take CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) ; :: thesis: ( CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) is ComplexUnitarySpace-like & CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) is ComplexLinearSpace-like & CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) is Abelian & CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) is add-associative & CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) is right_zeroed & CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) is right_complementable & CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) is strict )
thus CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) is ComplexUnitarySpace-like by Def13, Lm8; :: thesis: ( CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) is ComplexLinearSpace-like & CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) is Abelian & CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) is add-associative & CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) is right_zeroed & CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) is right_complementable & CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) is strict )
thus CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) is ComplexLinearSpace-like :: thesis: ( CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) is Abelian & CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) is add-associative & CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) is right_zeroed & CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) is right_complementable & CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) is strict )

proof

thus for a being Complex
for v, w being VECTOR of holds a * (v + w) = (a * v) + (a * w) :: according to CLVECT_1:def 2 :: thesis: ( ( for b₁, b₂ being Element of COMPLEX
for b₃ being Element of the carrier of CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) holds (b₁ + b₂) * b₃ = (b₁ * b₃) + (b₂ * b₃) ) & ( for b₁, b₂ being Element of COMPLEX
for b₃ being Element of the carrier of CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) holds (b₁ * b₂) * b₃ = b₁ * (b₂ * b₃) ) & ( for b₁ being Element of the carrier of CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) holds 1r * b₁ = b₁ ) )

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 ;
thus a * (v + w) = a * (v' + w')
.= (a * v') + (a * w') by CLVECT_1:def 2
.= (a * v) + (a * w) ; :: thesis: verum

end;

thus for a, b being Complex
for v being VECTOR of holds (a + b) * v = (a * v) + (b * v) :: thesis: ( ( for b₁, b₂ being Element of COMPLEX
for b₃ being Element of the carrier of CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) holds (b₁ * b₂) * b₃ = b₁ * (b₂ * b₃) ) & ( for b₁ being Element of the carrier of CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) holds 1r * b₁ = b₁ ) )

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 ;
thus (a + b) * v = (a + b) * v'
.= (a * v') + (b * v') by CLVECT_1:def 2
.= (a * v) + (b * v) ; :: thesis: verum

end;

thus for a, b being Complex
for v being VECTOR of holds (a * b) * v = a * (b * v) :: thesis: for b₁ being Element of the carrier of CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) holds 1r * b₁ = b₁

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 ;
thus (a * b) * v = (a * b) * v'
.= a * (b * v') by CLVECT_1:def 2
.= a * (b * v) ; :: thesis: verum

end;

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;

A1: for x, y being VECTOR of
for x', y' being VECTOR of st x = x' & y = y' holds
( x + y = x' + y' & ( for a being Complex holds a * x = a * x' ) ) ;
thus for v, w being VECTOR of holds v + w = w + v :: according to RLVECT_1:def 5 :: thesis: ( CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) is add-associative & CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) is right_zeroed & CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) is right_complementable & CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) is strict )

proof

let v, w be VECTOR of ; :: thesis: v + w = w + v
reconsider v' = v, w' = w as VECTOR of ;
thus v + w = w' + v' by A1
.= w + v ; :: thesis: verum

end;

thus for u, v, w being VECTOR of holds (u + v) + w = u + (v + w) :: according to RLVECT_1:def 6 :: thesis: ( CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) is right_zeroed & CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) is right_complementable & CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) is strict )

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 ;
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 holds v + (0. CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #)) = v :: according to RLVECT_1:def 7 :: thesis: ( CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) is right_complementable & CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) is strict )

proof

let v be VECTOR of ; :: thesis: v + (0. CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #)) = v
reconsider v' = v as VECTOR of ;
thus v + (0. CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #)) = v' + (0. ((0). V0))
.= v by RLVECT_1:10 ; :: thesis: verum

end;

thus CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) is right_complementable :: thesis: CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) is strict

proof

let v be VECTOR of ; :: according to ALGSTR_0:def 16 :: thesis: v is right_complementable
reconsider v' = v as VECTOR of ;
consider w' being VECTOR of such that
A2: v' + w' = 0. ((0). V0) by ALGSTR_0:def 11;
reconsider w = w' as VECTOR of ;
take w ; :: according to ALGSTR_0:def 11 :: thesis: v + w = 0. CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #)
thus v + w = 0. CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) by A2; :: thesis: verum

end;

thus CUNITSTR(# the carrier of ((0). V0),(0. ((0). V0)),the addF of ((0). V0),the Mult of ((0). V0),nilfunc #) is strict ; :: thesis: verum