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 <> 0 implies f(/)c - g = (f-c(#)g) (/) c
  proof
    assume
A1: c <> 0;
A2: dom(f(/)c - g) = dom(f(/)c) /\ dom g by VALUED_1:12
    .= dom f /\ dom g by VALUED_2:28;
A3: dom(f-c(#)g) = dom f /\ dom(c(#)g) by VALUED_1:12
    .= dom f /\ dom g by VALUED_1:def 5;
    hence dom(f(/)c - g) = dom((f-c(#)g) (/) c) by A2,VALUED_2:28;
    let x be object;
    assume
A4: x in dom(f(/)c - g);
    hence (f(/)c - g).x = (f(/)c).x - g.x by VALUED_1:13
    .= f.x/c - g.x by VALUED_2:29
    .= f.x/c - c*g.x/c by A1,XCMPLX_1:89
    .= f.x/c - (c(#)g).x/c by VALUED_1:6
    .= (f.x-(c(#)g).x) / c
    .= ((f-c(#)g)).x / c by A2,A3,A4,VALUED_1:13
    .= ((f-c(#)g) (/) c).x by VALUED_2:29;
  end;
