
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 convergent(b1) & lim b3 = b2
holds dist(b3,b2) is convergent;
:: BHSP_2:th 24
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 convergent(b1) & lim b3 = b2
holds dist(b3,b2) is convergent & lim dist(b3,b2) = 0;
:: BHSP_2:th 25
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4, b5 being Function-like quasi_total Relation of NAT,the carrier of b1
st b4 is convergent(b1) & lim b4 = b2 & b5 is convergent(b1) & lim b5 = b3
holds ||.b4 + b5.|| is convergent &
lim ||.b4 + b5.|| = ||.b2 + b3.||;
:: BHSP_2:th 26
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4, b5 being Function-like quasi_total Relation of NAT,the carrier of b1
st b4 is convergent(b1) & lim b4 = b2 & b5 is convergent(b1) & lim b5 = b3
holds ||.(b4 + b5) - (b2 + b3).|| is convergent &
lim ||.(b4 + b5) - (b2 + b3).|| = 0;
:: BHSP_2:th 27
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4, b5 being Function-like quasi_total Relation of NAT,the carrier of b1
st b4 is convergent(b1) & lim b4 = b2 & b5 is convergent(b1) & lim b5 = b3
holds ||.b4 - b5.|| is convergent &
lim ||.b4 - b5.|| = ||.b2 - b3.||;
:: BHSP_2:th 28
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4, b5 being Function-like quasi_total Relation of NAT,the carrier of b1
st b4 is convergent(b1) & lim b4 = b2 & b5 is convergent(b1) & lim b5 = b3
holds ||.(b4 - b5) - (b2 - b3).|| is convergent &
lim ||.(b4 - b5) - (b2 - b3).|| = 0;
:: BHSP_2:th 29
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 Element of REAL
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
st b4 is convergent(b1) & lim b4 = b2
holds ||.b3 * b4.|| is convergent &
lim ||.b3 * b4.|| = ||.b3 * b2.||;
:: BHSP_2:th 30
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 Element of REAL
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
st b4 is convergent(b1) & lim b4 = b2
holds ||.(b3 * b4) - (b3 * b2).|| is convergent &
lim ||.(b3 * b4) - (b3 * b2).|| = 0;
:: BHSP_2:th 31
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 convergent(b1) & lim b3 = b2
holds ||.- b3.|| is convergent &
lim ||.- b3.|| = ||.- b2.||;
:: BHSP_2:th 32
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 convergent(b1) & lim b3 = b2
holds ||.(- b3) - - b2.|| is convergent &
lim ||.(- b3) - - b2.|| = 0;
:: BHSP_2:th 33
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
st b4 is convergent(b1) & lim b4 = b2
holds ||.(b4 + b3) - (b2 + b3).|| is convergent &
lim ||.(b4 + b3) - (b2 + b3).|| = 0;
:: BHSP_2:th 34
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
st b4 is convergent(b1) & lim b4 = b2
holds ||.b4 - b3.|| is convergent &
lim ||.b4 - b3.|| = ||.b2 - b3.||;
:: BHSP_2:th 35
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
st b4 is convergent(b1) & lim b4 = b2
holds ||.(b4 - b3) - (b2 - b3).|| is convergent &
lim ||.(b4 - b3) - (b2 - b3).|| = 0;
:: BHSP_2:th 36
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4, b5 being Function-like quasi_total Relation of NAT,the carrier of b1
st b4 is convergent(b1) & lim b4 = b2 & b5 is convergent(b1) & lim b5 = b3
holds dist(b4 + b5,b2 + b3) is convergent &
lim dist(b4 + b5,b2 + b3) = 0;
:: BHSP_2:th 37
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4, b5 being Function-like quasi_total Relation of NAT,the carrier of b1
st b4 is convergent(b1) & lim b4 = b2 & b5 is convergent(b1) & lim b5 = b3
holds dist(b4 - b5,b2 - b3) is convergent &
lim dist(b4 - b5,b2 - b3) = 0;
:: BHSP_2:th 38
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 Element of REAL
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
st b4 is convergent(b1) & lim b4 = b2
holds dist(b3 * b4,b3 * b2) is convergent &
lim dist(b3 * b4,b3 * b2) = 0;
:: BHSP_2:th 39
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
st b4 is convergent(b1) & lim b4 = b2
holds dist(b4 + b3,b2 + b3) is convergent &
lim dist(b4 + b3,b2 + b3) = 0;
:: BHSP_2:funcnot 4 => BHSP_2:func 4
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
let a2 be Element of the carrier of a1;
let a3 be Element of REAL;
func Ball(A2,A3) -> Element of bool the carrier of a1 equals
{b1 where b1 is Element of the carrier of a1: ||.a2 - b1.|| < a3};
end;
:: BHSP_2:def 5
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 Element of REAL holds
Ball(b2,b3) = {b4 where b4 is Element of the carrier of b1: ||.b2 - b4.|| < b3};
:: BHSP_2:funcnot 5 => BHSP_2:func 5
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
let a2 be Element of the carrier of a1;
let a3 be Element of REAL;
func cl_Ball(A2,A3) -> Element of bool the carrier of a1 equals
{b1 where b1 is Element of the carrier of a1: ||.a2 - b1.|| <= a3};
end;
:: BHSP_2:def 6
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 Element of REAL holds
cl_Ball(b2,b3) = {b4 where b4 is Element of the carrier of b1: ||.b2 - b4.|| <= b3};
:: BHSP_2:funcnot 6 => BHSP_2:func 6
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
let a2 be Element of the carrier of a1;
let a3 be Element of REAL;
func Sphere(A2,A3) -> Element of bool the carrier of a1 equals
{b1 where b1 is Element of the carrier of a1: ||.a2 - b1.|| = a3};
end;
:: BHSP_2:def 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 Element of REAL holds
Sphere(b2,b3) = {b4 where b4 is Element of the carrier of b1: ||.b2 - b4.|| = b3};
:: BHSP_2:th 40
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4 being Element of REAL holds
b2 in Ball(b3,b4)
iff
||.b3 - b2.|| < b4;
:: BHSP_2:th 41
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4 being Element of REAL holds
b2 in Ball(b3,b4)
iff
dist(b3,b2) < b4;
:: BHSP_2:th 42
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 Element of REAL
st 0 < b3
holds b2 in Ball(b2,b3);
:: BHSP_2:th 43
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4 being Element of the carrier of b1
for b5 being Element of REAL
st b2 in Ball(b3,b5) & b4 in Ball(b3,b5)
holds dist(b2,b4) < 2 * b5;
:: BHSP_2:th 44
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4 being Element of the carrier of b1
for b5 being Element of REAL
st b2 in Ball(b3,b5)
holds b2 - b4 in Ball(b3 - b4,b5);
:: BHSP_2:th 45
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4 being Element of REAL
st b2 in Ball(b3,b4)
holds b2 - b3 in Ball(0. b1,b4);
:: BHSP_2:th 46
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4, b5 being Element of REAL
st b2 in Ball(b3,b4) & b4 <= b5
holds b2 in Ball(b3,b5);
:: BHSP_2:th 47
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4 being Element of REAL holds
b2 in cl_Ball(b3,b4)
iff
||.b3 - b2.|| <= b4;
:: BHSP_2:th 48
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4 being Element of REAL holds
b2 in cl_Ball(b3,b4)
iff
dist(b3,b2) <= b4;
:: BHSP_2:th 49
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 Element of REAL
st 0 <= b3
holds b2 in cl_Ball(b2,b3);
:: BHSP_2:th 50
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4 being Element of REAL
st b2 in Ball(b3,b4)
holds b2 in cl_Ball(b3,b4);
:: BHSP_2:th 51
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4 being Element of REAL holds
b2 in Sphere(b3,b4)
iff
||.b3 - b2.|| = b4;
:: BHSP_2:th 52
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4 being Element of REAL holds
b2 in Sphere(b3,b4)
iff
dist(b3,b2) = b4;
:: BHSP_2:th 53
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4 being Element of REAL
st b2 in Sphere(b3,b4)
holds b2 in cl_Ball(b3,b4);
:: BHSP_2:th 54
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 Element of REAL holds
Ball(b2,b3) c= cl_Ball(b2,b3);
:: BHSP_2:th 55
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 Element of REAL holds
Sphere(b2,b3) c= cl_Ball(b2,b3);
:: BHSP_2:th 56
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 Element of REAL holds
(Ball(b2,b3)) \/ Sphere(b2,b3) = cl_Ball(b2,b3);