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;
