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 Th83:
  f1 <##> f2 = f2 <##> f1
proof
  dom(f1<##>f2) = dom f1 /\ dom f2 by Def47;
  hence
A1: dom(f1<##>f2) = dom(f2<##>f1) by Def47;
  let x be object;
  assume
A2: x in dom(f1<##>f2);
  hence (f1<##>f2).x = f1.x (#) f2.x by Def47
    .= (f2<##>f1).x by A1,A2,Def47;
end;
