reserve

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

theorem Th23:
  for k being Element of NAT for X being non empty set st 2 <= k &
  k + 2 c= card X for F being IncProjMap over G_(k,X), G_(k,X) st F is
  automorphism for K being Subset of the Points of G_(k,X) holds K is STAR
  implies F.:K is STAR & F"K is STAR
proof
  let k be Element of NAT;
  let X be non empty set such that
A1: 2 <= k and
A2: k + 2 c= card X;
  let F be IncProjMap over G_(k,X), G_(k,X) such that
A3: F is automorphism;
A4: k - 1 is Element of NAT by A1,NAT_1:21,XXREAL_0:2;
  then reconsider k1 = k-1 as Nat;
A5: succ Segm k1 = Segm(k1 + 1) by NAT_1:38;
  2 - 1 <= k - 1 by A1,XREAL_1:9;
  then
A6: Segm 1 c= Segm k1 by NAT_1:39;
A7: 1 <= k by A1,XXREAL_0:2;
  then
A8: Segm 1 c= Segm k by NAT_1:39;
  let K be Subset of the Points of G_(k,X);
  assume
A9: K is STAR;
  then
A10: K is maximal_clique by A1,A2,Th14;
  then
A11: K is clique;
  k + 1 <= k + 2 by XREAL_1:7;
  then Segm(k + 1) c= Segm(k + 2) by NAT_1:39;
  then
A12: k + 1 c= card X by A2;
  then
A13: the
 Points of G_(k,X) = {A where A is Subset of X: card A = k} by A1,Def1;
A14: the Lines of G_(k,X) = {L where L is Subset of X: card L = k + 1} by A1
,A12,Def1;
A15: k + 0 <= k + 1 by XREAL_1:7;
  then 1 <= k + 1 by A7,XXREAL_0:2;
  then
A16: Segm 1 c= Segm(k + 1) by NAT_1:39;
A17: not F"K is TOP
  proof
    assume F"K is TOP;
    then consider B being Subset of X such that
A18: card B = k + 1 & F"K = {A where A is Subset of X: card A = k & A
    c= B};
    consider X1 being set such that
A19: X1 c= B & card X1 = 1 by A16,A18,CARD_FIL:36;
A20: B is finite by A18;
    then
A21: card(B \ X1) = (k + 1) - 1 by A18,A19,CARD_2:44;
    then consider X2 being set such that
A22: X2 c= B \ X1 and
A23: card X2 = 1 by A8,CARD_FIL:36;
    consider m being Nat such that
A24: m = k - 1 by A4;
A25: card(B \ X2) = (k + 1) - 1 by A18,A20,A22,A23,CARD_2:44,XBOOLE_1:106;
    then B \ X2 in the Points of G_(k,X) by A13;
    then consider B2 being POINT of G_(k,X) such that
A26: B \ X2 = B2;
    card((B \ X1) \ X2) = k - 1 by A20,A21,A22,A23,CARD_2:44;
    then consider X3 being set such that
A27: X3 c= (B \ X1) \ X2 and
A28: card X3 = 1 by A6,CARD_FIL:36;
A29: X3 c= B \ (X2 \/ X1) by A27,XBOOLE_1:41;
    then
A30: card(B \ X3) = (k + 1) - 1 by A18,A20,A28,CARD_2:44,XBOOLE_1:106;
    then B \ X3 in the Points of G_(k,X) by A13;
    then consider B3 being POINT of G_(k,X) such that
A31: B \ X3 = B3;
    B in the Lines of G_(k,X) by A14,A18;
    then consider L2 being LINE of G_(k,X) such that
A32: B = L2;
    B \ X1 in the Points of G_(k,X) by A13,A21;
    then consider B1 being POINT of G_(k,X) such that
A33: B \ X1 = B1;
    consider S being Subset of X such that
A34: card S = k - 1 and
A35: K = {A where A is Subset of X: card A = k & S c= A} by A9;
    consider A1 being POINT of G_(k,X) such that
A36: A1 = F.B1;
A37: X3 c= (B \ X2) \ X1 by A29,XBOOLE_1:41;
A38: B \ X1 <> B \ X2 & B \ X2 <> B \ X3 & B \ X1 <> B \ X3
    proof
      assume B \ X1 = B \ X2 or B \ X2 = B \ X3 or B \ X1 = B \ X3;
      then X2 = {} or X3 = {} or X3 = {} by A22,A27,A37,XBOOLE_1:38,106;
      hence contradiction by A23,A28;
    end;
    consider A3 being POINT of G_(k,X) such that
A39: A3 = F.B3;
A40: B \ X3 c= B by XBOOLE_1:106;
    then B3 in F"K by A18,A30,A31;
    then
A41: A3 in K by A39,FUNCT_1:def 7;
    then
A42: ex A13 being Subset of X st A3 = A13 & card A13 = k & S c= A13 by A35;
A43: B \ X1 c= B by XBOOLE_1:106;
    then B1 in F"K by A18,A21,A33;
    then
A44: A1 in K by A36,FUNCT_1:def 7;
    then
A45: ex A11 being Subset of X st A1 = A11 & card A11 = k & S c= A11 by A35;
    then
A46: card A1 = (k - 1) + 1;
    consider A2 being POINT of G_(k,X) such that
A47: A2 = F.B2;
A48: B \ X2 c= B by XBOOLE_1:106;
    then B2 in F"K by A18,A25,A26;
    then
A49: A2 in K by A47,FUNCT_1:def 7;
    then consider L3a being LINE of G_(k,X) such that
A50: {A1,A2} on L3a by A11,A44;
A51: card A1 = (k + 1) - 1 by A45;
A52: F is incidence_preserving by A3;
A53: card(A1 /\ A2 /\ A3) c= card(A1 /\ A2) by CARD_1:11,XBOOLE_1:17;
    consider L1 being LINE of G_(k,X) such that
A54: L1 = F.L2;
    B1 on L2 by A1,A12,A43,A33,A32,Th10;
    then
A55: A1 on L1 by A52,A36,A54;
    then
A56: A1 c= L1 by A1,A12,Th10;
    L1 in the Lines of G_(k,X);
    then
A57: ex l12 being Subset of X st L1 = l12 & card l12 = k + 1 by A14;
    B3 on L2 by A1,A12,A40,A31,A32,Th10;
    then
A58: A3 on L1 by A52,A39,A54;
    then
A59: A3 c= L1 by A1,A12,Th10;
    then A1 \/ A3 c= L1 by A56,XBOOLE_1:8;
    then
A60: card(A1 \/ A3) c= k + 1 by A57,CARD_1:11;
A61: ex A12 being Subset of X st A2 = A12 & card A12 = k & S c= A12 by A49,A35;
    then
A62: card A2 = (k - 1) + 1;
    B2 on L2 by A1,A12,A48,A26,A32,Th10;
    then
A63: A2 on L1 by A52,A47,A54;
    then
A64: A2 c= L1 by A1,A12,Th10;
    then A1 \/ A2 c= L1 by A56,XBOOLE_1:8;
    then
A65: card(A1 \/ A2) c= k + 1 by A57,CARD_1:11;
A66: the point-map of F is bijective & the Points of G_(k,X) = dom(the
    point-map of F) by A3,FUNCT_2:52;
    then
A67: A1 <> A2 by A38,A33,A26,A36,A47,FUNCT_1:def 4;
    then k + 1 c= card(A1 \/ A2) by A45,A61,Th1;
    then card(A1 \/ A2) = (k - 1) + 2*1 by A65,XBOOLE_0:def 10;
    then
A68: card(A1 /\ A2) = (k + 1) - 2 by A4,A61,A46,Th2;
    {A1,A2} on L1 by A55,A63,INCSP_1:1;
    then
A69: L1 = L3a by A67,A50,INCSP_1:def 10;
    consider L3b being LINE of G_(k,X) such that
A70: {A2,A3} on L3b by A11,A49,A41;
    A1 <> A3 by A66,A38,A33,A31,A36,A39,FUNCT_1:def 4;
    then k + 1 c= card(A1 \/ A3) by A45,A42,Th1;
    then card(A1 \/ A3) = (k - 1) + 2*1 by A60,XBOOLE_0:def 10;
    then
A71: card(A1 /\ A3) = (k + 1) - 2 by A4,A42,A46,Th2;
    A3 on L3b by A70,INCSP_1:1;
    then
A72: A3 c= L3b by A1,A12,Th10;
    A2 on L3b by A70,INCSP_1:1;
    then
A73: A2 c= L3b by A1,A12,Th10;
    L3b in the Lines of G_(k,X);
    then
A74: ex l13b being Subset of X st L3b = l13b & card l13b = k + 1 by A14;
    card(A1 /\ A2) in succ(k - 1) by A5,A67,A45,A61,Th1;
    then card(A1 /\ A2) c= m by A24,ORDINAL1:22;
    then
A75: card(A1 /\ A2 /\ A3) c= m by A53;
    S c= A1 /\ A2 by A45,A61,XBOOLE_1:19;
    then S c= (A1 /\ A2) /\ A3 by A42,XBOOLE_1:19;
    then m c= card(A1 /\ A2 /\ A3) by A34,A24,CARD_1:11;
    then
A76: k - 1 = card(A1 /\ A2 /\ A3) by A24,A75,XBOOLE_0:def 10;
    A1 on L3a by A50,INCSP_1:1;
    then
A77: A1 c= L3a by A1,A12,Th10;
A78: k - 1 <> (k + 1) - 3;
    A2 \/ A3 c= L1 by A64,A59,XBOOLE_1:8;
    then
A79: card(A2 \/ A3) c= k + 1 by A57,CARD_1:11;
A80: A2 <> A3 by A66,A38,A26,A31,A47,A39,FUNCT_1:def 4;
    then k + 1 c= card(A2 \/ A3) by A61,A42,Th1;
    then card(A2 \/ A3) = (k - 1) + 2*1 by A79,XBOOLE_0:def 10;
    then
A81: card(A2 /\ A3) = (k + 1) - 2 by A4,A42,A62,Th2;
    2 + 1 <= k + 1 & 2 <= k + 1 by A1,A15,XREAL_1:6,XXREAL_0:2;
    then
A82: card(A1 \/ A2 \/ A3) = (k + 1) + 1 by A61,A42,A76,A68,A51,A81,A71,A78,Th7;
A83: L3a <> L3b
    proof
      assume L3a = L3b;
      then A1 \/ A2 c= L3b by A77,A73,XBOOLE_1:8;
      then (A1 \/ A2) \/ A3 c= L3b by A72,XBOOLE_1:8;
      then Segm(k + 2) c= Segm(k + 1) by A82,A74,CARD_1:11;
      then k + 2 <= k + 1 by NAT_1:39;
      hence contradiction by XREAL_1:6;
    end;
    {A2,A3} on L1 by A63,A58,INCSP_1:1;
    hence contradiction by A80,A70,A83,A69,INCSP_1:def 10;
  end;
A84: not F.:K is TOP
  proof
A85: k - 1 <> (k + 1) - 3;
    assume F.:K is TOP;
    then consider B being Subset of X such that
A86: card B = k + 1 & F.:K = {A where A is Subset of X: card A = k & A
    c= B};
    B in the Lines of G_(k,X) by A14,A86;
    then consider L2 being LINE of G_(k,X) such that
A87: B = L2;
    the line-map of F is bijective by A3;
    then the Lines of G_(k,X) = rng(the line-map of F) by FUNCT_2:def 3;
    then consider l1 being object such that
A88: l1 in dom(the line-map of F) and
A89: L2 = (the line-map of F).l1 by FUNCT_1:def 3;
    consider L1 being LINE of G_(k,X) such that
A90: l1 = L1 by A88;
A91: L2 = F.L1 by A89,A90;
    consider X1 being set such that
A92: X1 c= B & card X1 = 1 by A16,A86,CARD_FIL:36;
A93: B is finite by A86;
    then
A94: card(B \ X1) = (k + 1) - 1 by A86,A92,CARD_2:44;
    then consider X2 being set such that
A95: X2 c= B \ X1 and
A96: card X2 = 1 by A8,CARD_FIL:36;
    consider m being Nat such that
A97: m = k - 1 by A4;
    card((B \ X1) \ X2) = k - 1 by A93,A94,A95,A96,CARD_2:44;
    then consider X3 being set such that
A98: X3 c= (B \ X1) \ X2 and
A99: card X3 = 1 by A6,CARD_FIL:36;
A100: X3 c= B \ (X2 \/ X1) by A98,XBOOLE_1:41;
    then
A101: card(B \ X3) = (k + 1) - 1 by A86,A93,A99,CARD_2:44,XBOOLE_1:106;
    then B \ X3 in the Points of G_(k,X) by A13;
    then consider B3 being POINT of G_(k,X) such that
A102: B \ X3 = B3;
    L1 in the Lines of G_(k,X);
    then
A103: ex l12 being Subset of X st L1 = l12 & card l12 = k + 1 by A14;
    B \ X1 in the Points of G_(k,X) by A13,A94;
    then consider B1 being POINT of G_(k,X) such that
A104: B \ X1 = B1;
A105: B \ X1 c= B by XBOOLE_1:106;
    then
A106: B1 on L2 by A1,A12,A104,A87,Th10;
    consider S being Subset of X such that
A107: card S = k - 1 and
A108: K = {A where A is Subset of X: card A = k & S c= A} by A9;
A109: F is incidence_preserving by A3;
A110: B \ X3 c= B by XBOOLE_1:106;
    then
A111: B3 on L2 by A1,A12,A102,A87,Th10;
A112: the point-map of F is bijective by A3;
    then
A113: the Points of G_(k,X) = rng(the point-map of F) by FUNCT_2:def 3;
    then consider a3 being object such that
A114: a3 in dom(the point-map of F) and
A115: B3 = (the point-map of F).a3 by FUNCT_1:def 3;
    consider A3 being POINT of G_(k,X) such that
A116: a3 = A3 by A114;
    consider a1 being object such that
A117: a1 in dom(the point-map of F) and
A118: B1 = (the point-map of F).a1 by A113,FUNCT_1:def 3;
    consider A1 being POINT of G_(k,X) such that
A119: a1 = A1 by A117;
    B3 in F.:K by A86,A101,A110,A102;
    then
    ex C3 being object
st C3 in dom(the point-map of F) & C3 in K & B3 = (the
    point-map of F).C3 by FUNCT_1:def 6;
    then
A120: A3 in K by A112,A114,A115,A116,FUNCT_1:def 4;
    then
A121: ex A13 being Subset of X st A3 = A13 & card A13 = k & S c= A13 by A108;
    B1 in F.:K by A86,A94,A105,A104;
    then
    ex C1 being object
st C1 in dom(the point-map of F) & C1 in K & B1 = (the
    point-map of F).C1 by FUNCT_1:def 6;
    then
A122: A1 in K by A112,A117,A118,A119,FUNCT_1:def 4;
    then
A123: ex A11 being Subset of X st A1 = A11 & card A11 = k & S c= A11 by A108;
    then
A124: card A1 = (k - 1) + 1;
A125: B1 = F.A1 by A118,A119;
    then A1 on L1 by A109,A106,A91;
    then
A126: A1 c= L1 by A1,A12,Th10;
A127: B3 = F.A3 by A115,A116;
    then A3 on L1 by A109,A111,A91;
    then
A128: A3 c= L1 by A1,A12,Th10;
    then A1 \/ A3 c= L1 by A126,XBOOLE_1:8;
    then
A129: card(A1 \/ A3) c= k + 1 by A103,CARD_1:11;
A130: X3 c= (B \ X2) \ X1 by A100,XBOOLE_1:41;
A131: B \ X1 <> B \ X2 & B \ X2 <> B \ X3 & B \ X1 <> B \ X3
    proof
      assume B \ X1 = B \ X2 or B \ X2 = B \ X3 or B \ X1 = B \ X3;
      then X2 = {} or X3 = {} or X3 = {} by A95,A98,A130,XBOOLE_1:38,106;
      hence contradiction by A96,A99;
    end;
    then k + 1 c= card(A1 \/ A3) by A104,A102,A118,A115,A119,A116,A123,A121,Th1
;
    then card(A1 \/ A3) = (k - 1) + 2*1 by A129,XBOOLE_0:def 10;
    then
A132: card(A1 /\ A3) = (k + 1) - 2 by A4,A121,A124,Th2;
A133: card(B \ X2) = (k + 1) - 1 by A86,A93,A95,A96,CARD_2:44,XBOOLE_1:106;
    then B \ X2 in the Points of G_(k,X) by A13;
    then consider B2 being POINT of G_(k,X) such that
A134: B \ X2 = B2;
A135: B \ X2 c= B by XBOOLE_1:106;
    then
A136: B2 on L2 by A1,A12,A134,A87,Th10;
    consider a2 being object such that
A137: a2 in dom(the point-map of F) and
A138: B2 = (the point-map of F).a2 by A113,FUNCT_1:def 3;
    consider A2 being POINT of G_(k,X) such that
A139: a2 = A2 by A137;
    B2 in F.:K by A86,A133,A135,A134;
    then
    ex C2 being object
st C2 in dom(the point-map of F) & C2 in K & B2 = (the
    point-map of F).C2 by FUNCT_1:def 6;
    then
A140: A2 in K by A112,A137,A138,A139,FUNCT_1:def 4;
    then
A141: ex A12 being Subset of X st A2 = A12 & card A12 = k & S c= A12 by A108;
    then
A142: card A2 = (k - 1) + 1;
A143: B2 = F.A2 by A138,A139;
    then A2 on L1 by A109,A136,A91;
    then
A144: A2 c= L1 by A1,A12,Th10;
    then A1 \/ A2 c= L1 by A126,XBOOLE_1:8;
    then
A145: card(A1 \/ A2) c= k + 1 by A103,CARD_1:11;
    k + 1 c= card(A1 \/ A2) by A131,A104,A134,A118,A138,A119,A139,A123,A141,Th1
;
    then card(A1 \/ A2) = (k - 1) + 2*1 by A145,XBOOLE_0:def 10;
    then
A146: card(A1 /\ A2) = (k + 1) - 2 by A4,A141,A124,Th2;
A147: A2 on L1 by A109,A136,A143,A91;
    A2 \/ A3 c= L1 by A144,A128,XBOOLE_1:8;
    then
A148: card(A2 \/ A3) c= k + 1 by A103,CARD_1:11;
    k + 1 c= card(A2 \/ A3) by A131,A134,A102,A138,A115,A139,A116,A141,A121,Th1
;
    then card(A2 \/ A3) = (k - 1) + 2*1 by A148,XBOOLE_0:def 10;
    then
A149: card(A2 /\ A3) = (k + 1) - 2 by A4,A121,A142,Th2;
A150: card A1 = (k + 1) - 1 by A123;
    consider L3a being LINE of G_(k,X) such that
A151: {A1,A2} on L3a by A11,A122,A140;
    card(A1 /\ A2) in k by A131,A104,A134,A118,A138,A119,A139,A123,A141,Th1;
    then
A152: card(A1 /\ A2) c= m by A5,A97,ORDINAL1:22;
    card(A1 /\ A2 /\ A3) c= card(A1 /\ A2) by CARD_1:11,XBOOLE_1:17;
    then
A153: card(A1 /\ A2 /\ A3) c= m by A152;
    S c= A1 /\ A2 by A123,A141,XBOOLE_1:19;
    then S c= (A1 /\ A2) /\ A3 by A121,XBOOLE_1:19;
    then m c= card(A1 /\ A2 /\ A3) by A107,A97,CARD_1:11;
    then
A154: k - 1 = card(A1 /\ A2 /\ A3) by A97,A153,XBOOLE_0:def 10;
    A1 on L3a by A151,INCSP_1:1;
    then
A155: A1 c= L3a by A1,A12,Th10;
    consider L3b being LINE of G_(k,X) such that
A156: {A2,A3} on L3b by A11,A140,A120;
    A3 on L3b by A156,INCSP_1:1;
    then
A157: A3 c= L3b by A1,A12,Th10;
    A2 on L3b by A156,INCSP_1:1;
    then
A158: A2 c= L3b by A1,A12,Th10;
    L3b in the Lines of G_(k,X);
    then
A159: ex l13b being Subset of X st L3b = l13b & card l13b = k + 1 by A14;
    2 + 1 <= k + 1 & 2 <= k + 1 by A1,A15,XREAL_1:6,XXREAL_0:2;
    then
A160: card(A1 \/ A2 \/ A3) = (k + 1) + 1 by A141,A121,A154,A146,A150,A149,A132
,A85,Th7;
A161: L3a <> L3b
    proof
      assume L3a = L3b;
      then A1 \/ A2 c= L3b by A155,A158,XBOOLE_1:8;
      then (A1 \/ A2) \/ A3 c= L3b by A157,XBOOLE_1:8;
      then Segm(k + 2) c= Segm(k + 1) by A160,A159,CARD_1:11;
      then k + 2 <= k + 1 by NAT_1:39;
      hence contradiction by XREAL_1:6;
    end;
    A1 on L1 by A109,A106,A125,A91;
    then {A1,A2} on L1 by A147,INCSP_1:1;
    then
A162: L1 = L3a by A131,A104,A134,A118,A138,A119,A139,A151,INCSP_1:def 10;
    A3 on L1 by A109,A111,A127,A91;
    then {A2,A3} on L1 by A147,INCSP_1:1;
    hence contradiction by A131,A134,A102,A138,A115,A139,A116,A156,A161,A162,
INCSP_1:def 10;
  end;
  F.:K is maximal_clique & F"K is maximal_clique by A3,A10,Th22;
  hence thesis by A1,A2,A84,A17,Th15;
end;
