let X be non empty set ; :: thesis: for Y being RealLinearSpace holds RLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(FuncExtMult (X,Y)) #) is RealLinearSpace
let Y be RealLinearSpace; :: thesis: RLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(FuncExtMult (X,Y)) #) is RealLinearSpace
A1: for a, b being Real
for v being VECTOR of RLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(FuncExtMult (X,Y)) #) holds (a * b) * v = a * (b * v) by Th8;
A2: for a, b being Real
for v being VECTOR of RLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(FuncExtMult (X,Y)) #) holds (a + b) * v = (a * v) + (b * v) by Th9;
set IT = RLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(FuncExtMult (X,Y)) #);
A3: for u, v, w being VECTOR of RLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(FuncExtMult (X,Y)) #) holds (u + v) + w = u + (v + w) by Th4;
A4: RLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(FuncExtMult (X,Y)) #) is right_complementable

proof

let u be VECTOR of RLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(FuncExtMult (X,Y)) #); :: according to ALGSTR_0:def 16 :: thesis: u is right_complementable
reconsider u9 = u as Element of Funcs (X, the carrier of Y) ;
reconsider w = (FuncExtMult (X,Y)) . [(- jj),u9] as VECTOR of RLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(FuncExtMult (X,Y)) #) ;
take w ; :: according to ALGSTR_0:def 11 :: thesis: u + w = 0. RLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(FuncExtMult (X,Y)) #)
thus u + w = 0. RLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(FuncExtMult (X,Y)) #) by Th6; :: thesis: verum

end;

A5: for a being Real
for u, v being VECTOR of RLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(FuncExtMult (X,Y)) #) holds a * (u + v) = (a * u) + (a * v) by Lm2;
A6: for v being VECTOR of RLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(FuncExtMult (X,Y)) #) holds 1 * v = v by Th7;
A7: for u being VECTOR of RLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(FuncExtMult (X,Y)) #) holds u + (0. RLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(FuncExtMult (X,Y)) #)) = u

proof

let u be VECTOR of RLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(FuncExtMult (X,Y)) #); :: thesis: u + (0. RLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(FuncExtMult (X,Y)) #)) = u
reconsider u9 = u as Element of Funcs (X, the carrier of Y) ;
thus u + (0. RLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(FuncExtMult (X,Y)) #)) = (FuncAdd (X,Y)) . ((FuncZero (X,Y)),u9) by Th3
.= u by Th5 ; :: thesis: verum

end;

for u, v being VECTOR of RLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(FuncExtMult (X,Y)) #) holds u + v = v + u by Th3;
hence RLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(FuncExtMult (X,Y)) #) is RealLinearSpace by A3, A7, A4, A5, A2, A1, A6, RLVECT_1:def 2, RLVECT_1:def 3, RLVECT_1:def 4, RLVECT_1:def 5, RLVECT_1:def 6, RLVECT_1:def 7, RLVECT_1:def 8; :: thesis: verum