set I = the carrier of R;
set scmult = the lmult of V | [: the carrier of R,(a * V):];
[: the carrier of R,(a * V):] c= [: the carrier of R, the carrier of V:] by ZFMISC_1:96;
then [: the carrier of R,(a * V):] c= dom the lmult of V by FUNCT_2:def 1;
then A1: dom ( the lmult of V | [: the carrier of R,(a * V):]) = [: the carrier of R,(a * V):] by RELAT_1:62;
for z being object st z in [: the carrier of R,(a * V):] holds
( the lmult of V | [: the carrier of R,(a * V):]) . z in a * V
proof
let z be object ; :: thesis: ( z in [: the carrier of R,(a * V):] implies ( the lmult of V | [: the carrier of R,(a * V):]) . z in a * V )
assume A2: z in [: the carrier of R,(a * V):] ; :: thesis: ( the lmult of V | [: the carrier of R,(a * V):]) . z in a * V
consider x, y being object such that
A3: ( x in the carrier of R & y in a * V & z = [x,y] ) by ;
reconsider i = x as Element of R by A3;
consider v being Element of V such that
A4: y = a * v by A3;
( the lmult of V | [: the carrier of R,(a * V):]) . z = i * (a * v) by
.= (i * a) * v by VECTSP_1:def 16
.= (a * i) * v
.= a * (i * v) by VECTSP_1:def 16 ;
hence ( the lmult of V | [: the carrier of R,(a * V):]) . z in a * V ; :: thesis: verum
end;
hence the lmult of V | [: the carrier of R,(a * V):] is Function of [: the carrier of R,(a * V):],(a * V) by ; :: thesis: verum