reserve V for non empty RealLinearSpace;

theorem Th21b:
for X be RealLinearSpace, f,h be VECTOR of X*', a be Real
 holds h = a*f iff for x be VECTOR of X holds h.x = a * f.x
proof
  let X be RealLinearSpace;
  let f,h be VECTOR of X*';
  let a be Real;
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 assume
A3: h = a*f;
    let x be Element of X;
    h1=a*f1 by A1,A3,RLSUB_1:14;
    hence h.x = a*f.x by FUNCSDOM:4;
  end;
  assume for x be Element of X holds h.x=a*f.x;
  then h1 = a*f1 by FUNCSDOM:4;
  hence h = a*f by A1,RLSUB_1:14;
end;
