reserve X for RealNormSpace;

theorem Th12:
  for r be Real,
      X, Y be non empty set,
      V be RealNormSpace,
      g be PartFunc of X,the carrier of V,
      g1 be PartFunc of Y,the carrier of V
        st g = g1 holds r(#)g1 = r(#)g
proof
  let r be Real,
      X, Y be non empty set,
      V be RealNormSpace,
      g be PartFunc of X,the carrier of V,
      g1 be PartFunc of Y,the carrier of V;
  assume A1: g = g1;
A2: dom (r(#)g) = dom (g) by VFUNCT_1:def 4
               .= dom (g1) by A1;
A3: r(#)g is PartFunc of Y,the carrier of V by A2,RELSET_1:5;
  for c be Element of Y st c in dom (r(#)g) holds
    (r(#)g)/.c = r * (g1/.c) by A1,VFUNCT_1:def 4;
  hence thesis by A3,A2,VFUNCT_1:def 4;
end;
