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

theorem Th24:
  for Lp,Lr be Sequence st dom Lp =dom Lr &
     for A st A in dom Lp holds
       (ex a,b be Ordinal,R be Relation st
          R =Lr.A & Lp.A = ClosedProd(R,a,b)) &
       Lr.A is Relation &
       (for R be Relation st R=Lr.A holds
          R preserves_No_Comparison_on Lp.A & R c= Lp.A)
   holds union rng Lr is Relation &
     for R st R = union rng Lr holds
       R preserves_No_Comparison_on union rng Lp & R c= union rng Lp
       & for A,a,b be Ordinal,S st
           A in dom Lp & S=Lr.A & Lp.A=ClosedProd(S,a,b)
         holds
  R /\ [:BeforeGames a,BeforeGames a:] = S /\ [:BeforeGames a,BeforeGames a:]
proof
  let Lp,Lr be Sequence such that
  A1:dom Lp =dom Lr and
  A2:for A be Ordinal st A in dom Lp holds
  (ex a,b be Ordinal,R be Relation st R=Lr.A & Lp.A=ClosedProd(R,a,b)) &
  Lr.A is Relation &
  (for R be Relation st R=Lr.A holds R preserves_No_Comparison_on Lp.A &
  R c= Lp.A);
  union rng Lr is Relation-like
  proof
    let y be object such that A3:y in union rng Lr;
    consider Y be set such that
    A4:y in Y & Y in rng Lr by A3,TARSKI:def 4;
    consider A be object such that
    A5:A in dom Lr & Lr.A = Y by A4,FUNCT_1:def 3;
    reconsider A as Ordinal by A5;
    ex a,b be Ordinal,R be Relation st
    R=Lr.A & Lp.A=ClosedProd(R,a,b) by A5,A2,A1;
    hence thesis by A4,A5,RELAT_1:def 1;
  end;
  hence union rng Lr is Relation;
  let R be Relation such that A6:R=union rng Lr;
  A7:R c= union rng Lp
  proof
    let y,z be object such that A8:[y,z] in R;
    consider Y be set such that
    A9:  [y,z]  in Y & Y in rng Lr by A6,A8,TARSKI:def 4;
    consider A be object such that
    A10:  A in dom Lr & Lr.A = Y by A9,FUNCT_1:def 3;
    reconsider A as Ordinal by A10;
    reconsider R=Lr.A as Relation by A10,A1,A2;
    R c= Lp.A in rng Lp by A10,A1,A2,FUNCT_1:def 3;
    hence thesis by A9,A10,TARSKI:def 4;
  end;
  R preserves_No_Comparison_on union rng Lp
  proof
    let x,y be object such that A11: [x,y] in union rng Lp;
    consider Y be set such that
    A12:[x,y]  in Y & Y in rng Lp by A11,TARSKI:def 4;
    consider A be object such that
    A13:A in dom Lp & Lp.A = Y by A12,FUNCT_1:def 3;
    consider a,b be Ordinal,T be Relation  such that
    A14:T=Lr.A & Lp.A=ClosedProd(T,a,b) by A13,A2;
    A15: T preserves_No_Comparison_on Lp.A & T c= Lp.A by A13,A14,A2;
    thus x <=R, y implies  L_x <<R, {y} & {x} <<R, R_y
    proof
      assume x<=R,y;
      then consider Y be set such that
      A16: [x,y] in Y & Y in rng Lr by A6,TARSKI:def 4;
      consider A be object such that
      A17: A in dom Lr & Lr.A = Y by A16,FUNCT_1:def 3;
      consider a,b be Ordinal,S be Relation such that
      A18: S=Lr.A & Lp.A=ClosedProd(S,a,b) by A17,A2,A1;
      A19: S preserves_No_Comparison_on ClosedProd(S,a,b) &
      S c= ClosedProd(S,a,b) by A17,A18,A1,A2;
      then A20: S is almost-No-order by XBOOLE_1:1;
      x<=S,y by A16,A17,A18;
      then A21: L_x <<S, {y} & {x} <<S, R_y by A19;
      A22: L_x <<R, {y}
      proof
        given l,r be object such that
        A23:l in L_x & r in {y} & l >=R,r;
        A24:not l >=S,r by A21,A23;
        consider Z be set such that
        A25:  [r,l]  in Z & Z in rng Lr by A23,A6,TARSKI:def 4;
        consider B be object such that
        A26:  B in dom Lr & Lr.B = Z by A25,FUNCT_1:def 3;
        consider a1,b1 be Ordinal,W be Relation  such that
        A27:  W=Lr.B & Lp.B=ClosedProd(W,a1,b1) by A26,A2,A1;
        A28: W preserves_No_Comparison_on ClosedProd(W,a1,b1) &
        W c= ClosedProd(W,a1,b1) by A26,A27,A1,A2;
        then A29: W is almost-No-order by XBOOLE_1:1;
        per cases;
        suppose a1 in a or (a1=a & b1 c= b);
          then ClosedProd(S,a1,b1) c= ClosedProd(S,a,b) by Th17;
          then S preserves_No_Comparison_on ClosedProd(S,a1,b1) by A19;
          then A30: S /\ ClosedProd(S,a1,b1) = W /\ ClosedProd(W,a1,b1)
          by A28,A20,A29,Th23;
          [r,l] in W /\ ClosedProd(W,a1,b1) by A25,A26,A27,A28,XBOOLE_0:def 4;
          hence thesis by A24,A30,XBOOLE_0:def 4;
        end;
        suppose A31:not a1 in a & not (a1=a & b1 c= b);
          then A32:a c= a1 by ORDINAL1:16;
          a in a1 or (a=a1 & b c= b1)
          proof
            assume A33:not a in a1 & not (a=a1 & b c= b1);
            then a1 c= a by ORDINAL1:16;
            hence thesis by A33,A31,A32,XBOOLE_0:def 10;
          end;
          then ClosedProd(W,a,b) c= ClosedProd(W,a1,b1) by Th17;
          then W preserves_No_Comparison_on ClosedProd(W,a,b) by A28;
          then A34: S /\ ClosedProd(S,a,b) = W /\ ClosedProd(W,a,b)
            by A19,A20,A29,Th23;
          [x,y] in W /\ ClosedProd(W,a,b)
            by A34,A16,A17,A18,A19,XBOOLE_0:def 4;
          then [x,y] in W by XBOOLE_0:def 4;
          then x<=W,y & [x,y] in ClosedProd(W,a1,b1) by A28;
          then L_x <<W, {y} by A28;
          then not l >=W,r by A23;
          hence thesis by A25,A26,A27;
        end;
      end;
      {x} <<R, R_y
      proof
        given l,r be object such that
        A35:l in {x} & r in R_y & l >=R,r;
        A36: not l >=S,r by A21,A35;
        consider Z be set such that
        A37:  [r,l]  in Z & Z in rng Lr by A35,A6,TARSKI:def 4;
        consider B be object such that
        A38:  B in dom Lr & Lr.B = Z by A37,FUNCT_1:def 3;
        consider a1,b1 be Ordinal,W be Relation  such that
        A39:  W=Lr.B & Lp.B=ClosedProd(W,a1,b1) by A38,A2,A1;
        A40: W  preserves_No_Comparison_on ClosedProd(W,a1,b1) &
              W c= ClosedProd(W,a1,b1) by A38,A39,A1,A2;
        then A41: W is almost-No-order by XBOOLE_1:1;
        per cases;
        suppose a1 in a or (a1=a & b1 c= b);
          then ClosedProd(S,a1,b1) c= ClosedProd(S,a,b) by Th17;
          then S preserves_No_Comparison_on ClosedProd(S,a1,b1) by A19;
          then A42: S /\ ClosedProd(S,a1,b1) = W /\ ClosedProd(W,a1,b1)
            by A40,A20,A41,Th23;
          [r,l] in W /\ ClosedProd(W,a1,b1) by A37,A38,A39,A40,XBOOLE_0:def 4;
          hence thesis by A36,A42,XBOOLE_0:def 4;
        end;
        suppose A43:not a1 in a & not (a1=a & b1 c= b);
          then A44:a c= a1 by ORDINAL1:16;
          a in a1 or (a=a1 & b c= b1)
          proof
            assume A45:not a in a1 & not (a=a1 & b c= b1);
            then a1 c= a by ORDINAL1:16;
            hence thesis by A45,A43,A44,XBOOLE_0:def 10;
          end;
          then ClosedProd(W,a,b) c= ClosedProd(W,a1,b1) by Th17;
          then W preserves_No_Comparison_on ClosedProd(W,a,b) by A40;
          then S /\ ClosedProd(S,a,b) = W /\ ClosedProd(W,a,b)
            by A19,A20,A41,Th23;
          then [x,y] in W /\ ClosedProd(W,a,b)
            by A16,A17,A18,A19,XBOOLE_0:def 4;
          then [x,y] in W by XBOOLE_0:def 4;
          then x<=W,y & [x,y] in ClosedProd(W,a1,b1) by A40;
          then {x} <<W, R_y by A40;
          then not l >=W,r by A35;
          hence thesis by A37,A38,A39;
        end;
      end;
      hence thesis by A22;
    end;
    assume A46: L_x <<R, {y} & {x} <<R, R_y;
    assume A47:not x <=R, y;
    A48:not x <=T,y
    proof
      assume A49: x<=T,y;
      T in rng Lr by A13,A14,A1,FUNCT_1:def 3;
      hence thesis by A47,A49,A6,TARSKI:def 4;
    end;
    per cases by A48,A15,A12,A13;
    suppose not L_x <<T, {y};
      then consider l,r be object such that
      A50:l in L_x & r in {y} & l >=T,r;
      T in rng Lr by A13,A14,A1,FUNCT_1:def 3;
      then r <=R,l by A50,A6,TARSKI:def 4;
      hence thesis by A46,A50;
    end;
    suppose not {x} <<T, R_y;
      then consider l,r be object such that
      A51:l in {x} & r in R_y & l >=T,r;
      T in rng Lr by A13,A14,A1,FUNCT_1:def 3;
      then r <=R,l by A51,A6,TARSKI:def 4;
      hence thesis by A46,A51;
    end;
  end;
  hence R preserves_No_Comparison_on union rng Lp & R c= union rng Lp by A7;
  let A,a,b be Ordinal,S be Relation such that
  A52:A in dom Lp & S=Lr.A & Lp.A=ClosedProd(S,a,b);
  A53: R /\ [:BeforeGames a,BeforeGames a:] c=
  S /\ [:BeforeGames a,BeforeGames a:]
  proof
    let x,y be object such that
    A54:[x,y] in R /\ [:BeforeGames a,BeforeGames a:];
    A55: [x,y] in [:BeforeGames a,BeforeGames a:]
    by A54,XBOOLE_0:def 4;
    [x,y] in R by A54,XBOOLE_0:def 4;
    then consider Y be set such that
    A56:  [x,y]  in Y & Y in rng Lr by A6,TARSKI:def 4;
    consider B be object such that
    A57:  B in dom Lr & Lr.B = Y by A56,FUNCT_1:def 3;
    reconsider B as Ordinal by A57;
    consider a1,b1 be Ordinal,W be Relation such that
    A58:  W=Lr.B & Lp.B=ClosedProd(W,a1,b1) by A57,A1,A2;
    reconsider W=Lr.B as Relation by A57,A1,A2;
    A59:S preserves_No_Comparison_on Lp.A & S c= Lp.A by A52,A2;
    then A60:S is almost-No-order by A52,XBOOLE_1:1;
    A61:W preserves_No_Comparison_on Lp.B & W c= Lp.B by A57,A1,A2;
    then A62:W is almost-No-order by A58,XBOOLE_1:1;
    per cases;
    suppose
      A63:not a1 in a & not (a1=a & b1 c= b);
      then A64:a c= a1 by ORDINAL1:16;
      a in a1 or (a=a1 & b c= b1)
      proof
        assume A65:not a in a1 & not (a=a1 & b c= b1);
        then a1 c= a by ORDINAL1:16;
        hence thesis by A65,A63,A64,XBOOLE_0:def 10;
      end;
      then ClosedProd(W,a,b) c= ClosedProd(W,a1,b1) by Th17;
      then W preserves_No_Comparison_on ClosedProd(W,a,b) by A61,A58;
      then A66: S /\ ClosedProd(S,a,b) = W /\ ClosedProd(W,a,b)
      by A62,A60,A52,A59,Th23;
      [x,y] in ClosedProd(W,a1,b1) by A56,A57,A61,A58;
      then A67:x in Day(W,a1) & y in Day(W,a1) by ZFMISC_1:87;
      A68: x in BeforeGames a & y in BeforeGames a by A55,ZFMISC_1:87;
      then consider Ox be Ordinal such that
      A69: Ox in a & x in Games Ox by Def5;
      consider Oy be Ordinal such that
      A70: Oy in a & y in Games Oy by A68,Def5;
      A71: Day(W,Ox) c= Day(W,a) by Th9,A69,ORDINAL1:def 2;
      A72: x in Day(W,Ox) by A67,A69,Th12;
      then born(W,x) c= Ox by Def8;
      then A73:born(W,x) in a & x in Day(W,a) by A69,A71,A72,ORDINAL1:12;
      A74: Day(W,Oy) c= Day(W,a) by A70,ORDINAL1:def 2,Th9;
      A75: y in Day(W,Oy) by A67,A70,Th12;
      then born(W,y) c= Oy by Def8;
      then born(W,y) in a & y in Day(W,a) by A70,A74,A75,ORDINAL1:12;
      then [x,y] in ClosedProd(W,a,b) by A73,Def10;
      then [x,y] in W /\ ClosedProd(W,a,b) by A56,A57,XBOOLE_0:def 4;
      then [x,y] in S by A66,XBOOLE_0:def 4;
      hence thesis by A55,XBOOLE_0:def 4;
    end;
    suppose a1 in a or (a1=a & b1 c= b);
      then ClosedProd(S,a1,b1) c= ClosedProd(S,a,b) by Th17;
      then S preserves_No_Comparison_on ClosedProd(S,a1,b1) by A52,A59;
      then A76: S /\ ClosedProd(S,a1,b1) = W /\ ClosedProd(W,a1,b1)
        by A61,A62,A60,A58,Th23;
      [x,y] in W /\ ClosedProd(W,a1,b1) by A56,A57,A61,A58,XBOOLE_0:def 4;
      then [x,y] in S by A76,XBOOLE_0:def 4;
      hence thesis by A55,XBOOLE_0:def 4;
    end;
  end;
  S /\ [:BeforeGames a,BeforeGames a:] c=
    R /\ [:BeforeGames a,BeforeGames a:]
  proof
    let x,y be object such that
    A77:[x,y] in S /\ [:BeforeGames a,BeforeGames a:];
    A78:[x,y] in [:BeforeGames a,BeforeGames a:] by A77,XBOOLE_0:def 4;
    A79:[x,y] in S by A77,XBOOLE_0:def 4;
    S in rng Lr by A52,A1,FUNCT_1:def 3;
    then [x,y] in R by A79,A6,TARSKI:def 4;
    hence thesis by A78,XBOOLE_0:def 4;
  end;
  hence thesis by A53,XBOOLE_0:def 10;
end;
