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

theorem Th11:
  R /\ [:BeforeGames A,BeforeGames A:] = S /\ [:BeforeGames A,BeforeGames A:]
    implies for a st a in Day(R,A) holds born(R,a) = born(S,a)
proof
  assume A1: R /\ [:BeforeGames A,BeforeGames A:] =
      S /\ [:BeforeGames A,BeforeGames A:];
  A2:Day(R,A) = Day(S,A) by A1,Th10;
  let x be object such that A3: x in Day(R,A);
  A4: x in Day(S,A) by A3,A1,Th10;
  born(R,x) c= A by A3,Def8;
  then Day(R,born(R,x)) = Day(S,born(R,x)) by A1,Th10;
  then x in Day(S,born(R,x)) by A3,Def8;
  then A5:born(S,x) c= born(R,x) by Def8;
  born(S,x) c= A by A3,A2,Def8;
  then Day(S,born(S,x)) = Day(R,born(S,x)) by A1,Th10;
  then x in Day(R,born(S,x)) by A4,Def8;
  then born(R,x) c= born(S,x) by Def8;
  hence thesis by A5,XBOOLE_0:def 10;
end;
