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

theorem Th20:
    R is almost-No-order & S is almost-No-order &
    R /\ OpenProd(R,A,B) = S /\ OpenProd(S,A,B)
  implies R /\ [:BeforeGames A,BeforeGames A:] =
          S /\ [:BeforeGames A,BeforeGames A:]
proof
  assume that A1:R is almost-No-order & S is almost-No-order and
  A2:R /\ OpenProd(R,A,B) = S /\ OpenProd(S,A,B);
  consider R0 be Ordinal such that
  A3:   R c= [:Day(R,R0),Day(R,R0):] by A1;
  consider S0 be Ordinal such that
  A4:   S c= [:Day(S,S0),Day(S,S0):] by A1;
  let y,z be object;
  thus [y,z] in R /\ [:BeforeGames A,BeforeGames A:] implies
  [y,z] in S /\ [:BeforeGames A,BeforeGames A:]
  proof
    assume [y,z] in R /\ [:BeforeGames A,BeforeGames A:];
    then A5: [y,z] in R & [y,z] in [:BeforeGames A,BeforeGames A:]
    by XBOOLE_0:def 4;
    then
    A6: y in BeforeGames A & z in BeforeGames A by ZFMISC_1:87;
    A7:y in Day(R,R0) & z in Day(R,R0) by A3,A5,ZFMISC_1:87;
    y in BeforeGames A & z in BeforeGames A by A5,ZFMISC_1:87;
    then consider Ay be Ordinal such that
    A8: Ay in A & y in Games Ay by Def5;
    consider Az be Ordinal such that
    A9: Az in A & z in Games Az by A6,Def5;
    A10: Day(R,Ay) c= Day(R,A) & Day(R,Az) c= Day(R,A)
      by Th9,A8,A9,ORDINAL1:def 2;
    A11:y in Day(R,Ay) & z in Day(R,Az) by A8,A7,A9,Th12;
    then born(R,y) c= Ay & born(R,z) c= Az by Def8;
    then A12:born(R,y) in A & born(R,z) in A by A8,A9,ORDINAL1:12;
    [y,z] in OpenProd(R,A,B) by A10,A11,A12,Def9;
    then [y,z] in R /\ OpenProd(R,A,B) by XBOOLE_0:def 4,A5;
    then [y,z] in S by A2,XBOOLE_0:def 4;
    hence thesis by A5,XBOOLE_0:def 4;
  end;
  assume [y,z] in S /\ [:BeforeGames A,BeforeGames A:];
  then A13: [y,z] in S & [y,z] in [:BeforeGames A,BeforeGames A:]
    by XBOOLE_0:def 4;
  then A14: y in BeforeGames A & z in BeforeGames A by ZFMISC_1:87;
  A15:y in Day(S,S0) & z in Day(S,S0) by A4,A13,ZFMISC_1:87;
  consider Ay be Ordinal such that
  A16: Ay in A & y in Games Ay by A14,Def5;
  consider Az be Ordinal such that
  A17: Az in A & z in Games Az by A14,Def5;
  A18: Day(S,Ay) c= Day(S,A) & Day(S,Az) c= Day(S,A)
    by Th9,A16,A17,ORDINAL1:def 2;
  A19:y in Day(S,Ay) & z in Day(S,Az) by A16,A15,A17,Th12;
  then born(S,y) c= Ay & born(S,z) c= Az by Def8;
  then A20:born(S,y) in A & born(S,z) in A by A16,A17,ORDINAL1:12;
  [y,z] in OpenProd(S,A,B) by A18,A19,A20,Def9;
  then [y,z] in S /\ OpenProd(S,A,B) by XBOOLE_0:def 4,A13;
  then [y,z] in R by A2,XBOOLE_0:def 4;
  hence thesis by A13,XBOOLE_0:def 4;
end;
