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

theorem
  for U1,U2,U3 being non-empty MSAlgebra over S holds U1,U2
  are_isomorphic & U2,U3 are_isomorphic implies U1,U3 are_isomorphic
proof
  let U1,U2,U3 be non-empty MSAlgebra over S;
  assume that
A1: U1,U2 are_isomorphic and
A2: U2,U3 are_isomorphic;
  consider F be ManySortedFunction of U1,U2 such that
A3: F is_isomorphism U1,U2 by A1;
  consider G be ManySortedFunction of U2,U3 such that
A4: G is_isomorphism U2,U3 by A2;
  reconsider H = G**F as ManySortedFunction of U1,U3;
  H is_isomorphism U1,U3 by A3,A4,Th15;
  hence thesis;
end;
