reserve UA for Universal_Algebra,
  f, g for Function of UA, UA;

theorem
  for f be Element of UAAut UA for g be Element of UAAutGroup UA st f =
  g holds f" = g"
proof
  let f be Element of UAAut UA;
  let g be Element of UAAutGroup UA;
  consider g1 be Element of UAAut UA such that
A1: g1 = g";
  assume f = g;
  then g1 * f = g * g" by A1,Def2;
  then g1 * f = 1_UAAutGroup UA by GROUP_1:def 5;
  then
A2: g1 * f = id the carrier of UA by Th8;
  f is_isomorphism by Def1;
  then f is_monomorphism;
  then
A3: f is one-to-one;
  rng f = dom f by Lm2
    .= the carrier of UA by Lm2;
  hence thesis by A1,A3,A2,FUNCT_2:30;
end;
