reserve U1,U2,U3 for Universal_Algebra,
  n,m for Nat,
  o1 for operation of U1,
  o2 for operation of U2,
  o3 for operation of U3,
  x,y for set;
reserve a for FinSequence of U1,
  f for Function of U1,U2;

theorem Th10:
  for h be Function of U1,U2, h1 be Function of U2,U1 st h
  is_isomorphism & h1 = h" holds h1 is_isomorphism
proof
  let h be Function of U1,U2,h1 be Function of U2,U1;
  assume that
A1: h is_isomorphism and
A2: h1=h";
A3: h1 is_homomorphism by A1,A2,Th9;
A4: h is one-to-one by A1,Th7;
  then rng h1 = dom h by A2,FUNCT_1:33
    .= the carrier of U1 by FUNCT_2:def 1;
  hence thesis by A2,A4,A3,Th7;
end;
