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 Th7:
  f is_isomorphism iff f is_homomorphism & rng f = the
  carrier of U2 & f is one-to-one
proof
  thus f is_isomorphism implies f is_homomorphism & rng f = the
  carrier of U2 & f is one-to-one
  proof
    assume f is_isomorphism;
    then f is_monomorphism & f is_epimorphism;
    hence thesis;
  end;
  assume f is_homomorphism & rng f = the carrier of U2 & f is one-to-one;
  then f is_monomorphism & f is_epimorphism;
  hence thesis;
end;
