reserve

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

theorem Th12:
  for k being Element of NAT for X being non empty set st 0 < k &
k + 1 c= card X holds for A1,A2,A3,A4,A5,A6 being POINT of G_(k,X) for L1,L2,L3
,L4 being LINE of G_(k,X) st A1 on L1 & A2 on L1 & A3 on L2 & A4 on L2 & A5 on
L1 & A5 on L2 & A1 on L3 & A3 on L3 & A2 on L4 & A4 on L4 & not A5 on L3 & not
A5 on L4 & L1 <> L2 & L3 <> L4 holds ex A6 being POINT of G_(k,X) st A6 on L3 &
  A6 on L4 & A6 = (A1 /\ A2) \/ (A3 /\ A4)
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;
A3: the Points of G_(k,X) = {A where A is Subset of X: card A = k} by A1,A2
,Def1;
A4: the Lines of G_(k,X) = {L where L is Subset of X: card L = k + 1} by A1,A2
,Def1;
  let A1,A2,A3,A4,A5,A6 be POINT of G_(k,X),L1,L2,L3,L4 be LINE of G_(k,X);
  assume that
A5: A1 on L1 and
A6: A2 on L1 and
A7: A3 on L2 and
A8: A4 on L2 and
A9: A5 on L1 and
A10: A5 on L2 and
A11: A1 on L3 and
A12: A3 on L3 and
A13: A2 on L4 and
A14: A4 on L4 and
A15: not A5 on L3 and
A16: not A5 on L4 and
A17: L1 <> L2 and
A18: L3 <> L4;
A19: A1 c= L1 & A2 c= L1 by A1,A2,A5,A6,Th10;
A20: A3 c= L2 & A4 c= L2 by A1,A2,A7,A8,Th10;
A21: A5 c= L1 & A5 c= L2 by A1,A2,A9,A10,Th10;
  A5 in the Points of G_(k,X);
  then
A22: ex B5 being Subset of X st B5 = A5 & card B5 = k by A3;
  A2 in the Points of G_(k,X);
  then
A23: ex B2 being Subset of X st B2 = A2 & card B2 = k by A3;
  then
A24: A2 is finite;
  reconsider k1= k - 1 as Element of NAT by A1,NAT_1:20;
  L3 in the Lines of G_(k,X);
  then
A25: ex K3 being Subset of X st K3 = L3 & card K3 = k + 1 by A4;
  then
A26: L3 is finite;
  L4 in the Lines of G_(k,X);
  then ex K4 being Subset of X st K4 = L4 & card K4 = k + 1 by A4;
  then card (L3 /\ L4) in Segm(k + 1) by A25,A18,Th1;
  then card (L3 /\ L4) in succ Segm k by NAT_1:38;
  then
A27: card (L3 /\ L4) c= k by ORDINAL1:22;
  A1 in the Points of G_(k,X);
  then
A28: ex B1 being Subset of X st B1 = A1 & card B1 = k by A3;
  then
A29: A1 is finite;
  G_(k,X) is Vebleian by A1,A2,Th11;
  then consider A6 being POINT of G_(k,X) such that
A30: A6 on L3 and
A31: A6 on L4 by A5,A6,A7,A8,A9,A10,A11,A12,A13,A14,A15,A16,A17;
  A6 in the Points of G_(k,X);
  then
A32: ex a6 being Subset of X st a6 = A6 & card a6 = k by A3;
  then
A33: A6 is finite;
A34: A6 c= L3 & A6 c= L4 by A1,A2,A30,A31,Th10;
  then A6 c= L3 /\ L4 by XBOOLE_1:19;
  then k c= card (L3 /\ L4) by A32,CARD_1:11;
  then card (L3 /\ L4) = k by A27,XBOOLE_0:def 10;
  then
A35: L3 /\ L4 = A6 by A32,A34,A26,CARD_2:102,XBOOLE_1:19;
  L2 in the Lines of G_(k,X);
  then
A36: ex K2 being Subset of X st K2 = L2 & card K2 = k + 1 by A4;
  then
A37: L2 is finite;
  A4 in the Points of G_(k,X);
  then
A38: ex B4 being Subset of X st B4 = A4 & card B4 = k by A3;
  then
A39: A4 is finite;
  L1 in the Lines of G_(k,X);
  then
A40: ex K1 being Subset of X st K1 = L1 & card K1 = k + 1 by A4;
  then
A41: L1 is finite;
  A3 in the Points of G_(k,X);
  then
A42: ex B3 being Subset of X st B3 = A3 & card B3 = k by A3;
  then
A43: A3 is finite;
A44: A3 c= L3 & A4 c= L4 by A1,A2,A12,A14,Th10;
  then
A45: A3 /\ A4 c= A6 by A35,XBOOLE_1:27;
A46: A1 c= L3 & A2 c= L4 by A1,A2,A11,A13,Th10;
  then
A47: A1 /\ A2 c= A6 by A35,XBOOLE_1:27;
  then
A48: (A1 /\ A2) \/ (A3 /\ A4) c= A6 by A45,XBOOLE_1:8;
A49: not A6 on L1 implies A6 = (A1 /\ A2) \/ (A3 /\ A4)
  proof
    assume
A50: not A6 on L1;
A51: not A6 on L2 implies A6 = (A1 /\ A2) \/ (A3 /\ A4)
    proof
A52:  A1 \/ A2 c= L1 by A19,XBOOLE_1:8;
      then
A53:  card(A1 \/ A2) c= k + 1 by A40,CARD_1:11;
A54:  A3 \/ A4 c= L2 by A20,XBOOLE_1:8;
      then
A55:  card(A3 \/ A4) c= k + 1 by A36,CARD_1:11;
A56:  card A3 = (k - 1) + 1 by A42;
      card ((A1 /\ A2) \/ (A3 /\ A4)) c= k by A32,A48,CARD_1:11;
      then card ((A1 /\ A2) \/ (A3 /\ A4)) in succ k by ORDINAL1:22;
      then card ((A1 /\ A2) \/ (A3 /\ A4)) in Segm(k1 + 1)
      or card ((A1 /\ A2) \/ (A3 /\
      A4)) = k by ORDINAL1:8;
      then card ((A1 /\ A2) \/ (A3 /\ A4)) in succ Segm k1 or
     card ((A1 /\ A2
      ) \/ (A3 /\ A4)) = k by NAT_1:38;
      then
A57:  card ((A1 /\ A2) \/ (A3 /\ A4)) c=Segm k1
      or card ((A1 /\ A2) \/ (
      A3 /\ A4)) = k by ORDINAL1:22;
A58:  card A1 = (k - 1) + 1 by A28;
      assume
A59:  not A6 on L2;
A60:  A1 <> A2 & A3 <> A4
      proof
        assume A1 = A2 or A3 = A4;
        then
        {A1,A6} on L3 & {A1,A6} on L4 or {A3,A6} on L3 & {A3,A6} on L4 by A11
,A12,A13,A14,A30,A31,INCSP_1:1;
        hence contradiction by A5,A7,A18,A50,A59,INCSP_1:def 10;
      end;
      then k + 1 c= card(A1 \/ A2) by A28,A23,Th1;
      then card (A1 \/ A2) = k1 + 2*1 by A53,XBOOLE_0:def 10;
      then
A61:  card (A1 /\ A2) = k1 by A23,A58,Th2;
      k + 1 c= card(A3 \/ A4) by A42,A38,A60,Th1;
      then card (A3 \/ A4) = k1 + 2*1 by A55,XBOOLE_0:def 10;
      then
A62:  card (A3 /\ A4) = k1 by A38,A56,Th2;
A63:  not card ((A1 /\ A2) \/ (A3 /\ A4)) = k - 1
      proof
A64:    A5 c= L1 /\ L2 by A21,XBOOLE_1:19;
A65:    (A1 /\ A2) /\ (A3 /\ A4) c= A1 /\ A2 by XBOOLE_1:17;
A66:    A1 /\ A2 /\ A3 /\ A4 = (A1 /\ A2) /\ (A3 /\ A4) by XBOOLE_1:16;
A67:    A1 /\ A2 c= A1 by XBOOLE_1:17;
        then
A68:    A1 = (A1 /\ A2 /\ A3 /\ A4) \/ (A1 \ (A1 /\ A2 /\ A3 /\ A4)) by A65,A66
,XBOOLE_1:1,45;
        assume
A69:    card ((A1 /\ A2) \/ (A3 /\ A4)) = k - 1;
        then card ((A1 /\ A2) \/ (A3 /\ A4)) = k1 + 2*0;
        then
A70:    card ((A1 /\ A2) /\ (A3 /\ A4)) = k1 by A61,A62,Th2;
        then
A71:    (A1 /\ A2) /\ (A3 /\ A4) = (A1 /\ A2) \/ (A3 /\ A4) by A29,A43,A69,
CARD_2:102,XBOOLE_1:29;
        then card (A1 \ (A1 /\ A2 /\ A3 /\ A4)) = k - (k - 1) by A28,A29,A69
,A67,A65,A66,CARD_2:44,XBOOLE_1:1;
        then consider x1 being object such that
A72:    A1 \ (A1 /\ A2 /\ A3 /\ A4) = {x1} by CARD_2:42;
A73:    A1 /\ A2 c= A2 by XBOOLE_1:17;
        then
A74:    A2 = (A1 /\ A2 /\ A3 /\ A4) \/ (A2 \ (A1 /\ A2 /\ A3 /\ A4)) by A65,A66
,XBOOLE_1:1,45;
        card (A2 \ (A1 /\ A2 /\ A3 /\ A4)) = k - (k - 1) by A23,A24,A69,A73,A65
,A71,A66,CARD_2:44,XBOOLE_1:1;
        then consider x2 being object such that
A75:    A2 \ (A1 /\ A2 /\ A3 /\ A4) = {x2} by CARD_2:42;
        x1 in {x1} by TARSKI:def 1;
        then
A76:    not x1 in A1 /\ A2 /\ A3 /\ A4 by A72,XBOOLE_0:def 5;
A77:    (A1 /\ A2) /\ (A3 /\ A4) c= A3 /\ A4 by XBOOLE_1:17;
A78:    A3 /\ A4 c= A4 by XBOOLE_1:17;
        then
A79:    A4 = (A1 /\ A2 /\ A3 /\ A4) \/ (A4 \ (A1 /\ A2 /\ A3 /\ A4)) by A77,A66
,XBOOLE_1:1,45;
        card (A4 \ (A1 /\ A2 /\ A3 /\ A4)) = k - (k - 1) by A38,A39,A69,A78,A77
,A71,A66,CARD_2:44,XBOOLE_1:1;
        then consider x4 being object such that
A80:    A4 \ (A1 /\ A2 /\ A3 /\ A4) = {x4} by CARD_2:42;
A81:    A3 /\ A4 c= A3 by XBOOLE_1:17;
        then
A82:    A3 = (A1 /\ A2 /\ A3 /\ A4) \/ (A3 \ (A1 /\ A2 /\ A3 /\ A4)) by A77,A66
,XBOOLE_1:1,45;
        card (A3 \ (A1 /\ A2 /\ A3 /\ A4)) = k - (k - 1) by A42,A43,A69,A81,A77
,A71,A66,CARD_2:44,XBOOLE_1:1;
        then consider x3 being object such that
A83:    A3 \ (A1 /\ A2 /\ A3 /\ A4) = {x3} by CARD_2:42;
        k + 1 c= card(A3 \/ A4) & card(A3 \/ A4) c= k + 1 by A42,A38,A36,A60
,A54,Th1,CARD_1:11;
        then card(A3 \/ A4) = k + 1 by XBOOLE_0:def 10;
        then A3 \/ A4 = L2 by A36,A20,A37,CARD_2:102,XBOOLE_1:8;
        then
A84:    L2 = (A1 /\ A2 /\ A3 /\ A4) \/ ({x3} \/ {x4}) by A83,A80,A82,A79,
XBOOLE_1:5;
        then
A85:    L2 = (A1 /\ A2 /\ A3 /\ A4) \/ {x3,x4} by ENUMSET1:1;
A86:    x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4
        proof
          assume x1 = x3 or x1 = x4 or x2 = x3 or x2 = x4;
          then {A1,A5} on L1 & {A1,A5} on L2 or {A1,A5} on L1 & {A1,A5} on L2
or {A2,A5} on L1 & {A2,A5} on L2 or {A2,A5} on L1 & {A2,A5} on L2 by A5,A6,A7
,A8,A9,A10,A72,A75,A83,A80,A68,A74,A82,A79,INCSP_1:1;
          hence contradiction by A11,A13,A15,A16,A17,INCSP_1:def 10;
        end;
        x2 in {x2} by TARSKI:def 1;
        then
A87:    not x2 in A1 /\ A2 /\ A3 /\ A4 by A75,XBOOLE_0:def 5;
        k + 1 c= card(A1 \/ A2) & card(A1 \/ A2) c= k + 1 by A28,A23,A40,A60
,A52,Th1,CARD_1:11;
        then card(A1 \/ A2) = k + 1 by XBOOLE_0:def 10;
        then A1 \/ A2 = L1 by A40,A19,A41,CARD_2:102,XBOOLE_1:8;
        then
A88:    L1 = (A1 /\ A2 /\ A3 /\ A4) \/ ({x1} \/ {x2}) by A72,A75,A68,A74,
XBOOLE_1:5;
        then
A89:    L1 = (A1 /\ A2 /\ A3 /\ A4) \/ {x1,x2} by ENUMSET1:1;
A90:    L1 /\ L2 c= A1 /\ A2 /\ A3 /\ A4
        proof
          assume not L1 /\ L2 c= A1 /\ A2 /\ A3 /\ A4;
          then consider x being object such that
A91:      x in L1 /\ L2 and
A92:      not x in A1 /\ A2 /\ A3 /\ A4;
          x in L1 by A91,XBOOLE_0:def 4;
          then
A93:      x in {x1,x2} by A89,A92,XBOOLE_0:def 3;
          x in L2 by A91,XBOOLE_0:def 4;
          then x1 in L2 or x2 in L2 by A93,TARSKI:def 2;
          then x1 in {x3,x4} or x2 in {x3,x4} by A85,A76,A87,XBOOLE_0:def 3;
          hence contradiction by A86,TARSKI:def 2;
        end;
A94:    A1 /\ A2 /\ A3 /\ A4 c= L2 by A84,XBOOLE_1:10;
        A1 /\ A2 /\ A3 /\ A4 c= L1 by A88,XBOOLE_1:10;
        then A1 /\ A2 /\ A3 /\ A4 c= L1 /\ L2 by A94,XBOOLE_1:19;
        then L1 /\ L2 = A1 /\ A2 /\ A3 /\ A4 by A90,XBOOLE_0:def 10;
        then card Segm k c= card Segm k1 by A22,A70,A66,A64,CARD_1:11;
        then
A95:    k <= k1 by NAT_1:40;
        k1 <= k1 + 1 by NAT_1:11;
        then k = k - 1 by A95,XXREAL_0:1;
        hence contradiction;
      end;
      k - 1 c= card ((A1 /\ A2) \/ (A3 /\ A4)) by A61,CARD_1:11,XBOOLE_1:7;
      then card ((A1 /\ A2) \/ (A3 /\ A4)) = k by A57,A63,XBOOLE_0:def 10;
      hence thesis by A32,A33,A47,A45,CARD_2:102,XBOOLE_1:8;
    end;
    A6 on L2 implies A6 = (A1 /\ A2) \/ (A3 /\ A4)
    proof
      assume
A96:  A6 on L2;
A97:  A4 = A6
      proof
        assume
A98:    A4 <> A6;
        {A4,A6} on L2 & {A4,A6} on L4 by A8,A14,A31,A96,INCSP_1:1;
        hence contradiction by A10,A16,A98,INCSP_1:def 10;
      end;
      A3 = A6
      proof
        assume
A99:   A3 <> A6;
        {A3,A6} on L2 & {A3,A6} on L3 by A7,A12,A30,A96,INCSP_1:1;
        hence contradiction by A10,A15,A99,INCSP_1:def 10;
      end;
      hence thesis by A46,A35,A97,XBOOLE_1:12,27;
    end;
    hence thesis by A51;
  end;
  A6 on L1 implies A6 = (A1 /\ A2) \/ (A3 /\ A4)
  proof
    assume
A100: A6 on L1;
A101: A1 = A6
    proof
      assume
A102: A1 <> A6;
      {A1,A6} on L1 & {A1,A6} on L3 by A5,A11,A30,A100,INCSP_1:1;
      hence contradiction by A9,A15,A102,INCSP_1:def 10;
    end;
    A2 = A6
    proof
      assume
A103: A2 <> A6;
      {A2,A6} on L1 & {A2,A6} on L4 by A6,A13,A31,A100,INCSP_1:1;
      hence contradiction by A9,A16,A103,INCSP_1:def 10;
    end;
    hence thesis by A44,A35,A101,XBOOLE_1:12,27;
  end;
  hence thesis by A30,A31,A49;
end;
