reserve x, y for object, X, X1, X2 for set;
reserve Y, Y1, Y2 for complex-functions-membered set,
  c, c1, c2 for Complex,
  f for PartFunc of X,Y,
  f1 for PartFunc of X1,Y1,
  f2 for PartFunc of X2, Y2,
  g, h, k for complex-valued Function;

theorem
  (<->f1) <##> f2 = <-> (f1<##>f2)
proof
  set f3 = f1<##>f2, f4 = <->f1;
A1: dom f3 = dom f1 /\ dom f2 & dom f4 = dom f1 by Def33,Def47;
  dom(f4<##>f2) = dom f4 /\ dom f2 by Def47;
  hence
A2: dom(f4<##>f2) = dom(<->f3) by A1,Def33;
  let x be object;
  assume
A3: x in dom(f4<##>f2);
  then
A4: x in dom f3 by A1,Def47;
  then
A5: x in dom <->f1 by A1,XBOOLE_0:def 4;
  thus (f4<##>f2).x = f4.x (#) f2.x by A3,Def47
    .= (-(f1.x)) (#) f2.x by A5,Def33
    .= -(f1.x) (#) f2.x by Th25
    .= -f3.x by A4,Def47
    .= (<->f3).x by A2,A3,Def33;
end;
