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
  for X,Y be set, A holds
    [X,Y] in Day A iff X << Y &
       for o be object st o in X \/ Y
          ex O st O in A & o in Day O
proof
  let X,Y be set, A be Ordinal;
  set S=No_Ord A;
  thus [X,Y] in Day A implies
  X << Y & for x be object st x in X \/ Y
    ex O st O in A & x in Day O
  proof
    assume A1:[X,Y] in Day A;
    then reconsider XY=[X,Y] as Surreal;
    L_XY << R_XY by Th45;
    hence X << Y;
    let x be object;
    assume x in X\/Y;
    then x in L_XY \/R_XY;
    then consider OL be Ordinal such that
    A2:OL in A & x in Day(S,OL) by A1,Th7;
    take OL;
    OL c= A by A2,ORDINAL1:def 2;
    hence thesis by A2,Th36;
  end;
  set XY=[X,Y];
  assume A3: X << Y & for x be object st x in X \/ Y ex O be Ordinal st
  O in A & x in Day O;
  for x be object st x in L_XY \/ R_XY
  ex O be Ordinal st O in A & x in Games O
  proof
    let x be object;
    assume x in L_XY \/ R_XY;
    then ex O be Ordinal st O in A & x in Day O by A3;
    hence thesis;
  end;
  then reconsider XY as Element of Games A by Th4;
  A4: L_XY << S, R_XY
  proof
    let l,r be object such that A5:l in L_XY & r in R_XY;
    assume A6: l >=S, r;
    then [r,l] in S c= [:Day(S,A),Day(S,A):] =
      ClosedProd(S,A,A) by Def12,Lm3;
    then r in Day A & l in Day A by Th33;
    then reconsider r,l as Surreal;
    r <=l by A6;
    hence thesis by A3,A5;
  end;
  for x be object st x in L_XY \/ R_XY
  ex O be Ordinal st O in A & x in Day(S,O)
  proof
    let x be object;
    assume x in L_XY \/ R_XY;
    then consider O be Ordinal such that
    A7: O in A & x in Day O by A3;
    O c= A by A7,ORDINAL1:def 2;
    then No_Ord A /\ [:BeforeGames O,BeforeGames O:] =
    No_Ord O /\ [:BeforeGames O,BeforeGames O:] by Th31;
    then Day(No_Ord O,O) = Day(S,O) by Th10;
    hence thesis by A7;
  end;
  hence thesis by Th7,A4;
end;
