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 Th33:
  for h be Function st (dom h = UAAut UA & for x be object st x in
  UAAut UA holds h.x = 0 .--> x) holds h is Homomorphism of UAAutGroup UA,
  MSAAutGroup (MSAlg UA)
proof
  let h be Function such that
A1: dom h = UAAut UA and
A2: for x be object st x in UAAut UA holds h.x = 0 .--> x;
  set H = MSAAutGroup (MSAlg UA);
  set G = UAAutGroup UA;
  rng h c= the carrier of H by A1,A2,Lm4;
  then reconsider h9 = h as Function of G,H by A1,FUNCT_2:def 1,RELSET_1:4;
  now
    let a, b be Element of UAAutGroup UA;
    thus h9.(a * b) = (h9.a) * (h9.b)
    proof
      reconsider a9 = a, b9 = b as Element of UAAut UA;
A3:   h9.(b9 * a9) = 0 .--> (b9 * a9) by A2,Th6;
      reconsider A = 0 .--> a9, B = 0 .--> b9 as ManySortedFunction of MSAlg
      UA, MSAlg UA by Th32;
      reconsider ha = h9.a, hb = h9.b as Element of MSAAut MSAlg UA;
      reconsider A9 = A, B9 = B as Element of MSAAutGroup MSAlg UA by Th27;
A4:   ha = A9 & hb = B9 by A2;
      thus h9.(a * b) = h9.(b9 * a9) by Def2
        .= MSAlg (b9 * a9) by A3,MSUHOM_1:def 3
        .= (MSAlg b9) ** (MSAlg a9) by MSUHOM_1:26
        .= B ** MSAlg a9 by MSUHOM_1:def 3
        .= B ** A by MSUHOM_1:def 3
        .= h9.a * h9.b by A4,Def6;
    end;
  end;
  hence thesis by GROUP_6:def 6;
end;
