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 f3 by Def33;
A2: dom(f1 <--> f2) = dom f1 /\ dom f2 & dom <->f1 = dom f1 by Def33,Def46;
  hence
A3: dom(<->f3) = dom(f4<++>f2) by A1,Def45;
  let x be object;
  assume
A4: x in dom(<->f3);
  then
A5: x in dom f4 by A2,A1,XBOOLE_0:def 4;
  thus (<->f3).x = -f3.x by A4,Def33
    .= -(f1.x-f2.x) by A1,A4,Def46
    .= -f1.x--f2.x by Th17
    .= f4.x + f2.x by A5,Def33
    .= (f4<++>f2).x by A3,A4,Def45;
end;
