reserve S for locally_directed OrderSortedSign;
reserve o for Element of the carrier' of S;

theorem Th21:
  for S being OrderSortedSign, U1 being non-empty OSAlgebra of S,
  R being OSCongruence of U1 st R = [| (the Sorts of U1), (the Sorts of U1) |]
  holds R is monotone
proof
  let S be OrderSortedSign, U1 be non-empty OSAlgebra of S, R be OSCongruence
  of U1 such that
A1: R = [| (the Sorts of U1), (the Sorts of U1) |];
  reconsider O1 = the Sorts of U1 as OrderSortedSet of S by OSALG_1:17;
  let o1,o2 be OperSymbol of S such that
A2: o1 <= o2;
  set s2 = the_result_sort_of o2, s1 = the_result_sort_of o1;
  s1 <= s2 by A2;
  then
A3: O1.s1 c= O1.s2 by OSALG_1:def 16;
  let x1 be Element of Args(o1,U1);
  let x2 be Element of Args(o2,U1) such that
  for y being Nat st y in dom x1 holds [x1.y,x2.y] in R.((the_arity_of o2)
  /.y);
A4: Den(o1,U1).x1 in (the Sorts of U1).s1 & Den(o2,U1).x2 in (the Sorts of
  U1). s2 by MSUALG_9:18;
  R.s2 = [:(the Sorts of U1).s2,(the Sorts of U1).s2:] by A1,PBOOLE:def 16;
  hence thesis by A3,A4,ZFMISC_1:87;
end;
