reserve R for non empty Poset,
  S1 for OrderSortedSign;

theorem
  for U1,U2,U3 being non-empty OSAlgebra of S1 holds U1,U2
  are_os_isomorphic & U2,U3 are_os_isomorphic implies U1,U3 are_os_isomorphic
proof
  let U1,U2,U3 be non-empty OSAlgebra of S1;
  assume that
A1: U1,U2 are_os_isomorphic and
A2: U2,U3 are_os_isomorphic;
  consider F be ManySortedFunction of U1,U2 such that
A3: F is_isomorphism U1,U2 and
A4: F is order-sorted by A1;
  consider G be ManySortedFunction of U2,U3 such that
A5: G is_isomorphism U2,U3 and
A6: G is order-sorted by A2;
  reconsider H = G**F as ManySortedFunction of U1,U3;
A7: H is_isomorphism U1,U3 by A3,A5,MSUALG_3:15;
A8: the Sorts of U3 is non-empty OrderSortedSet of S1 by OSALG_1:17;
  the Sorts of U1 is non-empty OrderSortedSet of S1 & the Sorts of U2 is
  non-empty OrderSortedSet of S1 by OSALG_1:17;
  then H is order-sorted by A4,A6,A8,Th5;
  hence thesis by A7;
end;
