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

theorem Th10:
  B c= A &
  R /\ [:BeforeGames A,BeforeGames A:] = S /\ [:BeforeGames A,BeforeGames A:]
    implies Day(R,B) = Day(S,B)
proof
  assume that A1:B c= A and
  A2: R /\ [:BeforeGames A,BeforeGames A:] =
      S /\ [:BeforeGames A,BeforeGames A:];
  BeforeGames B c= BeforeGames A by A1,Th5;
  then A3: [:BeforeGames B,BeforeGames B:] /\
  [:BeforeGames A,BeforeGames A:] =
  [:BeforeGames B,BeforeGames B:] by XBOOLE_1:28,ZFMISC_1:96;
  then R /\ [:BeforeGames B,BeforeGames B:] =
  S /\ [:BeforeGames A,BeforeGames A:]
  /\ [:BeforeGames B,BeforeGames B:] by A2,XBOOLE_1:16
  .= S /\ [:BeforeGames B,BeforeGames B:] by A3,XBOOLE_1:16;
  hence thesis by Lm1;
end;
