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[#](c1*c2)).x by A2,A3,Def39;
end;
