reserve

  k,n for Nat,
  x,y,X,Y,Z for set;

theorem Th15:
  for k being Element of NAT for X being non empty set st 2 <= k &
  k + 2 c= card X for K being Subset of the Points of G_(k,X) holds K is
  maximal_clique implies K is STAR or K is TOP
proof
A1: succ 0 = 0 + 1;
A2: succ 2 = 2 + 1;
  let k be Element of NAT;
  let X be non empty set;
  assume that
A3: 2 <= k and
A4: k + 2 c= card X;
  k + 1 <= k + 2 by XREAL_1:7;
  then
A5: Segm(k + 1) c= Segm(k + 2) by NAT_1:39;
  then
A6: k + 1 c= card X by A4;
  then
A7: the Points of G_(k,X) = {A where A is Subset of X: card A = k} by A3,Def1;
A8: succ Segm(k + 1) = Segm(k + 1 + 1) by NAT_1:38;
A9: 1 <= k by A3,XXREAL_0:2;
  let K be Subset of the Points of G_(k,X);
A10: succ Segm k = Segm(k + 1) by NAT_1:38;
  0 c= card K;
  then 0 in succ(card K) by ORDINAL1:22;
  then
A11: card K = 0 or 0 in card K by ORDINAL1:8;
  assume
A12: K is maximal_clique;
  then
A13: K is clique;
A14: the Lines of G_(k,X) = {L where L is Subset of X: card L = k + 1} by A3,A6
,Def1;
  k <= k + 1 by NAT_1:11;
  then
A15: Segm k c= Segm(k + 1) by NAT_1:39;
  then
A16: k c= card X by A6;
  K <> {}
  proof
    consider A1 being set such that
A17: A1 c= X and
A18: card A1 = k by A16,CARD_FIL:36;
    A1 in the Points of G_(k,X) by A7,A17,A18;
    then consider A being POINT of G_(k,X) such that
A19: A = A1;
    card A <> card X
    proof
      assume card A = card X;
      then k + 1 c= k by A4,A5,A18,A19;
      then k in k by A10,ORDINAL1:21;
      hence contradiction;
    end;
    then card A in card X by A16,A18,A19,CARD_1:3;
    then X \ A <> {} by CARD_1:68;
    then consider x being object such that
A20: x in X \ A by XBOOLE_0:def 1;
    {x} c= X by A20,ZFMISC_1:31;
    then
A21: A \/ {x} c= X by A17,A19,XBOOLE_1:8;
A22: not x in A by A20,XBOOLE_0:def 5;
    A is finite by A18,A19;
    then card(A \/ {x}) = k + 1 by A18,A19,A22,CARD_2:41;
    then A \/ {x} in the Lines of G_(k,X) by A14,A21;
    then consider L being LINE of G_(k,X) such that
A23: L = A \/ {x};
    consider U being Subset of the Points of G_(k,X) such that
A24: U = {A};
    A c= L by A23,XBOOLE_1:7;
    then
A25: A on L by A3,A6,Th10;
A26: U is clique
    proof
      let B,C be POINT of G_(k,X);
      assume B in U & C in U;
      then B on L & C on L by A25,A24,TARSKI:def 1;
      then {B,C} on L by INCSP_1:1;
      hence thesis;
    end;
    assume
A27: K = {};
    then K c= U;
    hence contradiction by A12,A27,A24,A26;
  end;
  then 1 c= card K by A1,A11,ORDINAL1:21;
  then 1 in succ card K by ORDINAL1:22;
  then
A28: card K = 1 or 1 in card K by ORDINAL1:8;
A29: k - 1 is Element of NAT by A3,NAT_1:21,XXREAL_0:2;
    then reconsider k1 = k-1 as Nat;
A30: card K <> 1
  proof
    assume card K = 1;
    then consider A3 being object such that
A31: K = {A3} by CARD_2:42;
A32: A3 in K by A31,TARSKI:def 1;
    then consider A being POINT of G_(k,X) such that
A33: A = A3;
    A3 in the Points of G_(k,X) by A32;
    then
A34: ex A4 being Subset of X st A = A4 & card A4 = k by A7,A33;
    then reconsider AA = A as finite set;
A35: A is finite by A34;
A36: card A <> card X
    proof
      assume card A = card X;
      then k + 1 c= k by A4,A5,A34;
      then k in k by A10,ORDINAL1:21;
      hence contradiction;
    end;
    card A c= card X by A6,A15,A34;
    then card A in card X by A36,CARD_1:3;
    then X \ A <> {} by CARD_1:68;
    then consider x being object such that
A37: x in X \ A by XBOOLE_0:def 1;
A38: {x} c= X by A37,ZFMISC_1:31;
    then
A39: A \/ {x} c= X by A34,XBOOLE_1:8;
    not x in A by A37,XBOOLE_0:def 5;
    then card(A \/ {x}) = k + 1 by A34,A35,CARD_2:41;
    then A \/ {x} in the Lines of G_(k,X) by A14,A39;
    then consider L being LINE of G_(k,X) such that
A40: L = A \/ {x};
    k - 1 <= (k - 1) + 1 by A29,NAT_1:11;
    then Segm k1 c= Segm card AA by A34,NAT_1:39;
    then consider B2 being set such that
A41: B2 c= A and
A42: card B2 = k - 1 by CARD_FIL:36;
A43: B2 is finite by A3,A42,NAT_1:21,XXREAL_0:2;
    B2 c= X by A34,A41,XBOOLE_1:1;
    then
A44: B2 \/ {x} c= X by A38,XBOOLE_1:8;
    not x in B2 by A37,A41,XBOOLE_0:def 5;
    then card(B2 \/ {x}) = (k - 1) + 1 by A42,A43,CARD_2:41;
    then B2 \/ {x} in the Points of G_(k,X) by A7,A44;
    then consider A1 being POINT of G_(k,X) such that
A45: A1 = B2 \/ {x};
A46: {x} c= L by A40,XBOOLE_1:7;
A47: A c= L by A40,XBOOLE_1:7;
    then B2 c= L by A41;
    then A1 c= L by A45,A46,XBOOLE_1:8;
    then
A48: A1 on L by A3,A6,Th10;
    {x} c= A1 by A45,XBOOLE_1:7;
    then x in A1 by ZFMISC_1:31;
    then
A49: A <> A1 by A37,XBOOLE_0:def 5;
    consider U being Subset of the Points of G_(k,X) such that
A50: U = {A,A1};
A51: A on L by A3,A6,A47,Th10;
A52: U is clique
    proof
      let B1,B2 be POINT of G_(k,X);
      assume B1 in U & B2 in U;
      then B1 on L & B2 on L by A51,A48,A50,TARSKI:def 2;
      then {B1,B2} on L by INCSP_1:1;
      hence thesis;
    end;
    A in U by A50,TARSKI:def 2;
    then
A53: K c= U by A31,A33,ZFMISC_1:31;
    A1 in U by A50,TARSKI:def 2;
    then U <> K by A31,A33,A49,TARSKI:def 1;
    hence contradiction by A12,A53,A52;
  end;
  succ 1 = 1 + 1;
  then
A54: 2 c= card K by A30,A28,ORDINAL1:21;
  then consider R being set such that
A55: R c= K and
A56: card R = 2 by CARD_FIL:36;
  consider A1,B1 being object such that
A57: A1 <> B1 and
A58: R = {A1,B1} by A56,CARD_2:60;
A59: A1 in R by A58,TARSKI:def 2;
  then
A60: A1 in the Points of G_(k,X) by A55,TARSKI:def 3;
  then consider A being POINT of G_(k,X) such that
A61: A = A1;
A62: B1 in R by A58,TARSKI:def 2;
  then
A63: B1 in the Points of G_(k,X) by A55,TARSKI:def 3;
  then consider B being POINT of G_(k,X) such that
A64: B = B1;
  consider L being LINE of G_(k,X) such that
A65: {A,B} on L by A13,A55,A59,A62,A61,A64;
  L in the Lines of G_(k,X);
  then
A66: ex L1 being Subset of X st L1 = L & card L1 = k + 1 by A14;
  then
A67: L is finite;
  A on L by A65,INCSP_1:1;
  then
A68: A c= L by A3,A6,Th10;
  then
A69: A /\ B c= L by XBOOLE_1:108;
  then
A70: A /\ B c= X by A66,XBOOLE_1:1;
  B on L by A65,INCSP_1:1;
  then
A71: B c= L by A3,A6,Th10;
  then
A72: A \/ B c= L by A68,XBOOLE_1:8;
  then
A73: card(A \/ B) c= k + 1 by A66,CARD_1:11;
A74: ex B2 being Subset of X st B2 = B & card B2 = k by A7,A63,A64;
  then
A75: B is finite;
A76: ex A2 being Subset of X st A2 = A & card A2 = k by A7,A60,A61;
  then
A77: A is finite;
A78: k + 1 c= card(A \/ B) by A57,A61,A64,A76,A74,Th1;
  then
A79: card(A \/ B) = k + 1 by A73,XBOOLE_0:def 10;
  then
A80: A \/ B = L by A68,A71,A66,A67,CARD_2:102,XBOOLE_1:8;
A81: not (ex C being POINT of G_(k,X) st C in K & C on L & A <> C & B <> C)
  implies K is STAR
  proof
A82: card L <> card X
    proof
      assume card L = card X;
      then k + 1 in k + 1 by A4,A8,A66,ORDINAL1:21;
      hence contradiction;
    end;
    card L c= card X by A4,A5,A66;
    then card L in card X by A82,CARD_1:3;
    then X \ L <> {} by CARD_1:68;
    then consider x being object such that
A83: x in X \ L by XBOOLE_0:def 1;
A84: ( not x in A)& not x in B by A68,A71,A83,XBOOLE_0:def 5;
A85: A /\ {x} = {} & B /\ {x} = {}
    proof
      assume A /\ {x} <> {} or B /\ {x} <> {};
      then consider z1 being object such that
A86:  z1 in A /\ {x} or z1 in B /\ {x} by XBOOLE_0:def 1;
      z1 in A & z1 in {x} or z1 in B & z1 in {x} by A86,XBOOLE_0:def 4;
      hence contradiction by A84,TARSKI:def 1;
    end;
A87: card A = (k - 1) + 1 & card(A \/ B) = (k - 1) + 2*1 by A76,A73,A78,
XBOOLE_0:def 10;
    then
A88: card(A /\ B) = k - 1 by A29,A74,Th2;
    then card(A \ (A /\ B)) = k - (k - 1) by A76,A77,CARD_2:44,XBOOLE_1:17;
    then consider z1 being object such that
A89: A \ (A /\ B) = {z1} by CARD_2:42;
    card(B \ (A /\ B)) = k - (k - 1) by A74,A75,A88,CARD_2:44,XBOOLE_1:17;
    then consider z2 being object such that
A90: B \ (A /\ B) = {z2} by CARD_2:42;
A91: B = (A /\ B) \/ {z2} by A90,XBOOLE_1:17,45;
A92: z2 in {z2} by TARSKI:def 1;
A93: card(A \/ B) = (k - 1) + 2*1 by A73,A78,XBOOLE_0:def 10;
A94: not x in A /\ B by A69,A83,XBOOLE_0:def 5;
    card A = (k - 1) + 1 by A76;
    then card(A /\ B) = k - 1 by A29,A74,A93,Th2;
    then
A95: card((A /\ B) \/ {x}) = (k - 1) + 1 by A68,A67,A94,CARD_2:41;
    {x} c= X by A83,ZFMISC_1:31;
    then
A96: (A /\ B) \/ {x} c= X by A70,XBOOLE_1:8;
    then (A /\ B) \/ {x} in the Points of G_(k,X) by A7,A95;
    then consider C being POINT of G_(k,X) such that
A97: C = (A /\ B) \/ {x};
A98: B \/ C c= X by A74,A96,A97,XBOOLE_1:8;
A99: A \/ C c= X by A76,A96,A97,XBOOLE_1:8;
A100: 1 + 1 <= k + 1 by A9,XREAL_1:7;
A101: A /\ B c= B by XBOOLE_1:17;
    B /\ C = (B /\ {x}) \/ (B /\ (B /\ A)) by A97,XBOOLE_1:23;
    then B /\ C = (B /\ {x}) \/ (B /\ B /\ A) by XBOOLE_1:16;
    then card(B /\ C) = k - 1 by A29,A74,A85,A87,Th2;
    then card(B \/ C) = (k - 1) + 2*1 by A29,A74,A95,A97,Th2;
    then B \/ C in the Lines of G_(k,X) by A14,A98;
    then consider L2 being LINE of G_(k,X) such that
A102: L2 = B \/ C;
    A /\ C = (A /\ {x}) \/ (A /\ (A /\ B)) by A97,XBOOLE_1:23;
    then A /\ C = (A /\ {x}) \/ (A /\ A /\ B) by XBOOLE_1:16;
    then card(A /\ C) = k - 1 by A29,A74,A85,A87,Th2;
    then card(A \/ C) = (k - 1) + 2*1 by A29,A76,A95,A97,Th2;
    then A \/ C in the Lines of G_(k,X) by A14,A99;
    then consider L1 being LINE of G_(k,X) such that
A103: L1 = A \/ C;
A104: {A,B,C} is clique
    proof
      let Z1,Z2 be POINT of G_(k,X);
      assume that
A105: Z1 in {A,B,C} and
A106: Z2 in {A,B,C};
A107: Z2 = A or Z2 = B or Z2 = C by A106,ENUMSET1:def 1;
      Z1 = A or Z1 = B or Z1 = C by A105,ENUMSET1:def 1;
      then Z1 c= A \/ B & Z2 c= A \/ B or Z1 c= A \/ C & Z2 c= A \/ C or Z1
      c= B \/ C & Z2 c= B \/ C by A107,XBOOLE_1:11;
      then Z1 on L & Z2 on L or Z1 on L1 & Z2 on L1 or Z1 on L2 & Z2 on L2 by
A3,A6,A80,A103,A102,Th10;
      then {Z1,Z2} on L or {Z1,Z2} on L1 or {Z1,Z2} on L2 by INCSP_1:1;
      hence thesis;
    end;
A108: C <> A & C <> B by A84,A97,XBOOLE_1:11,ZFMISC_1:31;
A109: 3 c= card K
    proof
      assume not 3 c= card K;
      then card K in 3 by ORDINAL1:16;
      then card K c= 2 by A2,ORDINAL1:22;
      then card K = 2 & K is finite by A54,XBOOLE_0:def 10;
      then
A110: K = {A,B} by A55,A56,A58,A61,A64,CARD_2:102;
      A in {A,B,C} & B in {A,B,C} by ENUMSET1:def 1;
      then
A111: {A,B} c= {A,B,C} by ZFMISC_1:32;
      C in {A,B,C} by ENUMSET1:def 1;
      then not {A,B,C} c= {A,B} by A108,TARSKI:def 2;
      hence contradiction by A12,A104,A110,A111;
    end;
    card {A,B} <> card K
    proof
      assume card{A,B} = card K;
      then 3 in 3 by A2,A56,A58,A61,A64,A109,ORDINAL1:22;
      hence contradiction;
    end;
    then card{A,B} in card K by A54,A56,A58,A61,A64,CARD_1:3;
    then K \ {A,B} <> {} by CARD_1:68;
    then consider E2 being object such that
A112: E2 in K \ {A,B} by XBOOLE_0:def 1;
A113: card A = (k - 1) + 1 by A76;
    then
A114: card(A /\ B) = (k + 1) - 2 by A29,A74,A93,Th2;
A115: card B = (k - 1) + 1 by A74;
A116: A /\ B c= A by XBOOLE_1:17;
A117: not E2 in {A,B} by A112,XBOOLE_0:def 5;
    then
A118: E2 <> A by TARSKI:def 2;
    E2 in the Points of G_(k,X) by A112;
    then consider E1 being Subset of X such that
A119: E1 = E2 and
A120: card E1 = k by A7;
    consider E being POINT of G_(k,X) such that
A121: E = E1 by A112,A119;
A122: A = (A /\ B) \/ {z1} by A89,XBOOLE_1:17,45;
A123: z1 in {z1} by TARSKI:def 1;
    then
A124: not z1 in A /\ B by A89,XBOOLE_0:def 5;
A125: card A = (k + 1) - 1 & 2 + 1 <= k + 1 by A3,A76,XREAL_1:7;
    consider S being set such that
A126: S = {D where D is Subset of X: card D = k & A /\ B c= D};
A127: E2 in K by A112,XBOOLE_0:def 5;
    then consider K1 being LINE of G_(k,X) such that
A128: {A,E} on K1 by A13,A55,A59,A61,A119,A121;
    consider K2 being LINE of G_(k,X) such that
A129: {B,E} on K2 by A13,A55,A62,A64,A127,A119,A121;
    E on K2 by A129,INCSP_1:1;
    then
A130: E c= K2 by A3,A6,Th10;
    K2 in the Lines of G_(k,X);
    then
A131: ex M2 being Subset of X st K2 = M2 & card M2 = k + 1 by A14;
    B on K2 by A129,INCSP_1:1;
    then B c= K2 by A3,A6,Th10;
    then B \/ E c= K2 by A130,XBOOLE_1:8;
    then
A132: card(B \/ E) c= k + 1 by A131,CARD_1:11;
A133: E2 <> B by A117,TARSKI:def 2;
    then k + 1 c= card(B \/ E) by A74,A119,A120,A121,Th1;
    then card(B \/ E) = (k - 1) + 2*1 by A132,XBOOLE_0:def 10;
    then
A134: card(B /\ E) = (k + 1) - 2 by A29,A120,A121,A115,Th2;
    assume
    not (ex C being POINT of G_(k,X) st C in K & C on L & A <> C & B <> C);
    then
A135: not E on L by A127,A118,A133,A119,A121;
A136: not card(A \/ B \/ E) = k + 1
    proof
      assume
A137: card(A \/ B \/ E) = k + 1;
      then A \/ B c= A \/ B \/ E & A \/ B \/ E is finite by XBOOLE_1:7;
      then
A138: A \/ B = A \/ B \/ E by A79,A137,CARD_2:102;
      E c= A \/ B \/ E by XBOOLE_1:7;
      then E c= L by A72,A138;
      hence contradiction by A3,A6,A135,Th10;
    end;
    E on K1 by A128,INCSP_1:1;
    then
A139: E c= K1 by A3,A6,Th10;
    K1 in the Lines of G_(k,X);
    then
A140: ex M1 being Subset of X st K1 = M1 & card M1 = k + 1 by A14;
    A on K1 by A128,INCSP_1:1;
    then A c= K1 by A3,A6,Th10;
    then A \/ E c= K1 by A139,XBOOLE_1:8;
    then
A141: card(A \/ E) c= k + 1 by A140,CARD_1:11;
    k + 1 c= card(A \/ E) by A76,A118,A119,A120,A121,Th1;
    then card(A \/ E) = (k - 1) + 2*1 by A141,XBOOLE_0:def 10;
    then card(A /\ E) = (k + 1) - 2 by A29,A120,A121,A113,Th2;
    then card(A /\ B /\ E) = (k + 1) - 2 & card(A \/ B \/ E) = (k + 1) + 1 or
card(A /\ B /\ E) = (k + 1) - 3 & card(A \/ B \/ E) = k + 1 by A74,A120,A121
,A134,A125,A100,A114,Th7;
    then
A142: A /\ B = A /\ B /\ E by A68,A67,A88,A136,CARD_2:102,XBOOLE_1:17;
    then
A143: A /\ B c= E by XBOOLE_1:17;
    E is finite by A120,A121;
    then card(E \ (A /\ B)) = k - (k - 1) by A88,A120,A121,A142,CARD_2:44
,XBOOLE_1:17;
    then consider z4 being object such that
A144: E \ (A /\ B) = {z4} by CARD_2:42;
A145: E = (A /\ B) \/ {z4} by A142,A144,XBOOLE_1:17,45;
A146: K c= S
    proof
      assume not K c= S;
      then consider D2 being object such that
A147: D2 in K and
A148: not D2 in S;
      D2 in the Points of G_(k,X) by A147;
      then consider D1 being Subset of X such that
A149: D1 = D2 and
A150: card D1 = k by A7;
      consider D being POINT of G_(k,X) such that
A151: D = D1 by A147,A149;
      consider K11 being LINE of G_(k,X) such that
A152: {A,D} on K11 by A13,A55,A59,A61,A147,A149,A151;
      D on K11 by A152,INCSP_1:1;
      then
A153: D c= K11 by A3,A6,Th10;
      K11 in the Lines of G_(k,X);
      then
A154: ex R11 being Subset of X st R11 = K11 & card R11 = k + 1 by A14;
A155: card D = (k - 1) + 1 by A150,A151;
      consider K13 being LINE of G_(k,X) such that
A156: {E,D} on K13 by A13,A127,A119,A121,A147,A149,A151;
      consider K12 being LINE of G_(k,X) such that
A157: {B,D} on K12 by A13,A55,A62,A64,A147,A149,A151;
      A on K11 by A152,INCSP_1:1;
      then A c= K11 by A3,A6,Th10;
      then A \/ D c= K11 by A153,XBOOLE_1:8;
      then
A158: card(A \/ D) c= k + 1 by A154,CARD_1:11;
      A <> D by A126,A116,A148,A149,A150,A151;
      then k + 1 c= card(A \/ D) by A76,A150,A151,Th1;
      then card(A \/ D) = (k - 1) + 2*1 by A158,XBOOLE_0:def 10;
      then
A159: card(A /\ D) = k - 1 by A29,A76,A155,Th2;
      not A /\ B c= D by A126,A148,A149,A150,A151;
      then ex y being object st y in A /\ B & not y in D;
      then A /\ B <> (A /\ B) /\ D by XBOOLE_0:def 4;
      then
A160: card((A /\ B) /\ D) <> card(A /\ B) by A77,CARD_2:102,XBOOLE_1:17;
      D on K13 by A156,INCSP_1:1;
      then
A161: D c= K13 by A3,A6,Th10;
      K13 in the Lines of G_(k,X);
      then
A162: ex R13 being Subset of X st R13 = K13 & card R13 = k + 1 by A14;
      D on K12 by A157,INCSP_1:1;
      then
A163: D c= K12 by A3,A6,Th10;
      K12 in the Lines of G_(k,X);
      then
A164: ex R12 being Subset of X st R12 = K12 & card R12 = k + 1 by A14;
      B on K12 by A157,INCSP_1:1;
      then B c= K12 by A3,A6,Th10;
      then B \/ D c= K12 by A163,XBOOLE_1:8;
      then
A165: card(B \/ D) c= k + 1 by A164,CARD_1:11;
      B <> D by A126,A101,A148,A149,A150,A151;
      then k + 1 c= card(B \/ D) by A74,A150,A151,Th1;
      then card(B \/ D) = (k - 1) + 2*1 by A165,XBOOLE_0:def 10;
      then
A166: card(B /\ D) = k - 1 by A29,A74,A155,Th2;
      E on K13 by A156,INCSP_1:1;
      then E c= K13 by A3,A6,Th10;
      then E \/ D c= K13 by A161,XBOOLE_1:8;
      then
A167: card(E \/ D) c= k + 1 by A162,CARD_1:11;
      E <> D by A126,A143,A148,A149,A150,A151;
      then k + 1 c= card(E \/ D) by A120,A121,A150,A151,Th1;
      then card(E \/ D) = (k - 1) + 2*1 by A167,XBOOLE_0:def 10;
      then
A168: card(E /\ D) = k - 1 by A29,A120,A121,A155,Th2;
A169: z1 in D & z2 in D & z4 in D
      proof
        assume not z1 in D or not z2 in D or not z4 in D;
        then A /\ D = ((A /\ B) \/ {z1}) /\ D & {z1} misses D or B /\ D = ((A
/\ B) \/ {z2}) /\ D & {z2} misses D or E /\ D = ((A /\ B) \/ {z4}) /\ D & {z4}
        misses D by A89,A90,A142,A144,XBOOLE_1:17,45,ZFMISC_1:50;
        then A /\ D = ((A /\ B) /\ D) \/ ({z1} /\ D) & {z1} /\ D = {} or B /\
D = ((A /\ B) /\ D) \/ ({z2} /\ D) & {z2} /\ D = {} or E /\ D = ((A /\ B) /\ D)
        \/ ({z4} /\ D) & {z4} /\ D = {} by XBOOLE_0:def 7,XBOOLE_1:23;
        hence contradiction by A29,A74,A87,A159,A166,A168,A160,Th2;
      end;
      then {z1,z2} c= D & {z4} c= D by ZFMISC_1:31,32;
      then {z1,z2} \/ {z4} c= D by XBOOLE_1:8;
      then (A /\ B) /\ D c= D & {z1,z2,z4} c= D by ENUMSET1:3,XBOOLE_1:17;
      then
A170: ((A /\ B) /\ D) \/ {z1,z2,z4} c= D by XBOOLE_1:8;
A171: z4 in E \ (A /\ B) & (A /\ B) /\ D c= A /\ B by A144,TARSKI:def 1
,XBOOLE_1:17;
A172: {z1,z2,z4} misses (A /\ B) /\ D
      proof
        assume not {z1,z2,z4} misses (A /\ B) /\ D;
        then {z1,z2,z4} /\ ((A /\ B) /\ D) <> {} by XBOOLE_0:def 7;
        then consider m being object such that
A173:   m in {z1,z2,z4} /\ ((A /\ B) /\ D) by XBOOLE_0:def 1;
        m in {z1,z2,z4} by A173,XBOOLE_0:def 4;
        then
A174:   m = z1 or m = z2 or m = z4 by ENUMSET1:def 1;
        m in (A /\ B) /\ D by A173,XBOOLE_0:def 4;
        hence contradiction by A89,A90,A123,A92,A171,A174,XBOOLE_0:def 5;
      end;
      reconsider r = card((A /\ B) /\ D) as Nat by A77;
A175: not z1 in (A /\ B) /\ D by A124,XBOOLE_0:def 4;
      A /\ D = ((A /\ B) \/ {z1}) /\ D by A89,XBOOLE_1:17,45;
      then A /\ D = ((A /\ B) /\ D) \/ ({z1} /\ D) by XBOOLE_1:23;
      then A /\ D = ((A /\ B) /\ D) \/ {z1} by A169,ZFMISC_1:46;
      then
A176: card(A /\ D) = r + 1 by A77,A175,CARD_2:41;
      card{z1,z2,z4} = 3 by A57,A61,A64,A122,A91,A118,A133,A119,A121,A145,
CARD_2:58;
      then card(((A /\ B) /\ D) \/ {z1,z2,z4}) = (k - 2) + 3 by A77,A159,A176
,A172,CARD_2:40;
      then k + 1 c= k by A150,A151,A170,CARD_1:11;
      then k in k by A10,ORDINAL1:21;
      hence contradiction;
    end;
    S c= the Points of G_(k,X)
    proof
      let Z be object;
      assume Z in S;
      then ex Z1 being Subset of X st Z = Z1 & card Z1 = k & A /\ B c= Z1 by
A126;
      hence thesis by A7;
    end;
    then consider S1 being Subset of the Points of G_(k,X) such that
A177: S1 = S;
A178: S1 is STAR by A70,A126,A88,A177;
    then S1 is maximal_clique by A3,A4,Th14;
    then S1 is clique;
    hence thesis by A12,A146,A177,A178;
  end;
  reconsider k2 = k - 2 as Element of NAT by A3,NAT_1:21;

A179: succ Segm k2 = Segm(k2+ 1) by NAT_1:38;
  (ex C being POINT of G_(k,X) st C in K & C on L & A <> C & B <> C)
  implies K is TOP
  proof
A180: 1 + 1 <= k + 1 by A9,XREAL_1:7;
A181: card B = (k - 1) + 1 by A74;
A182: card A = (k - 1) + 1 by A76;
    assume ex C being POINT of G_(k,X) st C in K & C on L & A <> C & B <> C;
    then consider C being POINT of G_(k,X) such that
A183: C in K and
A184: C on L and
A185: A <> C and
A186: B <> C;
A187: C c= L by A3,A6,A184,Th10;
    then A \/ C c= L by A68,XBOOLE_1:8;
    then
A188: card(A \/ C) c= k + 1 by A66,CARD_1:11;
    B \/ C c= L by A71,A187,XBOOLE_1:8;
    then
A189: card(B \/ C) c= k + 1 by A66,CARD_1:11;
    C in the Points of G_(k,X);
    then
A190: ex C2 being Subset of X st C2 = C & card C2 = k by A7;
    then k + 1 c= card(B \/ C) by A74,A186,Th1;
    then card(B \/ C) = (k - 1) + 2*1 by A189,XBOOLE_0:def 10;
    then
A191: card(B /\ C) = (k + 1) - 2 by A29,A190,A181,Th2;
    k + 1 c= card(A \/ C ) by A76,A185,A190,Th1;
    then card(A \/ C) = (k - 1) + 2*1 by A188,XBOOLE_0:def 10;
    then
A192: card(A /\ C) = (k + 1) - 2 by A29,A190,A182,Th2;
A193: card(A \/ B) = (k - 1) + 2*1 by A73,A78,XBOOLE_0:def 10;
    then
A194: A \/ B = L by A68,A71,A66,A67,CARD_2:102,XBOOLE_1:8;
A195: A \/ B c= A \/ B \/ C by XBOOLE_1:7;
    A \/ B \/ C c= L by A72,A187,XBOOLE_1:8;
    then
A196: card(A \/ B \/ C) = k + 1 by A193,A194,A195,XBOOLE_0:def 10;
A197: card A = (k + 1) - 1 & 2 + 1 <= k + 1 by A3,A76,XREAL_1:7;
    consider T being set such that
A198: T = {D where D is Subset of X: card D = k & D c= L};
    card(A /\ B) = k - 1 by A29,A74,A193,A182,Th2;
    then
A199: card(A /\ B /\ C) = (k + 1) - 3 & card(A \/ B \/ C) = k + 1 or card(
    A /\ B /\ C) = (k + 1) - 2 & card(A \/ B \/ C) = (k + 1) + 1 by A74,A190
,A192,A197,A191,A180,Th7;
A200: K c= T
    proof
      let D2 be object;
      assume that
A201: D2 in K and
A202: not D2 in T;
      D2 in the Points of G_(k,X) by A201;
      then consider D1 being Subset of X such that
A203: D1 = D2 and
A204: card D1 = k by A7;
      consider D being POINT of G_(k,X) such that
A205: D = D1 by A201,A203;
      not D c= L by A198,A202,A203,A204,A205;
      then consider x being object such that
A206: x in D and
A207: not x in L;
A208: card {x} = 1 by CARD_1:30;
A209: D is finite by A204,A205;
A210: card D = (k - 1) + 1 by A204,A205;
      {x} c= D by A206,ZFMISC_1:31;
      then
A211: card(D \ {x}) = k - 1 by A204,A205,A209,A208,CARD_2:44;
      consider L13 being LINE of G_(k,X) such that
A212: {C,D} on L13 by A13,A183,A201,A203,A205;
      D on L13 by A212,INCSP_1:1;
      then
A213: D c= L13 by A3,A6,Th10;
      L13 in the Lines of G_(k,X);
      then
A214: ex L23 being Subset of X st L23 = L13 & card L23 = k + 1 by A14;
      C on L13 by A212,INCSP_1:1;
      then C c= L13 by A3,A6,Th10;
      then C \/ D c= L13 by A213,XBOOLE_1:8;
      then
A215: card(C \/ D) c= k + 1 by A214,CARD_1:11;
A216: not x in C by A187,A207;
A217: C /\ D c= D \ {x}
      proof
        let z be object;
        assume
A218:   z in C /\ D;
        then z <> x by A216,XBOOLE_0:def 4;
        then
A219:   not z in {x} by TARSKI:def 1;
        z in D by A218,XBOOLE_0:def 4;
        hence thesis by A219,XBOOLE_0:def 5;
      end;
      C <> D by A187,A198,A202,A203,A204,A205;
      then k + 1 c= card(C \/ D) by A190,A204,A205,Th1;
      then card(C \/ D) = (k - 1) + 2*1 by A215,XBOOLE_0:def 10;
      then card(C /\ D) = k - 1 by A29,A190,A210,Th2;
      then
A220: C /\ D = D \ {x} by A209,A211,A217,CARD_2:102;
      consider L12 being LINE of G_(k,X) such that
A221: {B,D} on L12 by A13,A55,A62,A64,A201,A203,A205;
      consider L11 being LINE of G_(k,X) such that
A222: {A,D} on L11 by A13,A55,A59,A61,A201,A203,A205;
      D on L11 by A222,INCSP_1:1;
      then
A223: D c= L11 by A3,A6,Th10;
      L11 in the Lines of G_(k,X);
      then
A224: ex L21 being Subset of X st L21 = L11 & card L21 = k + 1 by A14;
      A on L11 by A222,INCSP_1:1;
      then A c= L11 by A3,A6,Th10;
      then A \/ D c= L11 by A223,XBOOLE_1:8;
      then
A225: card(A \/ D) c= k + 1 by A224,CARD_1:11;
A226: not x in A by A68,A207;
A227: A /\ D c= D \ {x}
      proof
        let z be object;
        assume
A228:   z in A /\ D;
        then z <> x by A226,XBOOLE_0:def 4;
        then
A229:   not z in {x} by TARSKI:def 1;
        z in D by A228,XBOOLE_0:def 4;
        hence thesis by A229,XBOOLE_0:def 5;
      end;
      A <> D by A68,A198,A202,A203,A204,A205;
      then k + 1 c= card(A \/ D) by A76,A204,A205,Th1;
      then
A230: card(A \/ D) = (k - 1) + 2*1 by A225,XBOOLE_0:def 10;
      then card(A /\ D) = k - 1 by A29,A76,A210,Th2;
      then
A231: A /\ D = D \ {x} by A209,A211,A227,CARD_2:102;
      D on L12 by A221,INCSP_1:1;
      then
A232: D c= L12 by A3,A6,Th10;
      L12 in the Lines of G_(k,X);
      then
A233: ex L22 being Subset of X st L22 = L12 & card L22 = k + 1 by A14;
      B on L12 by A221,INCSP_1:1;
      then B c= L12 by A3,A6,Th10;
      then B \/ D c= L12 by A232,XBOOLE_1:8;
      then
A234: card(B \/ D) c= k + 1 by A233,CARD_1:11;
A235: not x in B by A71,A207;
A236: B /\ D c= D \ {x}
      proof
        let z be object;
        assume
A237:   z in B /\ D;
        then z <> x by A235,XBOOLE_0:def 4;
        then
A238:   not z in {x} by TARSKI:def 1;
        z in D by A237,XBOOLE_0:def 4;
        hence thesis by A238,XBOOLE_0:def 5;
      end;
      B <> D by A71,A198,A202,A203,A204,A205;
      then k + 1 c= card(B \/ D) by A74,A204,A205,Th1;
      then card(B \/ D) = (k - 1) + 2*1 by A234,XBOOLE_0:def 10;
      then card(B /\ D) = k - 1 by A29,A74,A210,Th2;
      then B /\ D = D \ {x} by A209,A211,A236,CARD_2:102;
      then A /\ D = (A /\ D) /\ (B /\ D) by A231;
      then A /\ D = A /\ (D /\ B) /\ D by XBOOLE_1:16;
      then A /\ D = A /\ B /\ D /\ D by XBOOLE_1:16;
      then A /\ D = A /\ B /\ (D /\ D) by XBOOLE_1:16;
      then A /\ D = (A /\ B /\ D) /\ (C /\ D) by A231,A220;
      then A /\ D = A /\ B /\ (D /\ C) /\ D by XBOOLE_1:16;
      then A /\ D = A /\ B /\ C /\ D /\ D by XBOOLE_1:16;
      then A /\ D = A /\ B /\ C /\ (D /\ D) by XBOOLE_1:16;
      then card(A /\ B /\ C /\ D) = k - 1 by A29,A76,A210,A230,Th2;
      then k - 1 c= k2 by A196,A199,CARD_1:11,XBOOLE_1:17;
      then k - 1 in k - 1 by A179,ORDINAL1:22;
      hence contradiction;
    end;
    T c= the Points of G_(k,X)
    proof
      let e be object;
      assume e in T;
      then ex E being Subset of X st e = E & card E = k & E c= L by A198;
      hence thesis by A7;
    end;
    then consider T1 being Subset of the Points of G_(k,X) such that
A239: T1 = T;
A240: T1 is TOP by A66,A198,A239;
    then T1 is maximal_clique by A3,A4,Th14;
    then T1 is clique;
    hence thesis by A12,A200,A239,A240;
  end;
  hence thesis by A81;
end;
