set S = RLSStruct(# (Funcs A,REAL ),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #);
X1: RLSStruct(# (Funcs A,REAL ),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #) is Abelian

proof

let u, v be Element of RLSStruct(# (Funcs A,REAL ),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #); :: according to RLVECT_1:def 5 :: thesis: u + v = v + u
thus u + v = v + u by Th16; :: thesis: verum

end;

X2: RLSStruct(# (Funcs A,REAL ),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #) is add-associative

proof

let u, v, w be Element of RLSStruct(# (Funcs A,REAL ),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #); :: according to RLVECT_1:def 6 :: thesis: (u + v) + w = u + (v + w)
thus (u + v) + w = u + (v + w) by Th17; :: thesis: verum

end;

X4: RLSStruct(# (Funcs A,REAL ),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #) is right_zeroed

proof

let u be Element of RLSStruct(# (Funcs A,REAL ),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #); :: according to RLVECT_1:def 7 :: thesis: u + (0. RLSStruct(# (Funcs A,REAL ),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #)) = u
reconsider u' = u as Element of Funcs A,REAL ;
thus u + (0. RLSStruct(# (Funcs A,REAL ),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #)) = (RealFuncAdd A) . (RealFuncZero A),u' by Th16
.= u by Th21 ; :: thesis: verum

end;

X5: RLSStruct(# (Funcs A,REAL ),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #) is right_complementable

proof

let u be Element of RLSStruct(# (Funcs A,REAL ),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #); :: according to ALGSTR_0:def 16 :: thesis: u is right_complementable
reconsider u' = u as Element of Funcs A,REAL ;
reconsider w = (RealFuncExtMult A) . [(- 1),u'] as VECTOR of RLSStruct(# (Funcs A,REAL ),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #) ;
take w ; :: according to ALGSTR_0:def 11 :: thesis: u + w = 0. RLSStruct(# (Funcs A,REAL ),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #)
thus u + w = 0. RLSStruct(# (Funcs A,REAL ),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #) by Th22; :: thesis: verum

end;

RLSStruct(# (Funcs A,REAL ),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #) is RealLinearSpace-like

proof

thus for a being real number
for v, w being Element of RLSStruct(# (Funcs A,REAL ),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #) 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 RLSStruct(# (Funcs A,REAL ),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #) holds (b₁ + b₂) * b₃ = (b₁ * b₃) + (b₂ * b₃) ) & ( for b₁, b₂ being set
for b₃ being Element of the carrier of RLSStruct(# (Funcs A,REAL ),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #) holds (b₁ * b₂) * b₃ = b₁ * (b₂ * b₃) ) & ( for b₁ being Element of the carrier of RLSStruct(# (Funcs A,REAL ),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #) holds 1 * b₁ = b₁ ) )

proof

let a be real number ; :: thesis: for v, w being Element of RLSStruct(# (Funcs A,REAL ),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #) holds a * (v + w) = (a * v) + (a * w)
reconsider a = a as Real by XREAL_0:def 1;
for v, w being Element of RLSStruct(# (Funcs A,REAL ),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #) holds a * (v + w) = (a * v) + (a * w) by Lm2;
hence for v, w being Element of RLSStruct(# (Funcs A,REAL ),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #) holds a * (v + w) = (a * v) + (a * w) ; :: thesis: verum

end;

thus for a, b being real number
for v being Element of RLSStruct(# (Funcs A,REAL ),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #) holds (a + b) * v = (a * v) + (b * v) :: thesis: ( ( for b₁, b₂ being set
for b₃ being Element of the carrier of RLSStruct(# (Funcs A,REAL ),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #) holds (b₁ * b₂) * b₃ = b₁ * (b₂ * b₃) ) & ( for b₁ being Element of the carrier of RLSStruct(# (Funcs A,REAL ),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #) holds 1 * b₁ = b₁ ) )

proof

let a, b be real number ; :: thesis: for v being Element of RLSStruct(# (Funcs A,REAL ),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #) holds (a + b) * v = (a * v) + (b * v)
reconsider a = a, b = b as Real by XREAL_0:def 1;
for v being Element of RLSStruct(# (Funcs A,REAL ),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #) holds (a + b) * v = (a * v) + (b * v) by Th25;
hence for v being Element of RLSStruct(# (Funcs A,REAL ),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #) holds (a + b) * v = (a * v) + (b * v) ; :: thesis: verum

end;

thus for a, b being real number
for v being Element of RLSStruct(# (Funcs A,REAL ),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #) holds (a * b) * v = a * (b * v) :: thesis: for b₁ being Element of the carrier of RLSStruct(# (Funcs A,REAL ),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #) holds 1 * b₁ = b₁

proof

let a, b be real number ; :: thesis: for v being Element of RLSStruct(# (Funcs A,REAL ),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #) holds (a * b) * v = a * (b * v)
reconsider a = a, b = b as Real by XREAL_0:def 1;
for v being Element of RLSStruct(# (Funcs A,REAL ),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #) holds (a * b) * v = a * (b * v) by Th24;
hence for v being Element of RLSStruct(# (Funcs A,REAL ),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #) holds (a * b) * v = a * (b * v) ; :: thesis: verum

end;

let v be Element of RLSStruct(# (Funcs A,REAL ),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #); :: thesis: 1 * v = v
thus 1 * v = v by Th23; :: thesis: verum

end;

hence RLSStruct(# (Funcs A,REAL ),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #) is strict RealLinearSpace by X1, X2, X4, X5; :: thesis: verum