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
  x in Day A implies born x = born(No_Ord A,x)
proof
  set bx=born x;
  set bS=born(No_Ord A,x);
  assume A1: x in Day A;
  then A2:x in Day(No_Ord A,bS) by Def8;
  bS c= A by A1,Def8;
  then x in Day bS by Th36,A2;
  then A3:bx c= bS by Def18;
  bx c= A by Def18,A1;
  then A4: No_Ord bx /\ [:BeforeGames bx,BeforeGames bx:] =
  No_Ord A /\ [:BeforeGames bx,BeforeGames bx:] by Th31;
  A5:x in Day bx by Def18;
  then born(No_Ord bx,x) = bS by A4,Th11;
  then bS c= bx by A5,Def8;
  hence thesis by A3,XBOOLE_0:def 10;
end;
