reserve A,A1,A2,B,B1,B2,C,O for Ordinal,
      R,S for Relation,
      a,b,c,o,l,r for object;

theorem Th21:
    R is almost-No-order & S is almost-No-order &
    R /\ OpenProd(R,A,B) = S /\ OpenProd(S,A,B) &
    R preserves_No_Comparison_on ClosedProd(R,A,B) &
    S preserves_No_Comparison_on ClosedProd(S,A,B)
  implies R /\ ClosedProd(R,A,B) = S /\ ClosedProd(S,A,B)
proof
  assume A1: R is almost-No-order & S is almost-No-order &
  R /\ OpenProd(R,A,B) = S /\ OpenProd(S,A,B) &
  R preserves_No_Comparison_on ClosedProd(R,A,B) &
  S preserves_No_Comparison_on ClosedProd(S,A,B);
  per cases by ORDINAL1:16;
  suppose B c= A;
    then R /\ ClosedProd(R,A,B) c= S /\ ClosedProd(S,A,B) &
    S /\ ClosedProd(S,A,B) c= R /\ ClosedProd(R,A,B) by A1,Lm2;
    hence thesis by XBOOLE_0:def 10;
  end;
  suppose A in B;
    then ClosedProd(R,A,B) c= OpenProd(R,A,B) &
    OpenProd(R,A,B) c= ClosedProd(R,A,B) & ClosedProd(S,A,B)
    c= OpenProd(S,A,B) &
    OpenProd(S,A,B) c= ClosedProd(S,A,B) by Th16,Th19;
    then ClosedProd(R,A,B) = OpenProd(R,A,B) &
    ClosedProd(S,A,B) = OpenProd(S,A,B) by XBOOLE_0:def 10;
    hence thesis by A1;
  end;
end;
