reserve
  x for object, X for set,
  i, n, m for Nat,
  r, s for Real,
  c, c1, c2, d for Complex,
  f, g for complex-valued Function,
  g1 for n-element complex-valued FinSequence,
  f1 for n-element real-valued FinSequence,
  T for non empty TopSpace,
  p for Element of TOP-REAL n;

theorem
  (c-d)(#)f = c(#)f - d(#)f
  proof
    dom(c(#)f - d(#)f) = dom(c(#)f) /\ dom(d(#)f) by VALUED_1:12
    .= dom f /\ dom(d(#)f) by VALUED_1:def 5
    .= dom f /\ dom f by VALUED_1:def 5;
    hence
A1: dom((c-d)(#)f) = dom(c(#)f - d(#)f) by VALUED_1:def 5;
    let x be object;
    assume
A2: x in dom((c-d)(#)f);
    thus ((c-d)(#)f).x = (c-d)*f.x by VALUED_1:6
    .= c*f.x - d*f.x
    .= c*f.x - (d(#)f).x by VALUED_1:6
    .= (c(#)f).x - (d(#)f).x by VALUED_1:6
    .= (c(#)f - d(#)f).x by A1,A2,VALUED_1:13;
  end;
