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 Th31:
  for A,B,X be Ordinal st X c= A & X c= B holds
     No_Ord A /\ [:BeforeGames X,BeforeGames X:] =
     No_Ord B /\ [:BeforeGames X,BeforeGames X:]
proof
  let A,B,X be Ordinal such that A1: X c= A & X c= B;
  set SA=No_Ord A,SB=No_Ord B;
  [:Day(SA,A),Day(SA,A):] = ClosedProd(SA,A,A) by Lm3;
  then A2: SA preserves_No_Comparison_on ClosedProd(SA,A,A) by Def12;
  A c= X or X  in A by ORDINAL1:16;
  then X=A or X in A by A1,XBOOLE_0:def 10;
  then ClosedProd(SA,X,X) c= ClosedProd(SA,A,A) by Th17;
  then A3: SA preserves_No_Comparison_on ClosedProd(SA,X,X) by A2;
   [:Day(SB,B),Day(SB,B):] = ClosedProd(SB,B,B) by Lm3;
  then A4: SB preserves_No_Comparison_on ClosedProd(SB,B,B) by Def12;
  B c= X or X  in B by ORDINAL1:16;
  then X=B or X in B by A1,XBOOLE_0:def 10;
  then ClosedProd(SB,X,X) c= ClosedProd(SB,B,B) by Th17;
  then SB preserves_No_Comparison_on ClosedProd(SB,X,X) by A4;
  then A5:SA /\ ClosedProd(SA,X,X) = SB /\ ClosedProd(SB,X,X) by Th23,A3;
  ClosedProd(SA,X,X) = OpenProd(SA,X,succ X) &
  ClosedProd(SB,X,X) = OpenProd(SB,X,succ X) by ORDINAL1:6,Th29;
  hence thesis by A5,Th20;
end;
