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 Th29:
  id the Sorts of U1 = 1_MSAAutGroup U1
proof
  set T = the Sorts of U1;
  set f = the Element of MSAAutGroup U1;
  reconsider g = id T as Element of MSAAutGroup U1 by Th24;
  consider g1 be ManySortedFunction of T, T such that
A1: g1 = g;
  f is Element of MSAAut U1;
  then consider f1 be ManySortedFunction of T, T such that
A2: f1 = f;
  g * f = f1 ** g1 by A1,A2,Def6
    .= f by A1,A2,MSUALG_3:3;
  hence thesis by GROUP_1:7;
end;
