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 v being Element of (Maps (X,L)) holds (1. R) * v = v

let v be Element of (Maps (X,L)); :: thesis: (1. R) * v = v
reconsider f = v as Function of X,L by A, FUNCT_2:66;
thus (1. R) * v = the lmult of (Maps (X,L)) . ((1. R),v) by VECTSP_1:def 12
.= (1. R) '*' f by A, dM
.= v ; :: thesis: verum

end;

hence Maps (X,L) is scalar-unital by VECTSP_1:def 17; :: thesis: verum