Article RUSUB_1, MML version 4.99.1005
:: RUSUB_1:modenot 1 => RUSUB_1:mode 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
mode Subspace of A1 -> non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR means
the carrier of it c= the carrier of a1 &
0. it = 0. a1 &
the addF of it = (the addF of a1) || the carrier of it &
the Mult of it = (the Mult of a1) | [:REAL,the carrier of it:] &
the scalar of it = (the scalar of a1) || the carrier of it;
end;
:: RUSUB_1:dfs 1
definiens
let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
To prove
a2 is Subspace of a1
it is sufficient to prove
thus the carrier of a2 c= the carrier of a1 &
0. a2 = 0. a1 &
the addF of a2 = (the addF of a1) || the carrier of a2 &
the Mult of a2 = (the Mult of a1) | [:REAL,the carrier of a2:] &
the scalar of a2 = (the scalar of a1) || the carrier of a2;
:: RUSUB_1:def 1
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR holds
b2 is Subspace of b1
iff
the carrier of b2 c= the carrier of b1 &
0. b2 = 0. b1 &
the addF of b2 = (the addF of b1) || the carrier of b2 &
the Mult of b2 = (the Mult of b1) | [:REAL,the carrier of b2:] &
the scalar of b2 = (the scalar of b1) || the carrier of b2;
:: RUSUB_1:th 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Subspace of b1
for b4 being set
st b4 in b2 & b2 is Subspace of b3
holds b4 in b3;
:: RUSUB_1:th 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3 being set
st b3 in b2
holds b3 in b1;
:: RUSUB_1:th 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3 being Element of the carrier of b2 holds
b3 is Element of the carrier of b1;
:: RUSUB_1:th 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1 holds
0. b2 = 0. b1;
:: RUSUB_1:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Subspace of b1 holds
0. b2 = 0. b3;
:: RUSUB_1:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3, b4 being Element of the carrier of b1
for b5, b6 being Element of the carrier of b2
st b5 = b4 & b6 = b3
holds b5 + b6 = b4 + b3;
:: RUSUB_1:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2
for b5 being Element of REAL
st b4 = b3
holds b5 * b4 = b5 * b3;
:: RUSUB_1:th 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3, b4 being Element of the carrier of b1
for b5, b6 being Element of the carrier of b2
st b5 = b3 & b6 = b4
holds b5 .|. b6 = b3 .|. b4;
:: RUSUB_1:th 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2
st b4 = b3
holds - b3 = - b4;
:: RUSUB_1:th 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3, b4 being Element of the carrier of b1
for b5, b6 being Element of the carrier of b2
st b5 = b4 & b6 = b3
holds b5 - b6 = b4 - b3;
:: RUSUB_1:th 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1 holds
0. b1 in b2;
:: RUSUB_1:th 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Subspace of b1 holds
0. b2 in b3;
:: RUSUB_1:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1 holds
0. b2 in b1;
:: RUSUB_1:th 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3, b4 being Element of the carrier of b1
st b3 in b2 & b4 in b2
holds b3 + b4 in b2;
:: RUSUB_1:th 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3 being Element of the carrier of b1
for b4 being Element of REAL
st b3 in b2
holds b4 * b3 in b2;
:: RUSUB_1:th 16
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3 being Element of the carrier of b1
st b3 in b2
holds - b3 in b2;
:: RUSUB_1:th 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3, b4 being Element of the carrier of b1
st b3 in b2 & b4 in b2
holds b3 - b4 in b2;
:: RUSUB_1: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 bool the carrier of b1
for b3 being non empty set
for b4 being Element of b3
for b5 being Function-like quasi_total Relation of [:b3,b3:],b3
for b6 being Function-like quasi_total Relation of [:REAL,b3:],b3
for b7 being Function-like quasi_total Relation of [:b3,b3:],REAL
st b2 = b3 &
b4 = 0. b1 &
b5 = (the addF of b1) || b2 &
b6 = (the Mult of b1) | [:REAL,b2:] &
b7 = (the scalar of b1) || b2
holds UNITSTR(#b3,b4,b5,b6,b7#) is Subspace of b1;
:: RUSUB_1:th 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR holds
b1 is Subspace of b1;
:: RUSUB_1:th 20
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like strict RealUnitarySpace-like UNITSTR
st b1 is Subspace of b2 & b2 is Subspace of b1
holds b1 = b2;
:: RUSUB_1:th 21
theorem
for b1, b2, b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
st b1 is Subspace of b2 & b2 is Subspace of b3
holds b1 is Subspace of b3;
:: RUSUB_1:th 22
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Subspace of b1
st the carrier of b2 c= the carrier of b3
holds b2 is Subspace of b3;
:: RUSUB_1:th 23
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Subspace of b1
st for b4 being Element of the carrier of b1
st b4 in b2
holds b4 in b3
holds b2 is Subspace of b3;
:: RUSUB_1:exreg 1
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
cluster non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like strict RealUnitarySpace-like Subspace of a1;
end;
:: RUSUB_1:th 24
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being strict Subspace of b1
st the carrier of b2 = the carrier of b3
holds b2 = b3;
:: RUSUB_1:th 25
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being strict Subspace of b1
st for b4 being Element of the carrier of b1 holds
b4 in b2
iff
b4 in b3
holds b2 = b3;
:: RUSUB_1:th 26
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like strict RealUnitarySpace-like UNITSTR
for b2 being strict Subspace of b1
st the carrier of b2 = the carrier of b1
holds b2 = b1;
:: RUSUB_1:th 27
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like strict RealUnitarySpace-like UNITSTR
for b2 being strict Subspace of b1
st for b3 being Element of the carrier of b1 holds
b3 in b2
iff
b3 in b1
holds b2 = b1;
:: RUSUB_1:th 28
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3 being Element of bool the carrier of b1
st the carrier of b2 = b3
holds b3 is linearly-closed(b1);
:: RUSUB_1: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 bool the carrier of b1
st b2 <> {} & b2 is linearly-closed(b1)
holds ex b3 being strict Subspace of b1 st
b2 = the carrier of b3;
:: RUSUB_1:funcnot 1 => RUSUB_1:func 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
func (0). A1 -> strict Subspace of a1 means
the carrier of it = {0. a1};
end;
:: RUSUB_1:def 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being strict Subspace of b1 holds
b2 = (0). b1
iff
the carrier of b2 = {0. b1};
:: RUSUB_1:funcnot 2 => RUSUB_1:func 2
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
func (Omega). A1 -> strict Subspace of a1 equals
UNITSTR(#the carrier of a1,the ZeroF of a1,the addF of a1,the Mult of a1,the scalar of a1#);
end;
:: RUSUB_1:def 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR holds
(Omega). b1 = UNITSTR(#the carrier of b1,the ZeroF of b1,the addF of b1,the Mult of b1,the scalar of b1#);
:: RUSUB_1:th 30
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1 holds
(0). b2 = (0). b1;
:: RUSUB_1:th 31
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Subspace of b1 holds
(0). b2 = (0). b3;
:: RUSUB_1:th 32
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1 holds
(0). b2 is Subspace of b1;
:: RUSUB_1:th 33
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1 holds
(0). b1 is Subspace of b2;
:: RUSUB_1:th 34
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Subspace of b1 holds
(0). b2 is Subspace of b3;
:: RUSUB_1:th 35
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like strict RealUnitarySpace-like UNITSTR holds
b1 is Subspace of (Omega). b1;
:: RUSUB_1:funcnot 3 => RUSUB_1:func 3
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 Subspace of a1;
func A2 + A3 -> Element of bool the carrier of a1 equals
{a2 + b1 where b1 is Element of the carrier of a1: b1 in a3};
end;
:: RUSUB_1:def 4
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 Subspace of b1 holds
b2 + b3 = {b2 + b4 where b4 is Element of the carrier of b1: b4 in b3};
:: RUSUB_1:modenot 2 => RUSUB_1:mode 2
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
let a2 be Subspace of a1;
mode Coset of A2 -> Element of bool the carrier of a1 means
ex b1 being Element of the carrier of a1 st
it = b1 + a2;
end;
:: RUSUB_1:dfs 5
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
let a2 be Subspace of a1;
let a3 be Element of bool the carrier of a1;
To prove
a3 is Coset of a2
it is sufficient to prove
thus ex b1 being Element of the carrier of a1 st
a3 = b1 + a2;
:: RUSUB_1:def 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3 being Element of bool the carrier of b1 holds
b3 is Coset of b2
iff
ex b4 being Element of the carrier of b1 st
b3 = b4 + b2;
:: RUSUB_1:th 36
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3 being Element of the carrier of b1 holds
0. b1 in b3 + b2
iff
b3 in b2;
:: RUSUB_1:th 37
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3 being Element of the carrier of b1 holds
b3 in b3 + b2;
:: RUSUB_1:th 38
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1 holds
(0. b1) + b2 = the carrier of b2;
:: RUSUB_1:th 39
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 holds
b2 + (0). b1 = {b2};
:: RUSUB_1:th 40
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 holds
b2 + (Omega). b1 = the carrier of b1;
:: RUSUB_1:th 41
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3 being Element of the carrier of b1 holds
0. b1 in b3 + b2
iff
b3 + b2 = the carrier of b2;
:: RUSUB_1:th 42
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3 being Element of the carrier of b1 holds
b3 in b2
iff
b3 + b2 = the carrier of b2;
:: RUSUB_1:th 43
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3 being Element of the carrier of b1
for b4 being Element of REAL
st b3 in b2
holds (b4 * b3) + b2 = the carrier of b2;
:: RUSUB_1:th 44
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3 being Element of the carrier of b1
for b4 being Element of REAL
st b4 <> 0 & (b4 * b3) + b2 = the carrier of b2
holds b3 in b2;
:: RUSUB_1:th 45
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3 being Element of the carrier of b1 holds
b3 in b2
iff
(- b3) + b2 = the carrier of b2;
:: RUSUB_1:th 46
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3, b4 being Element of the carrier of b1 holds
b3 in b2
iff
b4 + b2 = (b4 + b3) + b2;
:: RUSUB_1:th 47
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3, b4 being Element of the carrier of b1 holds
b3 in b2
iff
b4 + b2 = (b4 - b3) + b2;
:: RUSUB_1:th 48
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3, b4 being Element of the carrier of b1 holds
b4 in b3 + b2
iff
b3 + b2 = b4 + b2;
:: RUSUB_1:th 49
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3 being Element of the carrier of b1 holds
b3 + b2 = (- b3) + b2
iff
b3 in b2;
:: RUSUB_1:th 50
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3, b4, b5 being Element of the carrier of b1
st b3 in b4 + b2 & b3 in b5 + b2
holds b4 + b2 = b5 + b2;
:: RUSUB_1:th 51
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3, b4 being Element of the carrier of b1
st b3 in b4 + b2 & b3 in (- b4) + b2
holds b4 in b2;
:: RUSUB_1:th 52
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3 being Element of the carrier of b1
for b4 being Element of REAL
st b4 <> 1 & b4 * b3 in b3 + b2
holds b3 in b2;
:: RUSUB_1:th 53
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3 being Element of the carrier of b1
for b4 being Element of REAL
st b3 in b2
holds b4 * b3 in b3 + b2;
:: RUSUB_1:th 54
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3 being Element of the carrier of b1 holds
- b3 in b3 + b2
iff
b3 in b2;
:: RUSUB_1:th 55
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3, b4 being Element of the carrier of b1 holds
b3 + b4 in b4 + b2
iff
b3 in b2;
:: RUSUB_1:th 56
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3, b4 being Element of the carrier of b1 holds
b4 - b3 in b4 + b2
iff
b3 in b2;
:: RUSUB_1:th 57
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3, b4 being Element of the carrier of b1 holds
b3 in b4 + b2
iff
ex b5 being Element of the carrier of b1 st
b5 in b2 & b3 = b4 + b5;
:: RUSUB_1:th 58
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3, b4 being Element of the carrier of b1 holds
b3 in b4 + b2
iff
ex b5 being Element of the carrier of b1 st
b5 in b2 & b3 = b4 - b5;
:: RUSUB_1:th 59
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3, b4 being Element of the carrier of b1 holds
ex b5 being Element of the carrier of b1 st
b3 in b5 + b2 & b4 in b5 + b2
iff
b3 - b4 in b2;
:: RUSUB_1:th 60
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3, b4 being Element of the carrier of b1
st b4 + b2 = b3 + b2
holds ex b5 being Element of the carrier of b1 st
b5 in b2 & b4 + b5 = b3;
:: RUSUB_1:th 61
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3, b4 being Element of the carrier of b1
st b4 + b2 = b3 + b2
holds ex b5 being Element of the carrier of b1 st
b5 in b2 & b4 - b5 = b3;
:: RUSUB_1:th 62
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being strict Subspace of b1
for b4 being Element of the carrier of b1 holds
b4 + b2 = b4 + b3
iff
b2 = b3;
:: RUSUB_1:th 63
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being strict Subspace of b1
for b4, b5 being Element of the carrier of b1
st b5 + b2 = b4 + b3
holds b2 = b3;
:: RUSUB_1:th 64
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3 being Coset of b2 holds
b3 is linearly-closed(b1)
iff
b3 = the carrier of b2;
:: RUSUB_1:th 65
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being strict Subspace of b1
for b4 being Coset of b2
for b5 being Coset of b3
st b4 = b5
holds b2 = b3;
:: RUSUB_1:th 66
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 holds
{b2} is Coset of (0). b1;
:: RUSUB_1:th 67
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
st b2 is Coset of (0). b1
holds ex b3 being Element of the carrier of b1 st
b2 = {b3};
:: RUSUB_1:th 68
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1 holds
the carrier of b2 is Coset of b2;
:: RUSUB_1:th 69
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR holds
the carrier of b1 is Coset of (Omega). b1;
:: RUSUB_1:th 70
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
st b2 is Coset of (Omega). b1
holds b2 = the carrier of b1;
:: RUSUB_1:th 71
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3 being Coset of b2 holds
0. b1 in b3
iff
b3 = the carrier of b2;
:: RUSUB_1:th 72
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3 being Coset of b2
for b4 being Element of the carrier of b1 holds
b4 in b3
iff
b3 = b4 + b2;
:: RUSUB_1:th 73
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3 being Coset of b2
for b4, b5 being Element of the carrier of b1
st b4 in b3 & b5 in b3
holds ex b6 being Element of the carrier of b1 st
b6 in b2 & b4 + b6 = b5;
:: RUSUB_1:th 74
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3 being Coset of b2
for b4, b5 being Element of the carrier of b1
st b4 in b3 & b5 in b3
holds ex b6 being Element of the carrier of b1 st
b6 in b2 & b4 - b6 = b5;
:: RUSUB_1:th 75
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3, b4 being Element of the carrier of b1 holds
ex b5 being Coset of b2 st
b3 in b5 & b4 in b5
iff
b3 - b4 in b2;
:: RUSUB_1:th 76
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3 being Element of the carrier of b1
for b4, b5 being Coset of b2
st b3 in b4 & b3 in b5
holds b4 = b5;