reserve UA for Universal_Algebra;

theorem Th4:
  for x, y be Element of UAEndMonoid UA for f, g be Element of
  UAEnd UA st x = f & y = g holds x * y = g * f
proof
  reconsider i = id the carrier of UA as Element of UAEnd UA by Th2;
  let x, y be Element of UAEndMonoid UA;
  let f, g be Element of UAEnd UA;
  set H = multLoopStr (# UAEnd UA, UAEndComp UA,i #);
  1.H = i;
  then
A1: UAEndMonoid UA = H by Def3;
  assume x = f & y = g;
  hence thesis by A1,Def2;
end;
