reserve R for non empty Poset,
  S1 for OrderSortedSign;

theorem Th9:
  for U1,U2 being non-empty OSAlgebra of S1 holds U1,U2
  are_os_isomorphic implies U2,U1 are_os_isomorphic
proof
  let U1,U2 be non-empty OSAlgebra of S1;
A1: the Sorts of U1 is OrderSortedSet of S1 & the Sorts of U2 is
  OrderSortedSet of S1 by OSALG_1:17;
  assume U1,U2 are_os_isomorphic;
  then consider F be ManySortedFunction of U1,U2 such that
A2: F is_isomorphism U1,U2 and
A3: F is order-sorted;
  reconsider G = F"" as ManySortedFunction of U2,U1;
A4: G is_isomorphism U2,U1 by A2,MSUALG_3:14;
  F is "onto" & F is "1-1" by A2,MSUALG_3:13;
  then F"" is order-sorted by A3,A1,Th6;
  hence thesis by A4;
end;
