reserve UA for Universal_Algebra,
  f, g for Function of UA, UA;
reserve I for set,
  A, B, C for ManySortedSet of I;
reserve S for non void non empty ManySortedSign,
  U1, U2 for non-empty MSAlgebra over S;

theorem
  for f be Element of MSAAut U1 for g be Element of MSAAutGroup U1 st
  f = g holds f"" = g"
proof
  let f be Element of MSAAut U1;
  let g be Element of MSAAutGroup U1;
  consider g1 be Element of MSAAut U1 such that
A1: g1 = g";
  assume f = g;
  then g1 ** f = g * g" by A1,Def6;
  then g1 ** f = 1_MSAAutGroup U1 by GROUP_1:def 5;
  then
A2: g1 ** f = id the Sorts of U1 by Th29;
  f is "1-1" & f is "onto" by Lm3;
  hence thesis by A1,A2,Th17;
end;
