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

theorem Th23:
  for S being OrderSortedSign, U1 being monotone non-empty
  OSAlgebra of S, R being OSCongruence of U1 holds R is monotone
proof
  let S be OrderSortedSign, U1 be monotone non-empty OSAlgebra of S, R be
  OSCongruence of U1;
  let o1,o2 be OperSymbol of S such that
A1: o1 <= o2;
  let x1 be Element of Args(o1,U1);
  Args(o1,U1) c= Args(o2,U1) by A1,OSALG_1:26;
  then reconsider x3 = x1 as Element of Args(o2,U1);
  let x2 be Element of Args(o2,U1);
  assume
  for y being Nat st y in dom x1 holds [x1.y,x2.y] in R.((the_arity_of o2)/.y);
  then
A2: [Den(o2,U1).x3,Den(o2,U1).x2] in R.(the_result_sort_of o2) by
MSUALG_4:def 4;
  x1 in Args(o1,U1);
  then
A3: x1 in dom Den(o1,U1) by FUNCT_2:def 1;
  Den(o1,U1) c= Den(o2,U1) by A1,OSALG_1:27;
  hence thesis by A2,A3,GRFUNC_1:2;
end;
