take X ; :: thesis: ( X is RealNormSpace-like & X is RealLinearSpace-like & X is Abelian & X is add-associative & X is right_zeroed & X is right_complementable & X is strict )
thus X is RealNormSpace-like by Def2, Lm5; :: thesis: ( X is RealLinearSpace-like & X is Abelian & X is add-associative & X is right_zeroed & X is right_complementable & X is strict )
A1: for x, y being VECTOR of X
for x', y' being VECTOR of ((0). V) st x = x' & y = y' holds
( x + y = x' + y' & ( for a being Real holds a * x = a * x' ) ) ;
thus X is RealLinearSpace-like :: thesis: ( X is Abelian & X is add-associative & X is right_zeroed & X is right_complementable & X is strict )

proof

thus for a being real number
for v, w being VECTOR of X holds a * (v + w) = (a * v) + (a * w) :: according to RLVECT_1:def 9 :: thesis: ( ( for b₁, b₂ being set
for b₃ being Element of the carrier of X holds (b₁ + b₂) * b₃ = (b₁ * b₃) + (b₂ * b₃) ) & ( for b₁, b₂ being set
for b₃ being Element of the carrier of X holds (b₁ * b₂) * b₃ = b₁ * (b₂ * b₃) ) & ( for b₁ being Element of the carrier of X holds 1 * b₁ = b₁ ) )

proof

let a be real number ; :: thesis: for v, w being VECTOR of X holds a * (v + w) = (a * v) + (a * w)
let v, w be VECTOR of X; :: thesis: a * (v + w) = (a * v) + (a * w)
reconsider v' = v, w' = w as VECTOR of ((0). V) ;
thus a * (v + w) = a * (v' + w')
.= (a * v') + (a * w') by RLVECT_1:def 9
.= (a * v) + (a * w) ; :: thesis: verum

end;

thus for a, b being real number
for v being VECTOR of X holds (a + b) * v = (a * v) + (b * v) :: thesis: ( ( for b₁, b₂ being set
for b₃ being Element of the carrier of X holds (b₁ * b₂) * b₃ = b₁ * (b₂ * b₃) ) & ( for b₁ being Element of the carrier of X holds 1 * b₁ = b₁ ) )

proof

let a, b be real number ; :: thesis: for v being VECTOR of X holds (a + b) * v = (a * v) + (b * v)
let v be VECTOR of X; :: thesis: (a + b) * v = (a * v) + (b * v)
reconsider v' = v as VECTOR of ((0). V) ;
thus (a + b) * v = (a + b) * v'
.= (a * v') + (b * v') by RLVECT_1:def 9
.= (a * v) + (b * v) ; :: thesis: verum

end;

thus for a, b being real number
for v being VECTOR of X holds (a * b) * v = a * (b * v) :: thesis: for b₁ being Element of the carrier of X holds 1 * b₁ = b₁

proof

let a, b be real number ; :: thesis: for v being VECTOR of X holds (a * b) * v = a * (b * v)
let v be VECTOR of X; :: thesis: (a * b) * v = a * (b * v)
reconsider v' = v as VECTOR of ((0). V) ;
thus (a * b) * v = (a * b) * v'
.= a * (b * v') by RLVECT_1:def 9
.= a * (b * v) ; :: thesis: verum

end;

let v be VECTOR of X; :: thesis: 1 * v = v
reconsider v' = v as VECTOR of ((0). V) ;
thus 1 * v = 1 * v'
.= v by RLVECT_1:def 9 ; :: thesis: verum

end;

thus for v, w being VECTOR of X holds v + w = w + v :: according to RLVECT_1:def 5 :: 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 v' = v, w' = w as VECTOR of ((0). V) ;
thus v + w = w' + v' 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 6 :: 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 u' = u, v' = v, w' = w as VECTOR of ((0). 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 X holds v + (0. X) = v :: according to RLVECT_1:def 7 :: thesis: ( X is right_complementable & X is strict )

proof

let v be VECTOR of X; :: thesis: v + (0. X) = v
reconsider v' = v as VECTOR of ((0). V) ;
thus v + (0. X) = v' + (0. ((0). V))
.= v by RLVECT_1:10 ; :: 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 v' = v as VECTOR of ((0). V) ;
consider w' being VECTOR of ((0). V) such that
A2: v' + w' = 0. ((0). V) by ALGSTR_0:def 11;
reconsider w = w' 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