reserve S for non void non empty ManySortedSign,
  U1,U2 for MSAlgebra over S,
  o for OperSymbol of S,
  n for Nat;

theorem Th13:
  for F be ManySortedFunction of U1,U2 holds F is_isomorphism U1,
  U2 iff F is_homomorphism U1,U2 & F is "onto" & F is "1-1"
proof
  let F be ManySortedFunction of U1,U2;
  thus F is_isomorphism U1,U2 implies F is_homomorphism U1,U2 & F is "onto" &
  F is "1-1"
  proof
    assume F is_isomorphism U1,U2;
    then F is_epimorphism U1,U2 & F is_monomorphism U1,U2;
    hence thesis;
  end;
  assume F is_homomorphism U1,U2 & F is "onto" & F is "1-1";
  then F is_epimorphism U1,U2 & F is_monomorphism U1,U2;
  hence thesis;
end;
