reserve X for RealNormSpace;

theorem
  for X, Y be non empty set,
      V be RealNormSpace,
      g, f be PartFunc of X,the carrier of V,
      g1, f1 be PartFunc of Y,the carrier of V
        st g = g1 & f = f1 holds g1 - f1 = g - f
proof
  let X, Y be non empty set,
      V be RealNormSpace,
      g, f be PartFunc of X,the carrier of V,
      g1, f1 be PartFunc of Y,the carrier of V;
  assume A1: g = g1 & f = f1;
A2: dom (g - f) = (dom g) /\ (dom f) by VFUNCT_1:def 2
               .= (dom g1) /\ (dom f1) by A1;
A3: g - f is PartFunc of Y,the carrier of V by A2,RELSET_1:5;
  for c be Element of Y st c in dom (g - f) holds
    (g - f)/.c = g1/.c - f1/.c by A1,VFUNCT_1:def 2;
  hence thesis by A3,A2,VFUNCT_1:def 2;
end;
