set V = Maps (X,L);
A: ( the carrier of (Maps (X,L)) = Funcs (X, the carrier of L) & the addF of (Maps (X,L)) = mapAdd (X,L) & the ZeroF of (Maps (X,L)) = X --> (0. L) & the lmult of (Maps (X,L)) = mapMult (X,L) ) by defmap;

now :: thesis: for x, y being Element of (Maps (X,L)) holds x + y = y + x

let x, y be Element of (Maps (X,L)); :: thesis: x + y = y + x
reconsider f = x, g = y as Function of X, the carrier of L by A, FUNCT_2:66;
thus x + y = f '+' g by A, dA
.= y + x by A, dA ; :: thesis: verum

end;

hence Maps (X,L) is Abelian by RLVECT_1:def 2; :: thesis: verum