reserve A,A1,A2,B,B1,B2,C,O for Ordinal,
      R,S for Relation,
      a,b,c,o,l,r for object;
reserve x,y,z,t,r,l for Surreal,
        X,Y,Z for set;

theorem Th34:
  A c= B implies
    No_Ord A = No_Ord B /\ ClosedProd(No_Ord B,A,A)
proof
  set R=No_Ord A, S= No_Ord B;
  assume A c= B;
  then A1: ClosedProd(S,A,A) c= ClosedProd(S,B,B) by Th30;
  A2: [:Day(S,B),Day(S,B):] = ClosedProd(S,B,B)&
  [:Day(R,A),Day(R,A):] = ClosedProd(R,A,A) by Lm3;
  then S preserves_No_Comparison_on ClosedProd(S,B,B) by Def12;
  then R preserves_No_Comparison_on ClosedProd(R,A,A) &
  S preserves_No_Comparison_on ClosedProd(S,A,A) by A2,A1,Def12;
  then A3:R /\ ClosedProd(R,A,A) = S /\ ClosedProd(S,A,A) by Th23;
  R c= ClosedProd(R,A,A) by A2,Def12;
  hence thesis by A3,XBOOLE_1:28;
end;
