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 Th11:
  for h be Function of U1,U2, h1 be Function of U2,U3 st h
is_isomorphism & h1 is_isomorphism holds h1 * h is_isomorphism
proof
  let h be Function of U1,U2, h1 be Function of U2,U3;
  assume that
A1: h is_isomorphism and
A2: h1 is_isomorphism;
  dom h1 = the carrier of U2 & rng h = the carrier of U2 by A1,Th8,
FUNCT_2:def 1;
  then
A3: rng (h1 * h) = rng h1 by RELAT_1:28
    .= the carrier of U3 by A2,Th8;
  h is_homomorphism & h1 is_homomorphism by A1,A2,Th7;
  then
A4: h1 * h is_homomorphism by Th6;
  h is one-to-one & h1 is one-to-one by A1,A2,Th7;
  hence thesis by A3,A4,Th7;
end;
