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 Th34:
  for h be Homomorphism of UAAutGroup UA, MSAAutGroup (MSAlg UA)
  st for x be object st x in UAAut UA holds h.x = 0 .--> x
  holds h is bijective
proof
  let h be Homomorphism of UAAutGroup UA, MSAAutGroup (MSAlg UA);
  set G = UAAutGroup UA;
  assume
A1: for x be object st x in UAAut UA holds h.x = 0 .--> x;
  for a, b be Element of G st h.a = h.b holds a = b
  proof
    let a, b be Element of G;
    assume
A2: h.a = h.b;
A3: h.b = 0 .--> b by A1
      .= [:{0}, {b}:];
    h.a = 0 .--> a by A1
      .= [:{0}, {a}:];
    then {a} = {b} by A2,A3,ZFMISC_1:110;
    hence thesis by ZFMISC_1:3;
  end;
  then
A4: h is one-to-one by GROUP_6:1;
  dom h = UAAut UA by FUNCT_2:def 1;
  then rng h = the carrier of MSAAutGroup (MSAlg UA) by A1,Lm4;
  then h is onto;
  hence h is bijective by A4;
end;
