reserve

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

theorem Th11:
  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 Vebleian
proof
  let k be Element of NAT;
  let X be non empty set;
  k <= k + 1 by NAT_1:11;
  then
A1: Segm k c= Segm(k + 1) by NAT_1:39;
  assume
A2: 0 < k & k + 1 c= card X;
  then
A3: the Points of G_(k,X) = {A where A is Subset of X: card A = k} by 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
A4: A1 on L1 and
A5: A2 on L1 and
A6: A3 on L2 and
A7: A4 on L2 and
A8: A5 on L1 & A5 on L2 and
A9: A1 on L3 and
A10: A3 on L3 and
A11: A2 on L4 and
A12: A4 on L4 and
A13: ( not A5 on L3)& not A5 on L4 and
A14: L1 <> L2;
A15: A2 c= L1 & A4 c= L2 by A2,A5,A7,Th10;
A16: A1 <> A3 & A2 <> A4
  proof
    assume A1 = A3 or A2 = A4;
    then
    {A1,A5} on L1 & {A1,A5} on L2 or {A2,A5} on L1 & {A2,A5} on L2 by A4,A5,A6
,A7,A8,INCSP_1:1;
    hence contradiction by A9,A11,A13,A14,INCSP_1:def 10;
  end;
A17: the Lines of G_(k,X) = {L where L is Subset of X: card L = k + 1} by A2
,Def1;
  A5 c= L1 & A5 c= L2 by A2,A8,Th10;
  then
A18: A5 c= L1 /\ L2 by XBOOLE_1:19;
  A5 in the Points of G_(k,X);
  then ex B5 being Subset of X st B5 = A5 & card B5 = k by A3;
  then
A19: k c= card (L1 /\ L2) by A18,CARD_1:11;
  L2 in the Lines of G_(k,X);
  then
A20: ex K2 being Subset of X st K2 = L2 & card K2 = k + 1 by A17;
  A3 in the Points of G_(k,X);
  then
A21: ex B3 being Subset of X st B3 = A3 & card B3 = k by A3;
  A1 in the Points of G_(k,X);
  then ex B1 being Subset of X st B1 = A1 & card B1 = k by A3;
  then
A22: k + 1 c= card (A1 \/ A3) by A21,A16,Th1;
A23: A1 c= L1 & A3 c= L2 by A2,A4,A6,Th10;
  L3 in the Lines of G_(k,X);
  then
A24: ex K3 being Subset of X st K3 = L3 & card K3 = k + 1 by A17;
  then
A25: L3 is finite;
  A4 in the Points of G_(k,X);
  then
A26: ex B4 being Subset of X st B4 = A4 & card B4 = k by A3;
  A2 in the Points of G_(k,X);
  then ex B2 being Subset of X st B2 = A2 & card B2 = k by A3;
  then
A27: k + 1 c= card (A2 \/ A4) by A26,A16,Th1;
  L4 in the Lines of G_(k,X);
  then
A28: ex K4 being Subset of X st K4 = L4 & card K4 = k + 1 by A17;
  then
A29: L4 is finite;
A30: A2 c= L4 & A4 c= L4 by A2,A11,A12,Th10;
  then A2 \/ A4 c= L4 by XBOOLE_1:8;
  then card (A2 \/ A4) c= k + 1 by A28,CARD_1:11;
  then card (A2 \/ A4) = k + 1 by A27,XBOOLE_0:def 10;
  then A2 \/ A4 = L4 by A28,A30,A29,CARD_2:102,XBOOLE_1:8;
  then
A31: L4 c= L1 \/ L2 by A15,XBOOLE_1:13;
  L1 in the Lines of G_(k,X);
  then
A32: ex K1 being Subset of X st K1 = L1 & card K1 = k + 1 by A17;
  then card (L1 /\ L2) in Segm(k + 1) by A20,A14,Th1;
  then card (L1 /\ L2) in succ Segm k by NAT_1:38;
  then card (L1 /\ L2) c= k by ORDINAL1:22;
  then card (L1 /\ L2) = k by A19,XBOOLE_0:def 10;
  then
A33: card (L1 \/ L2) = k + 2*1 by A32,A20,Th2;
A34: A1 c= L3 & A3 c= L3 by A2,A9,A10,Th10;
  then A1 \/ A3 c= L3 by XBOOLE_1:8;
  then card (A1 \/ A3) c= k + 1 by A24,CARD_1:11;
  then card (A1 \/ A3) = k + 1 by A22,XBOOLE_0:def 10;
  then A1 \/ A3 = L3 by A24,A34,A25,CARD_2:102,XBOOLE_1:8;
  then L3 c= L1 \/ L2 by A23,XBOOLE_1:13;
  then L3 \/ L4 c= L1 \/ L2 by A31,XBOOLE_1:8;
  then card (L3 \/ L4) c= k + 2 by A33,CARD_1:11;
  then card (L3 \/ L4) in succ (k + 2) by ORDINAL1:22;
  then card (L3 \/ L4) in Segm(k + 1 + 1) or
      card (L3 \/ L4) = k + 2 by ORDINAL1:8;
  then card (L3 \/ L4) in succ Segm(k + 1) or
      card (L3 \/ L4) = k + 2 by NAT_1:38;
  then
A35: card (L3 \/ L4) c= k + 1 or card (L3 \/ L4) = k + 2 by ORDINAL1:22;
  k + 1 c= card (L3 \/ L4) by A24,CARD_1:11,XBOOLE_1:7;
  then card (L3 \/ L4) = k + 1 + 2*0 or card (L3 \/ L4) = k + 2*1 by A35,
XBOOLE_0:def 10;
  then k c= card (L3 /\ L4) by A24,A28,A1,Th2;
  then consider B6 being set such that
A36: B6 c= L3 /\ L4 and
A37: card B6 = k by CARD_FIL:36;
A38: L3 /\ L4 c= L3 by XBOOLE_1:17;
  then L3 /\ L4 c= X by A24,XBOOLE_1:1;
  then reconsider A6 = B6 as Subset of X by A36,XBOOLE_1:1;
A39: A6 in the Points of G_(k,X) by A3,A37;
  L3 /\ L4 c= L4 by XBOOLE_1:17;
  then
A40: B6 c= L4 by A36;
  reconsider A6 as POINT of G_(k,X) by A39;
  take B6;
  A6 c= B6 & B6 c= L3 by A36,A38;
  hence thesis by A2,A40,Th10;
end;
