let V be non empty Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct ; :: thesis:
let V1 be non empty Subset of V; :: thesis:
let d1 be Element of V1; :: thesis:
let A be BinOp of V1; :: thesis:
let M be Function of [:REAL ,V1:],V1; :: thesis:
set D = V1;
assume that
A1: d1 = 0. V and
A2: A = the addF of V || V1 and
A3: M = the Mult of V | [:REAL ,V1:] ; :: thesis:
set W = RLSStruct(# V1,d1,A,M #);

let a be Real; :: thesis:
let x be VECTOR of RLSStruct(# V1,d1,A,M #); :: thesis:
thus a * x = the Mult of V . [a,x] by A3, FUNCT_1:72
.= the Mult of V . a,x ; :: thesis:

end;

let x, y be VECTOR of RLSStruct(# V1,d1,A,M #); :: thesis:
thus x + y = the addF of V . [x,y] by A2, FUNCT_1:72
.= the addF of V . x,y ; :: thesis:

end;

( RLSStruct(# V1,d1,A,M #) is Abelian & RLSStruct(# V1,d1,A,M #) is add-associative & RLSStruct(# V1,d1,A,M #) is right_zeroed & RLSStruct(# V1,d1,A,M #) is RealLinearSpace-like )

set Mv = the Mult of V;
set Av = the addF of V;

let x, y be VECTOR of RLSStruct(# V1,d1,A,M #); :: thesis:
reconsider x1 = x, y1 = y as VECTOR of V by TARSKI:def 3;
thus x + y = x1 + y1 by A5
.= y1 + x1
.= y + x by A5 ; :: thesis:

end;

let x, y, z be VECTOR of RLSStruct(# V1,d1,A,M #); :: thesis:
reconsider x1 = x, y1 = y, z1 = z as VECTOR of V by TARSKI:def 3;
thus (x + y) + z = the addF of V . (x + y),z1 by A5
.= (x1 + y1) + z1 by A5
.= x1 + (y1 + z1) by RLVECT_1:def 6
.= the addF of V . x1,(y + z) by A5
.= x + (y + z) by A5 ; :: thesis:

end;

let x be VECTOR of RLSStruct(# V1,d1,A,M #); :: thesis:
reconsider y = x as VECTOR of V by TARSKI:def 3;
thus x + (0. RLSStruct(# V1,d1,A,M #)) = y + (0. V) by A1, A5
.= x by RLVECT_1:def 7 ; :: thesis:

end;

let a1 be real number ; :: thesis:
let x, y be VECTOR of RLSStruct(# V1,d1,A,M #); :: thesis:
reconsider a = a1 as Real by XREAL_0:def 1;
reconsider x1 = x, y1 = y as VECTOR of V by TARSKI:def 3;
a * (x + y) = the Mult of V . a,(x + y) by A4
.= a * (x1 + y1) by A5
.= (a * x1) + (a * y1) by RLVECT_1:def 9
.= the addF of V . (the Mult of V . a,x1),(a * y) by A4
.= the addF of V . (a * x),(a * y) by A4
.= (a * x) + (a * y) by A5 ;
hence a1 * (x + y) = (a1 * x) + (a1 * y) ; :: thesis:

end;

let a1, b1 be real number ; :: thesis:
let x be VECTOR of RLSStruct(# V1,d1,A,M #); :: thesis:
reconsider a = a1, b = b1 as Real by XREAL_0:def 1;
reconsider y = x as VECTOR of V by TARSKI:def 3;
(a + b) * x = (a + b) * y by A4
.= (a * y) + (b * y) by RLVECT_1:def 9
.= the addF of V . (the Mult of V . a,y),(b * x) by A4
.= the addF of V . (a * x),(b * x) by A4
.= (a * x) + (b * x) by A5 ;
hence (a1 + b1) * x = (a1 * x) + (b1 * x) ; :: thesis:

end;

let a1, b1 be real number ; :: thesis:
let x be VECTOR of RLSStruct(# V1,d1,A,M #); :: thesis:
reconsider a = a1, b = b1 as Real by XREAL_0:def 1;
reconsider y = x as VECTOR of V by TARSKI:def 3;
(a * b) * x = (a * b) * y by A4
.= a * (b * y) by RLVECT_1:def 9
.= the Mult of V . a,(b * x) by A4
.= a * (b * x) by A4 ;
hence (a1 * b1) * x = a1 * (b1 * x) ; :: thesis:

end;

let x be VECTOR of RLSStruct(# V1,d1,A,M #); :: thesis:
reconsider y = x as VECTOR of V by TARSKI:def 3;
thus 1 * x = 1 * y by A4
.= x by RLVECT_1:def 9 ; :: thesis:

end;

hence ( RLSStruct(# V1,d1,A,M #) is Abelian & RLSStruct(# V1,d1,A,M #) is add-associative & RLSStruct(# V1,d1,A,M #) is right_zeroed & RLSStruct(# V1,d1,A,M #) is RealLinearSpace-like ) ; :: thesis: