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

theorem Th20:
  for S being OrderSortedSign, U1 being non-empty OSAlgebra of S
  holds [| the Sorts of U1, the Sorts of U1 |] is OSCongruence of U1
proof
  let S be OrderSortedSign, U1 be non-empty OSAlgebra of S;
  reconsider C = [| the Sorts of U1, the Sorts of U1 |] as MSCongruence of U1
  by MSUALG_5:18;
  C is os-compatible
  proof
    reconsider O1 = (the Sorts of U1) as OrderSortedSet of S by OSALG_1:17;
    let s1,s2 be Element of S such that
A1: s1 <= s2;
    reconsider s3 = s1, s4 = s2 as Element of S;
A2: O1.s3 c= O1.s4 by A1,OSALG_1:def 16;
A3: C.s1 = [:(the Sorts of U1).s1,(the Sorts of U1).s1:] & C.s2 = [:(the
    Sorts of U1).s2,(the Sorts of U1).s2:] by PBOOLE:def 16;
    let x,y be set;
    assume x in (the Sorts of U1).s1 & y in (the Sorts of U1).s1;
    hence [x,y] in C.s1 iff [x,y] in C.s2 by A2,A3,ZFMISC_1:87;
  end;
  hence thesis by Def2;
end;
