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
  <->(f<->g) = (<->f) <+> g
proof
  set f1 = f<->g, f2 = <->f;
A1: dom <->f1 = dom f1 by Def33;
A2: dom f1 = dom f /\ dom g & dom f2 = dom f by Def33,Th61;
  hence
A3: dom <->f1 = dom(f2<+>g) by A1,Def41;
  let x be object;
  assume
A4: x in dom <->f1;
  then
A5: x in dom f2 by A1,A2,XBOOLE_0:def 4;
  thus (<->f1).x = -f1.x by A4,Def33
    .= -(f.x-g.x) by A1,A4,Th62
    .= -f.x+g.x by Th11
    .= f2.x+g.x by A5,Def33
    .= (f2<+>g).x by A3,A4,Def41;
end;
