set W = CLSStruct(# the carrier of V,the U2 of V,the addF of V,the Mult of V #);
A1: for v, w being VECTOR of holds v + w = w + v

proof

let v, w be VECTOR of ; :: thesis: v + w = w + v
reconsider v' = v, w' = w as VECTOR of ;
v + w = v' + w' ;
hence v + w = w' + v'
.= w + v ;
:: thesis: verum

end;

A2: for u, v, w being VECTOR of holds (u + v) + w = u + (v + w)

proof

let u, v, w be VECTOR of ; :: thesis: (u + v) + w = u + (v + w)
reconsider u' = u, v' = v, w' = w as VECTOR of ;
(u' + v') + w' = u' + (v' + w') by RLVECT_1:def 6;
hence (u + v) + w = u + (v + w) ; :: thesis: verum

end;

A3: for v being VECTOR of holds v is right_complementable

proof

let v be VECTOR of ; :: thesis: v is right_complementable
reconsider v' = v as VECTOR of ;
v' is right_complementable by ALGSTR_0:def 16;
then consider w' being VECTOR of such that
A4: v' + w' = 0. V by ALGSTR_0:def 11;
reconsider w = w' as VECTOR of ;
take w ; :: according to ALGSTR_0:def 11 :: thesis: v + w = 0. CLSStruct(# the carrier of V,the U2 of V,the addF of V,the Mult of V #)
thus v + w = 0. CLSStruct(# the carrier of V,the U2 of V,the addF of V,the Mult of V #) by A4; :: thesis: verum

end;

A5: for v being VECTOR of holds v + (0. CLSStruct(# the carrier of V,the U2 of V,the addF of V,the Mult of V #)) = v

proof

let v be VECTOR of ; :: thesis: v + (0. CLSStruct(# the carrier of V,the U2 of V,the addF of V,the Mult of V #)) = v
reconsider v' = v as VECTOR of ;
thus v + (0. CLSStruct(# the carrier of V,the U2 of V,the addF of V,the Mult of V #)) = v' + (0. V)
.= v by RLVECT_1:10 ; :: thesis: verum

end;

A6: for a, b being Complex
for v being VECTOR of holds (a * b) * v = a * (b * v)

proof

let a, b be Complex; :: thesis: for v being VECTOR of holds (a * b) * v = a * (b * v)
let v be VECTOR of ; :: thesis: (a * b) * v = a * (b * v)
reconsider v' = v as VECTOR of ;
(a * b) * v' = a * (b * v') by CLVECT_1:def 2;
hence (a * b) * v = a * (b * v) ; :: thesis: verum

end;

A7: for a, b being Complex
for v being VECTOR of holds (a + b) * v = (a * v) + (b * v)

proof

let a, b be Complex; :: thesis: for v being VECTOR of holds (a + b) * v = (a * v) + (b * v)
let v be VECTOR of ; :: thesis: (a + b) * v = (a * v) + (b * v)
reconsider v' = v as VECTOR of ;
(a + b) * v' = (a * v') + (b * v') by CLVECT_1:def 2;
hence (a + b) * v = (a * v) + (b * v) ; :: thesis: verum

end;

A8: for a being Complex
for v, w being VECTOR of holds a * (v + w) = (a * v) + (a * w)

proof

let a be Complex; :: thesis: for v, w being VECTOR of holds a * (v + w) = (a * v) + (a * w)
let v, w be VECTOR of ; :: thesis: a * (v + w) = (a * v) + (a * w)
reconsider v' = v, w' = w as VECTOR of ;
a * (v' + w') = (a * v') + (a * w') by CLVECT_1:def 2;
hence a * (v + w) = (a * v) + (a * w) ; :: thesis: verum

end;

for v being VECTOR of holds 1r * v = v

proof

let v be VECTOR of ; :: thesis: 1r * v = v
reconsider v' = v as VECTOR of ;
thus 1r * v = 1r * v'
.= v by CLVECT_1:def 2 ; :: thesis: verum

end;

then reconsider W = CLSStruct(# the carrier of V,the U2 of V,the addF of V,the Mult of V #) as ComplexLinearSpace by A1, A2, A5, A3, A8, A7, A6, ALGSTR_0:def 16, CLVECT_1:def 2, RLVECT_1:def 5, RLVECT_1:def 6, RLVECT_1:def 7;
A9: the Mult of W = the Mult of V | [:COMPLEX ,the carrier of W:] by RELSET_1:34;
( 0. W = 0. V & the addF of W = the addF of V || the carrier of W ) by RELSET_1:34;
hence CLSStruct(# the carrier of V,the U2 of V,the addF of V,the Mult of V #) is strict Subspace of V by A9, CLVECT_1:def 5; :: thesis: verum