reserve r,s,t,u for Real;

theorem Th47:
  for X being LinearTopSpace, r being non zero Real holds rng mlt( r,X) = [#]X
proof
  let X be LinearTopSpace, r be non zero Real;
  thus rng mlt(r,X) c= [#]X;
  let y be object;
  assume y in [#]X;
  then reconsider y as Point of X;
  mlt(r,X).(r"*y) = r*(r"*y) by Def13
    .= r*r"*y by RLVECT_1:def 7
    .= 1*y by XCMPLX_0:def 7
    .= y by RLVECT_1:def 8;
  hence thesis by FUNCT_2:4;
end;
