take X ; :: thesis: ( X is reflexive & X is discerning & X is ComplexNormSpace-like & X is vector-distributive & X is scalar-distributive & X is scalar-associative & X is scalar-unital & X is Abelian & X is add-associative & X is right_zeroed & X is right_complementable & X is strict )
thus X is reflexive by Lm11; :: thesis: ( X is discerning & X is ComplexNormSpace-like & X is vector-distributive & X is scalar-distributive & X is scalar-associative & X is scalar-unital & X is Abelian & X is add-associative & X is right_zeroed & X is right_complementable & X is strict )
thus ( X is discerning & X is ComplexNormSpace-like ) by Lm11; :: thesis: ( X is vector-distributive & X is scalar-distributive & X is scalar-associative & X is scalar-unital & X is Abelian & X is add-associative & X is right_zeroed & X is right_complementable & X is strict )
thus ( X is vector-distributive & X is scalar-distributive & X is scalar-associative & X is scalar-unital ) :: thesis: ( X is Abelian & X is add-associative & X is right_zeroed & X is right_complementable & X is strict )

proof

thus for z being Complex
for v, w being VECTOR of X holds z * (v + w) = (z * v) + (z * w) :: according to CLVECT_1:def 2 :: thesis: ( X is scalar-distributive & X is scalar-associative & X is scalar-unital )

proof

let z be Complex; :: thesis: for v, w being VECTOR of X holds z * (v + w) = (z * v) + (z * w)
let v, w be VECTOR of X; :: thesis: z * (v + w) = (z * v) + (z * w)
reconsider v9 = v, w9 = w as VECTOR of ((0). the ComplexLinearSpace) ;
thus z * (v + w) = z * (v9 + w9)
.= (z * v9) + (z * w9) by Def2
.= (z * v) + (z * w) ; :: thesis: verum

end;

thus for z1, z2 being Complex
for v being VECTOR of X holds (z1 + z2) * v = (z1 * v) + (z2 * v) :: according to CLVECT_1:def 3 :: thesis: ( X is scalar-associative & X is scalar-unital )

proof

let z1, z2 be Complex; :: thesis: for v being VECTOR of X holds (z1 + z2) * v = (z1 * v) + (z2 * v)
let v be VECTOR of X; :: thesis: (z1 + z2) * v = (z1 * v) + (z2 * v)
reconsider v9 = v as VECTOR of ((0). the ComplexLinearSpace) ;
thus (z1 + z2) * v = (z1 + z2) * v9
.= (z1 * v9) + (z2 * v9) by Def3
.= (z1 * v) + (z2 * v) ; :: thesis: verum

end;

thus for z1, z2 being Complex
for v being VECTOR of X holds (z1 * z2) * v = z1 * (z2 * v) :: according to CLVECT_1:def 4 :: thesis: X is scalar-unital

proof

let z1, z2 be Complex; :: thesis: for v being VECTOR of X holds (z1 * z2) * v = z1 * (z2 * v)
let v be VECTOR of X; :: thesis: (z1 * z2) * v = z1 * (z2 * v)
reconsider v9 = v as VECTOR of ((0). the ComplexLinearSpace) ;
thus (z1 * z2) * v = (z1 * z2) * v9
.= z1 * (z2 * v9) by Def4
.= z1 * (z2 * v) ; :: thesis: verum

end;

let v be VECTOR of X; :: according to CLVECT_1:def 5 :: thesis: 1r * v = v
reconsider v9 = v as VECTOR of ((0). the ComplexLinearSpace) ;
thus 1r * v = 1r * v9
.= v by Def5 ; :: thesis: verum

end;

A1: for x, y being VECTOR of X
for x9, y9 being VECTOR of ((0). the ComplexLinearSpace) st x = x9 & y = y9 holds
( x + y = x9 + y9 & ( for z being Complex holds z * x = z * x9 ) ) ;
thus for v, w being VECTOR of X holds v + w = w + v :: according to RLVECT_1:def 2 :: thesis: ( X is add-associative & X is right_zeroed & X is right_complementable & X is strict )

proof

let v, w be VECTOR of X; :: thesis: v + w = w + v
reconsider v9 = v, w9 = w as VECTOR of ((0). the ComplexLinearSpace) ;
thus v + w = w9 + v9 by A1
.= w + v ; :: thesis: verum

end;

thus for u, v, w being VECTOR of X holds (u + v) + w = u + (v + w) :: according to RLVECT_1:def 3 :: thesis: ( X is right_zeroed & X is right_complementable & X is strict )

proof

let u, v, w be VECTOR of X; :: thesis: (u + v) + w = u + (v + w)
reconsider u9 = u, v9 = v, w9 = w as VECTOR of ((0). the ComplexLinearSpace) ;
thus (u + v) + w = (u9 + v9) + w9
.= u9 + (v9 + w9) by RLVECT_1:def 3
.= u + (v + w) ; :: thesis: verum

end;

thus for v being VECTOR of X holds v + (0. X) = v :: according to RLVECT_1:def 4 :: thesis: ( X is right_complementable & X is strict )

proof

let v be VECTOR of X; :: thesis: v + (0. X) = v
reconsider v9 = v as VECTOR of ((0). the ComplexLinearSpace) ;
thus v + (0. X) = v9 + (0. ((0). the ComplexLinearSpace))
.= v by RLVECT_1:4 ; :: thesis: verum

end;

thus X is right_complementable :: thesis: X is strict

proof

let v be VECTOR of X; :: according to ALGSTR_0:def 16 :: thesis: v is right_complementable
reconsider v9 = v as VECTOR of ((0). the ComplexLinearSpace) ;
consider w9 being VECTOR of ((0). the ComplexLinearSpace) such that
A2: v9 + w9 = 0. ((0). the ComplexLinearSpace) by ALGSTR_0:def 11;
reconsider w = w9 as VECTOR of X ;
take w ; :: according to ALGSTR_0:def 11 :: thesis: v + w = 0. X
thus v + w = 0. X by A2; :: thesis: verum

end;

thus X is strict ; :: thesis: verum