reserve

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

theorem
  for k being Element of NAT for X being non empty set st 0 < k & k + 1
  c= card X holds G_(k,X) is Desarguesian
proof
  let k be Element of NAT;
  let X be non empty set;
  assume that
A1: 0 < k and
A2: k + 1 c= card X;
  let o,b1,a1,b2,a2,b3,a3,r,s,t be POINT of G_(k,X);
  let C1,C2,C3,A1,A2,A3,B1,B2,B3 be LINE of G_(k,X);
  assume that
A3: {o,b1,a1} on C1 and
A4: {o,a2,b2} on C2 and
A5: {o,a3,b3} on C3 and
A6: {a3,a2,t} on A1 and
A7: {a3,r,a1} on A2 and
A8: {a2,s,a1} on A3 and
A9: {t,b2,b3} on B1 and
A10: {b1,r,b3} on B2 and
A11: {b1,s,b2} on B3 and
A12: C1,C2,C3 are_mutually_distinct and
A13: o<>a1 and
A14: o<>a2 & o<>a3 and
A15: o<>b1 and
A16: o<>b2 & o<>b3 and
A17: a1<>b1 and
A18: a2<>b2 and
A19: a3<>b3;
A20: o on C1 by A3,INCSP_1:2;
A21: b2 on C2 by A4,INCSP_1:2;
  then
A22: b2 c= C2 by A1,A2,Th10;
A23: a2 on C2 by A4,INCSP_1:2;
  then
A24: {a2,b2} on C2 by A21,INCSP_1:1;
A25: o on C2 by A4,INCSP_1:2;
  then
A26: {o,b2} on C2 by A21,INCSP_1:1;
A27: {o,a2} on C2 by A25,A23,INCSP_1:1;
A28: a3 on A1 & a2 on A1 by A6,INCSP_1:2;
A29: b3 on B2 by A10,INCSP_1:2;
A30: b3 on C3 by A5,INCSP_1:2;
  then
A31: b3 c= C3 by A1,A2,Th10;
A32: a3 on C3 by A5,INCSP_1:2;
  then
A33: {a3,b3} on C3 by A30,INCSP_1:1;
A34: o on C3 by A5,INCSP_1:2;
  then
A35: {o,b3} on C3 by A30,INCSP_1:1;
A36: {o,a3} on C3 by A34,A32,INCSP_1:1;
A37: a3 on A2 & a1 on A2 by A7,INCSP_1:2;
A38: b1 on B3 by A11,INCSP_1:2;
A39: C1 <> C3 by A12,ZFMISC_1:def 5;
A40: b1 on B2 by A10,INCSP_1:2;
A41: C2 <> C3 by A12,ZFMISC_1:def 5;
A42: b3 on B1 by A9,INCSP_1:2;
A43: C1 <> C2 by A12,ZFMISC_1:def 5;
A44: b2 on B1 by A9,INCSP_1:2;
A45: a1 on C1 by A3,INCSP_1:2;
  then
A46: a1 c= C1 by A1,A2,Th10;
A47: b1 on C1 by A3,INCSP_1:2;
  then
A48: {o,b1} on C1 by A20,INCSP_1:1;
A49: b2 on B3 by A11,INCSP_1:2;
A50: {a1,b1} on C1 by A47,A45,INCSP_1:1;
A51: not a1 on B2 & not a2 on B3 & not a3 on B1
  proof
    assume a1 on B2 or a2 on B3 or a3 on B1;
    then {a1,b1} on B2 or {a2,b2} on B3 or {a3,b3} on B1 by A42,A40,A49,
INCSP_1:1;
    then
    b3 on C1 or b1 on C2 or b2 on C3 by A17,A18,A19,A44,A29,A38,A50,A24,A33,
INCSP_1:def 10;
    then {o,b3} on C1 or {o,b1} on C2 or {o,b2} on C3 by A20,A25,A34,INCSP_1:1;
    hence contradiction by A15,A16,A48,A26,A35,A43,A41,A39,INCSP_1:def 10;
  end;
A52: s on A3 & s on B3 by A8,A11,INCSP_1:2;
A53: t on B1 by A9,INCSP_1:2;
A54: r on A2 & r on B2 by A7,A10,INCSP_1:2;
A55: t on A1 by A6,INCSP_1:2;
A56: a1 on A3 by A8,INCSP_1:2;
A57: a2 on A3 by A8,INCSP_1:2;
A58: the Lines of G_(k,X) = {L where L is Subset of X: card L = k + 1} by A1,A2
,Def1;
A59: {o,a1} on C1 by A20,A45,INCSP_1:1;
A60: not o on A1 & not o on B1 & not o on A2 & not o on B2 & not o on A3 &
  not o on B3
  proof
    assume o on A1 or o on B1 or o on A2 or o on B2 or o on A3 or o on B3;
    then {o,a2} on A1 & {o,a3} on A1 or {o,b2} on B1 & {o,b3} on B1 or {o,a1}
on A2 & {o,a3} on A2 or {o,b1} on B2 & {o,b3} on B2 or {o,a2} on A3 & {o,a1} on
A3 or {o,b2} on B3 & {o,b1} on B3 by A28,A37,A57,A56,A44,A42,A40,A29,A38,A49,
INCSP_1:1;
    then A1 = C2 & A1 = C3 or B1 = C2 & B1 = C3 or A2 = C1 & A2 = C3 or B2 =
C1 & B2 = C3 or A3 = C2 & A3 = C1 or B3 = C2 & B3 = C1 by A13,A14,A15,A16,A59
,A27,A36,A48,A26,A35,INCSP_1:def 10;
    hence contradiction by A12,ZFMISC_1:def 5;
  end;
  then consider salpha being POINT of G_(k,X) such that
A61: salpha on A3 & salpha on B3 and
A62: salpha = (a1 /\ b1) \/ (a2 /\ b2) by A1,A2,A20,A47,A45,A25,A23,A21,A57,A56
,A38,A49,A43,A51,Th12;
  consider ralpha being POINT of G_(k,X) such that
A63: ralpha on B2 & ralpha on A2 and
A64: ralpha = (a1 /\ b1) \/ (a3 /\ b3) by A1,A2,A20,A47,A45,A34,A32,A30,A37,A40
,A29,A39,A51,A60,Th12;
A65: (a1 /\ b1) \/ (a3 /\ b3) \/ (a2 /\ b2) = (a1 /\ b1) \/ ((a3 /\ b3) \/
  ( a2 /\ b2)) by XBOOLE_1:4;
  consider talpha being POINT of G_(k,X) such that
A66: talpha on A1 & talpha on B1 and
A67: talpha = (a2 /\ b2) \/ (a3 /\ b3) by A1,A2,A25,A23,A21,A34,A32,A30,A28,A44
,A42,A41,A51,A60,Th12;
A68: A1 <> B1 & A2 <> B2 by A6,A7,A51,INCSP_1:2;
A69: r = ralpha & s = salpha & t = talpha
  proof
A70: {s,salpha} on A3 & {s,salpha} on B3 by A52,A61,INCSP_1:1;
A71: {r,ralpha} on A2 & {r,ralpha} on B2 by A54,A63,INCSP_1:1;
    assume
A72: r <> ralpha or s <> salpha or t <> talpha;
    {t,talpha} on A1 & {t,talpha} on B1 by A55,A53,A66,INCSP_1:1;
    hence contradiction by A57,A51,A68,A72,A71,A70,INCSP_1:def 10;
  end;
  then r \/ s = (a3 /\ b3) \/ (a1 /\ b1) \/ (a1 /\ b1) \/ (a2 /\ b2) by A62,A64
,XBOOLE_1:4;
  then r \/ s = (a3 /\ b3) \/ ((a1 /\ b1) \/ (a1 /\ b1)) \/ (a2 /\ b2) by
XBOOLE_1:4;
  then
A73: r \/ s \/ t = (a1 /\ b1) \/ (a3 /\ b3) \/ (a2 /\ b2) by A67,A69,A65,
XBOOLE_1:7,12;
  a2 c= C2 by A1,A2,A23,Th10;
  then
A74: a2 \/ b2 c= C2 by A22,XBOOLE_1:8;
  r c= r \/ (s \/ t) by XBOOLE_1:7;
  then
A75: r c= r \/ s \/ t by XBOOLE_1:4;
  C1 in the Lines of G_(k,X);
  then consider C11 being Subset of X such that
A76: C11 = C1 & card C11 = k + 1 by A58;
  reconsider C1 as finite set by A76;
A77: b1 c= C1 by A1,A2,A47,Th10;
  then a1 \/ b1 c= C1 by A46,XBOOLE_1:8;
  then
A78: card(a1 \/ b1) c= k + 1 by A76,CARD_1:11;
A79: the Points of G_(k,X) = {A where A is Subset of X: card A = k} by A1,A2
,Def1;
  o in the Points of G_(k,X);
  then
A80: ex o1 being Subset of X st o1 = o & card o1 = k by A79;
  b1 in the Points of G_(k,X);
  then
A81: ex b11 being Subset of X st b11 = b1 & card b11 = k by A79;
  a3 in the Points of G_(k,X);
  then
A82: ex a13 being Subset of X st a13 = a3 & card a13 = k by A79;
  then
A83: card a3 = (k - 1) + 1;
  t in the Points of G_(k,X);
  then
A84: ex t1 being Subset of X st t1 = t & card t1 = k by A79;
  then
A85: t is finite;
  a2 in the Points of G_(k,X);
  then
A86: ex a12 being Subset of X st a12 = a2 & card a12 = k by A79;
  then
A87: card a2 = (k - 1) + 1;
  s in the Points of G_(k,X);
  then
A88: ex s1 being Subset of X st s1 = s & card s1 = k by A79;
  then
A89: s is finite;
  a1 in the Points of G_(k,X);
  then
A90: ex a11 being Subset of X st a11 = a1 & card a11 = k by A79;
  then k + 1 c= card(a1 \/ b1) by A81,A17,Th1;
  then
A91: card(a1 \/ b1) = (k - 1) + 2*1 by A78,XBOOLE_0:def 10;
A92: k - 1 is Element of NAT by A1,NAT_1:20;
  C2 in the Lines of G_(k,X);
  then ex C12 being Subset of X st C12 = C2 & card C12 = k + 1 by A58;
  then
A93: card(a2 \/ b2) c= k + 1 by A74,CARD_1:11;
  b2 in the Points of G_(k,X);
  then
A94: ex b12 being Subset of X st b12 = b2 & card b12 = k by A79;
  then k + 1 c= card(a2 \/ b2) by A86,A18,Th1;
  then
A95: card(a2 \/ b2) = (k - 1) + 2*1 by A93,XBOOLE_0:def 10;
  then
A96: card(a2 /\ b2) = k - 1 by A92,A94,A87,Th2;
A97: card(a2 /\ b2) = (k - 2) + 1 by A92,A94,A95,A87,Th2;
A98: card a1 = (k - 1) + 1 by A90;
  then
A99: card(a1 /\ b1) = k - 1 by A92,A81,A91,Th2;
  a3 c= C3 by A1,A2,A32,Th10;
  then
A100: a3 \/ b3 c= C3 by A31,XBOOLE_1:8;
  s c= s \/ (r \/ t) by XBOOLE_1:7;
  then
A101: s c= r \/ s \/ t by XBOOLE_1:4;
  0 + 1 < k + 1 by A1,XREAL_1:8;
  then 1 <= (k - 1) + 1 by NAT_1:13;
  then 1 <= k - 1 or k = 1 by A92,NAT_1:8;
  then
A102: 1 + 1 <= (k - 1) + 1 or k = 1 by XREAL_1:6;
A103: o c= C1 by A1,A2,A20,Th10;
A104: not k = 1
  proof
    assume
A105: k = 1;
    then consider z1 being object such that
A106: o = {z1} by A80,CARD_2:42;
    consider z3 being object such that
A107: b1 = {z3} by A81,A105,CARD_2:42;
    consider z2 being object such that
A108: a1 = {z2} by A90,A105,CARD_2:42;
    o \/ a1 c= C1 by A103,A46,XBOOLE_1:8;
    then (o \/ a1) \/ b1 c= C1 by A77,XBOOLE_1:8;
    then {z1,z2} \/ b1 c= C1 by A106,A108,ENUMSET1:1;
    then
A109: {z1,z2,z3} c= C1 by A107,ENUMSET1:3;
    card{z1,z2,z3} = 3 by A13,A15,A17,A106,A108,A107,CARD_2:58;
    then Segm 3 c= Segm card C1 by A109,CARD_1:11;
    hence contradiction by A76,A105,NAT_1:39;
  end;
  then
A110: k - 2 is Element of NAT by A102,NAT_1:21;
  C3 in the Lines of G_(k,X);
  then ex C13 being Subset of X st C13 = C3 & card C13 = k + 1 by A58;
  then
A111: card(a3 \/ b3) c= k + 1 by A100,CARD_1:11;
  b3 in the Points of G_(k,X);
  then
A112: ex b13 being Subset of X st b13 = b3 & card b13 = k by A79;
  then k + 1 c= card(a3 \/ b3) by A82,A19,Th1;
  then
A113: card(a3 \/ b3) = (k - 1) + 2*1 by A111,XBOOLE_0:def 10;
  then
A114: card(a3 /\ b3) = k - 1 by A92,A112,A83,Th2;
  r in the Points of G_(k,X);
  then
A115: ex r1 being Subset of X st r1 = r & card r1 = k by A79;
  then r \/ s c= X by A88,XBOOLE_1:8;
  then
A116: r \/ s \/ t c= X by A84,XBOOLE_1:8;
A117: card(a3 /\ b3) = (k - 2) + 1 by A92,A112,A113,A83,Th2;
A118: card(a1 /\ b1) = (k - 2) + 1 by A92,A81,A91,A98,Th2;
  card((a1 /\ b1) \/ (a2 /\ b2)) = (k - 2) + 2*1 by A62,A69,A88;
  then
A119: card((a1 /\ b1) /\ (a2 /\ b2)) = k - 2 by A110,A118,A97,Th2;
  card((a2 /\ b2) \/ (a3 /\ b3)) = (k - 2) + 2*1 by A67,A69,A84;
  then
A120: card((a2 /\ b2) /\ (a3 /\ b3)) = k - 2 by A110,A97,A117,Th2;
  card((a1 /\ b1) \/ (a3 /\ b3)) = (k - 2) + 2*1 by A64,A69,A115;
  then
A121: card((a1 /\ b1) /\ (a3 /\ b3)) = k - 2 by A110,A118,A117,Th2;
A122: t c= r \/ s \/ t by XBOOLE_1:7;
A123: k = 2 implies ex O being LINE of G_(k,X) st {r,s,t} on O
  proof
    assume k = 2;
    then
    card(r \/ s \/ t) = k + 1 by A99,A96,A114,A73,A121,A120,A119,Th7;
    then r \/ s \/ t in the Lines of G_(k,X) by A58,A116;
    then consider O being LINE of G_(k,X) such that
A124: O = r \/ s \/ t;
A125: t on O by A1,A2,A122,A124,Th10;
    r on O & s on O by A1,A2,A75,A101,A124,Th10;
    then {r,s,t} on O by A125,INCSP_1:2;
    hence thesis;
  end;
A126: r is finite by A115;
A127: 3 <= k implies ex O being LINE of G_(k,X) st {r,s,t} on O
  proof
A128: card(r \/ s \/ t) = k + 1 implies ex O being LINE of G_(k,X) st {r,s
    ,t} on O
    proof
      assume card(r \/ s \/ t) = k + 1;
      then r \/ s \/ t in the Lines of G_(k,X) by A58,A116;
      then consider O being LINE of G_(k,X) such that
A129: O = r \/ s \/ t;
A130: t on O by A1,A2,A122,A129,Th10;
      r on O & s on O by A1,A2,A75,A101,A129,Th10;
      then {r,s,t} on O by A130,INCSP_1:2;
      hence thesis;
    end;
A131: card(r \/ s \/ t) = k implies ex O being LINE of G_(k,X) st {r,s,t} on O
    proof
      assume
A132: card(r \/ s \/ t) = k;
      then
A133: t = r \/ s \/ t by A84,A126,A89,A85,CARD_2:102,XBOOLE_1:7;
      r = r \/ s \/ t & s = r \/ s \/ t by A115,A88,A126,A89,A85,A75,A101,A132,
CARD_2:102;
      then {r,s,t} on A1 by A55,A133,INCSP_1:2;
      hence thesis;
    end;
    assume 3 <= k;
    hence thesis by A99,A96,A114,A73,A102,A121,A120,A119,A128,A131,Th7;
  end;
  k = 2 or 2 <= k - 1 by A92,A104,A102,NAT_1:8;
  then k = 2 or 2 + 1 <= (k - 1) + 1 by XREAL_1:6;
  hence thesis by A123,A127;
end;
