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
  c1 <> 0 & c2 <> 0 implies f(/)c1 - g(/)c2 = (f(#)c2-g(#)c1) (/) (c1*c2)
  proof
    assume
A1: c1 <> 0 & c2 <> 0;
A2: dom (f(/)c1) = dom f by VALUED_2:28;
A3: dom (g(/)c2) = dom g by VALUED_2:28;
A4: dom (f(#)c2) = dom f by VALUED_1:def 5;
A5: dom (g(#)c1) = dom g by VALUED_1:def 5;
A6: dom (f(/)c1-g(/)c2) = dom (f(/)c1) /\ dom (g(/)c2) by VALUED_1:12;
A7: dom (f(#)c2-g(#)c1) = dom (f(#)c2) /\ dom (g(#)c1) by VALUED_1:12;
    hence dom (f(/)c1 - g(/)c2) = dom ((f(#)c2-g(#)c1) (/) (c1*c2))
    by A2,A3,A4,A5,A6,VALUED_2:28;
    let x be object;
    assume
A8: x in dom (f(/)c1 - g(/)c2);
    hence ((f(/)c1 - g(/)c2)).x = (f(/)c1).x - (g(/)c2).x by VALUED_1:13
    .= (f.x/c1) - (g(/)c2).x by VALUED_2:29
    .= (f.x/c1) - (g.x/c2) by VALUED_2:29
    .= (f.x*c2-g.x*c1) / (c1*c2) by A1,XCMPLX_1:130
    .= (f.x*c2-(g(#)c1).x) / (c1*c2) by VALUED_1:6
    .= ((f(#)c2).x-(g(#)c1).x) / (c1*c2) by VALUED_1:6
    .= ((f(#)c2-g(#)c1)).x / (c1*c2) by A2,A3,A4,A5,A6,A7,A8,VALUED_1:13
    .= ((f(#)c2-g(#)c1) (/) (c1*c2)).x by VALUED_2:29;
  end;
