set W = CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #);
( CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #) is Abelian & CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #) is add-associative & CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #) is right_zeroed & CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #) is right_complementable & CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #) is ComplexLinearSpace-like )

proof

A1: for z being Complex
for v, w being VECTOR of CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #)
for v', w' being VECTOR of V st v = v' & w = w' holds
( v + w = v' + w' & z * v = z * v' ) ;
thus for v, w being VECTOR of CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #) holds v + w = w + v :: according to RLVECT_1:def 5 :: thesis: ( CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #) is add-associative & CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #) is right_zeroed & CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #) is right_complementable & CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #) is ComplexLinearSpace-like )

proof

let v, w be VECTOR of CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #); :: thesis: v + w = w + v
reconsider v' = v, w' = w as VECTOR of V ;
thus v + w = w' + v' by A1
.= w + v ; :: thesis: verum

end;

thus for u, v, w being VECTOR of CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #) holds (u + v) + w = u + (v + w) :: according to RLVECT_1:def 6 :: thesis: ( CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #) is right_zeroed & CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #) is right_complementable & CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #) is ComplexLinearSpace-like )

proof

let u, v, w be VECTOR of CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #); :: thesis: (u + v) + w = u + (v + w)
reconsider u' = u, v' = v, w' = w as VECTOR of V ;
thus (u + v) + w = (u' + v') + w'
.= u' + (v' + w') by RLVECT_1:def 6
.= u + (v + w) ; :: thesis: verum

end;

thus for v being VECTOR of CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #) holds v + (0. CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #)) = v :: according to RLVECT_1:def 7 :: thesis: ( CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #) is right_complementable & CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #) is ComplexLinearSpace-like )

proof

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

end;

thus CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #) is right_complementable :: thesis: CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #) is ComplexLinearSpace-like

proof

let v be VECTOR of CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #); :: according to ALGSTR_0:def 16 :: thesis: v is right_complementable
reconsider v' = v as VECTOR of V ;
consider w' being VECTOR of V such that
A2: v' + w' = 0. V by ALGSTR_0:def 11;
reconsider w = w' as VECTOR of CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #) ;
take w ; :: according to ALGSTR_0:def 11 :: thesis: v + w = 0. CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #)
thus v + w = 0. CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #) by A2; :: thesis: verum

end;

thus for z being Complex
for v, w being VECTOR of CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #) holds z * (v + w) = (z * v) + (z * w) :: according to CLVECT_1:def 2 :: thesis: ( ( for z1, z2 being Complex
for v being VECTOR of CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #) holds (z1 + z2) * v = (z1 * v) + (z2 * v) ) & ( for z1, z2 being Complex
for v being VECTOR of CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #) holds (z1 * z2) * v = z1 * (z2 * v) ) & ( for v being VECTOR of CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #) holds 1r * v = v ) )

proof

let z be Complex; :: thesis: for v, w being VECTOR of CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #) holds z * (v + w) = (z * v) + (z * w)
let v, w be VECTOR of CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #); :: thesis: z * (v + w) = (z * v) + (z * w)
reconsider v' = v, w' = w as VECTOR of V ;
thus z * (v + w) = z * (v' + w')
.= (z * v') + (z * w') by Def2
.= (z * v) + (z * w) ; :: thesis: verum

end;

thus for z1, z2 being Complex
for v being VECTOR of CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #) holds (z1 + z2) * v = (z1 * v) + (z2 * v) :: thesis: ( ( for z1, z2 being Complex
for v being VECTOR of CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #) holds (z1 * z2) * v = z1 * (z2 * v) ) & ( for v being VECTOR of CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #) holds 1r * v = v ) )

proof

let z1, z2 be Complex; :: thesis: for v being VECTOR of CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #) holds (z1 + z2) * v = (z1 * v) + (z2 * v)
let v be VECTOR of CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #); :: thesis: (z1 + z2) * v = (z1 * v) + (z2 * v)
reconsider v' = v as VECTOR of V ;
thus (z1 + z2) * v = (z1 + z2) * v'
.= (z1 * v') + (z2 * v') by Def2
.= (z1 * v) + (z2 * v) ; :: thesis: verum

end;

thus for z1, z2 being Complex
for v being VECTOR of CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #) holds (z1 * z2) * v = z1 * (z2 * v) :: thesis: for v being VECTOR of CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #) holds 1r * v = v

proof

let z1, z2 be Complex; :: thesis: for v being VECTOR of CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #) holds (z1 * z2) * v = z1 * (z2 * v)
let v be VECTOR of CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #); :: thesis: (z1 * z2) * v = z1 * (z2 * v)
reconsider v' = v as VECTOR of V ;
thus (z1 * z2) * v = (z1 * z2) * v'
.= z1 * (z2 * v') by Def2
.= z1 * (z2 * v) ; :: thesis: verum

end;

thus for v being VECTOR of CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #) holds 1r * v = v :: thesis: verum

proof

let v be VECTOR of CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #); :: thesis: 1r * v = v
reconsider v' = v as VECTOR of V ;
thus 1r * v = 1r * v'
.= v by Def2 ; :: thesis: verum

end;

end;

then reconsider W = CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #) as ComplexLinearSpace ;
W is Subspace of V

proof

thus ( the carrier of W c= the carrier of V & 0. W = 0. V ) ; :: according to CLVECT_1:def 5 :: thesis: ( the addF of W = the addF of V || the carrier of W & the Mult of W = the Mult of V | [:COMPLEX ,the carrier of W:] )
thus ( the addF of W = the addF of V || the carrier of W & the Mult of W = the Mult of V | [:COMPLEX ,the carrier of W:] ) by RELSET_1:34; :: thesis: verum

end;

hence CLSStruct(# the carrier of V,the ZeroF of V,the addF of V,the Mult of V #) is strict Subspace of V ; :: thesis: verum