let X be RealLinearSpace; :: thesis: for f, h being VECTOR of (X *')
for a being Real holds
( h = a * f iff for x being VECTOR of X holds h . x = a * (f . x) )
let f, h be VECTOR of (X *'); :: thesis: for a being Real holds
( h = a * f iff for x being VECTOR of X holds h . x = a * (f . x) )
let a be Real; :: thesis: ( h = a * f iff for x being VECTOR of X holds h . x = a * (f . x) )
A1: X *' is Subspace of RealVectSpace the carrier of X by Th17, RSSPACE:11;
then reconsider f1 = f, h1 = h as VECTOR of (RealVectSpace the carrier of X) by RLSUB_1:10;

hereby :: thesis: ( ( for x being VECTOR of X holds h . x = a * (f . x) ) implies h = a * f )

assume A3: h = a * f ; :: thesis: for x being Element of X holds h . x = a * (f . x)
let x be Element of X; :: thesis: h . x = a * (f . x)
h1 = a * f1 by A1, A3, RLSUB_1:14;
hence h . x = a * (f . x) by FUNCSDOM:4; :: thesis: verum

end;

assume for x being Element of X holds h . x = a * (f . x) ; :: thesis: h = a * f
then h1 = a * f1 by FUNCSDOM:4;
hence h = a * f by A1, RLSUB_1:14; :: thesis: verum