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

theorem Th22:
  for S being OrderSortedSign, U1 being non-empty OSAlgebra of S,
  R being monotone MSEquivalence-like OrderSortedRelation of U1 holds R is
  MSCongruence-like
proof
  let S be OrderSortedSign, U1 be non-empty OSAlgebra of S, R be monotone
  MSEquivalence-like OrderSortedRelation of U1;
  for o be (Element of the carrier' of S), x,y be Element of Args(o,U1) st
(for n be Nat st n in dom x holds [x.n,y.n] in R.((the_arity_of o)/.n)) holds [
  Den(o,U1).x,Den(o,U1).y] in R.(the_result_sort_of o) by Def26;
  hence thesis by MSUALG_4:def 4;
end;
