reserve R for non empty Poset,
  S1 for OrderSortedSign;

theorem
  for U1,U2 being strict non-empty OSAlgebra of S1 st U1,U2
  are_os_isomorphic holds U1 is monotone iff U2 is monotone
proof
  let U1,U2 be strict non-empty OSAlgebra of S1;
  assume U1,U2 are_os_isomorphic;
  then consider F be ManySortedFunction of U1,U2 such that
A1: F is_isomorphism U1,U2 and
A2: F is order-sorted;
  reconsider O1 = the Sorts of U1, O2 = the Sorts of U2 as OrderSortedSet of
  S1 by OSALG_1:17;
  reconsider F1 = F as ManySortedFunction of O1,O2;
  F is "onto" & F is "1-1" by A1,MSUALG_3:13;
  then
A3: F1"" is order-sorted by A2,Th6;
A4: F is_epimorphism U1,U2 by A1,MSUALG_3:def 10;
  then
A5: F is_homomorphism U1,U2 by MSUALG_3:def 8;
  then Image F = U2 by A4,MSUALG_3:19;
  hence U1 is monotone implies U2 is monotone by A2,A5,Th13;
  reconsider F2 = F1"" as ManySortedFunction of U2,U1;
  assume
A6: U2 is monotone;
  F"" is_isomorphism U2,U1 by A1,MSUALG_3:14;
  then
A7: F2 is_epimorphism U2,U1 by MSUALG_3:def 10;
  then
A8: F2 is_homomorphism U2,U1 by MSUALG_3:def 8;
  then Image F2 = U1 by A7,MSUALG_3:19;
  hence thesis by A3,A8,A6,Th13;
end;
