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 [/] c1 [/] c2 = f [/] (c1*c2)
proof
  set f1 = f[/]c1;
A1: dom(f1[/]c2) = dom f1 by Def39;
  dom f1 = dom f by Def39;
  hence
A2: dom(f1[/]c2) = dom(f[/](c1*c2)) by A1,Def39;
  let x be object;
  assume
A3: x in dom(f1[/]c2);
  hence (f1[/]c2).x = f1.x (#) c2" by Def39
    .= f.x (#) c1" (#) c2" by A1,A3,Def39
    .= f.x (#) (c1" * c2") by Th16
    .= f.x (#) (c1*c2)" by XCMPLX_1:204
    .= (f[/](c1*c2)).x by A2,A3,Def39;
end;
