reserve UA for Universal_Algebra;
reserve S for non void non empty ManySortedSign,
  U1 for non-empty MSAlgebra over S;

theorem Th10:
  for x, y be Element of MSAEndMonoid U1 for f, g be Element of
  MSAEnd U1 st x = f & y = g holds x * y = g ** f
proof
  reconsider i = id the Sorts of U1 as Element of MSAEnd U1 by Th7;
  let x, y be Element of MSAEndMonoid U1;
  let f, g be Element of MSAEnd U1;
  set H = multLoopStr(# MSAEnd U1, MSAEndComp U1, i #);
  1.H = i;
  then
A1: MSAEndMonoid U1 = H by Def6;
  assume x = f & y = g;
  hence thesis by A1,Def5;
end;
