
theorem Th18:
  for F being Field for V being finite-dimensional VectSp of F for
  k being Nat st 1 <= k & k < dim V & 3 <= dim V holds PencilSpace(V,k) is non
  degenerated
proof
  let F be Field;
  let V be finite-dimensional VectSp of F;
  let k be Nat such that
A1: 1 <= k and
A2: k < dim V and
A3: 3 <= dim V;
  set S=PencilSpace(V,k);
  now
    let B be Block of S;
    the topology of S is non empty by A1,A2,Th10;
    then consider W1,W2 being Subspace of V such that
A4: W1 is Subspace of W2 and
A5: dim W1+1=k and
A6: dim W2=k+1 and
A7: B=pencil(W1,W2,k) by Def4;
A8: the carrier of W1 c= the carrier of V by VECTSP_4:def 2;
    per cases by A2,A3,Th1;
    suppose
      k+1 < dim V;
      then
A9:   (Omega).W2 <> (Omega).V by A6,VECTSP_9:28;
A10:  now
        assume
A11:    the carrier of V = the carrier of W2;
        (Omega).W2 is Subspace of V by Th4;
        hence contradiction by A9,A11,VECTSP_4:29;
      end;
      the carrier of W2 c= the carrier of V by VECTSP_4:def 2;
      then not the carrier of V c= the carrier of W2 by A10;
      then consider v being object such that
A12:  v in the carrier of V and
A13:  not v in the carrier of W2;
      reconsider v as Vector of V by A12;
      set X=W1+Lin{v};
A14:  now
        v in {v} by TARSKI:def 1;
        then v in Lin{v} by VECTSP_7:8;
        then v in X by VECTSP_5:2;
        then
A15:    v in the carrier of X;
        assume X in B;
        then X is Subspace of W2 by A7,Th8;
        then the carrier of X c= the carrier of W2 by VECTSP_4:def 2;
        hence contradiction by A13,A15;
      end;
      not v in W2 by A13;
      then dim X = k by A4,A5,Th13,VECTSP_4:8;
      hence the carrier of S <> B by A14,VECTSP_9:def 2;
    end;
    suppose
A16:  2 <= k & k+1>=dim V;
      set I = the Basis of W1;
      reconsider I1=I as finite Subset of W1;
      2-1 <= dim W1+1-1 by A5,A16,XREAL_1:9;
      then 1 <= card I1 by VECTSP_9:def 1;
      then I1 is non empty;
      then consider i being object such that
A17:  i in I;
      reconsider i as Vector of W1 by A17;
      reconsider J=I1\{i} as finite Subset of V by A8,XBOOLE_1:1;
      I is linearly-independent by VECTSP_7:def 3;
      then I\{i} is linearly-independent by VECTSP_7:1,XBOOLE_1:36;
      then reconsider JJ=I\{i} as linearly-independent Subset of V by
VECTSP_9:11;
      J c= the carrier of Lin J
      proof
        let a be object;
        assume a in J;
        then a in Lin J by VECTSP_7:8;
        hence thesis;
      end;
      then reconsider JJJ=JJ as linearly-independent Subset of Lin J by
VECTSP_9:12;
      Lin JJJ = Lin J by VECTSP_9:17;
      then
A18:  J is Basis of Lin J by VECTSP_7:def 3;
A19:  card I = dim W1 by VECTSP_9:def 1;
      set T = the Linear_Compl of W1;
A20:  V is_the_direct_sum_of W1,T by VECTSP_5:38;
      then
A21:  W1/\T=(0).V by VECTSP_5:def 4;
      k+1 <= dim V by A2,NAT_1:13;
      then dim V = k+1 by A16,XXREAL_0:1;
      then k+1 - (dim W1 + 1) = dim W1 + dim T - (dim W1 + 1) by A20,
VECTSP_9:34;
      then consider u,v being Vector of T such that
A22:  u<>v and
A23:  {u,v} is linearly-independent and
A24:  (Omega).T=Lin{u,v} by A5,VECTSP_9:31;
      the carrier of T c= the carrier of V & u in the carrier of T by
VECTSP_4:def 2;
      then reconsider u1=u,v1=v as Vector of V;
      reconsider Y={u,v} as linearly-independent Subset of V by A23,VECTSP_9:11
;
A25:  Y={u,v};
      Lin (I\{i}) is Subspace of Lin I by VECTSP_7:13,XBOOLE_1:36;
      then
A26:  Lin J is Subspace of W1 by VECTSP_9:17;
      the carrier of ((Lin J)/\Lin{u1,v1}) c= the carrier of (0).V
      proof
        let a be object;
        assume a in the carrier of ((Lin J)/\Lin{u1,v1});
        then
A27:    a in (Lin J)/\Lin{u1,v1};
        then a in Lin {u1,v1} by VECTSP_5:3;
        then a in Lin{u,v} by VECTSP_9:17;
        then a in the carrier of the ModuleStr of T by A24;
        then
A28:    a in T;
        a in Lin J by A27,VECTSP_5:3;
        then a in W1 by A26,VECTSP_4:8;
        then a in W1 /\ T by A28,VECTSP_5:3;
        hence thesis by A21;
      end;
      then
A29:  (0).V is Subspace of (Lin J)/\Lin{u1,v1} & (Lin J)/\Lin{u1,v1} is
      Subspace of (0).V by VECTSP_4:27,39;
      card J = card I1 - card{i} by A17,EULER_1:4
        .= dim W1 - 1 by A19,CARD_1:30;
      then dim Lin J = dim W1 - 1 by A18,VECTSP_9:def 1;
      then
A30:  dim ((Lin J)+Lin{u1,v1}) = dim W1 - 1 + 2 by A22,A25,A29,Th14,VECTSP_4:25
;
A31:  Lin I = (Omega).W1 by VECTSP_7:def 3;
A32:  i in W1;
      now
A33:    now
          reconsider IV=I as Subset of V by A8,XBOOLE_1:1;
          assume
A34:      i in (Lin J)+Lin{u1,v1};
          IV c= the carrier of (Lin J)+Lin{u1,v1}
          proof
            let a be object;
            {i} c= I by A17,ZFMISC_1:31;
            then
A35:        I\{i}\/{i}=I by XBOOLE_1:45;
            assume
A36:        a in IV;
            per cases by A36,A35,XBOOLE_0:def 3;
            suppose
              a in J;
              then
A37:          a in Lin J by VECTSP_7:8;
              then a in V by VECTSP_4:9;
              then reconsider o=a as Vector of V;
              o in (Lin J)+Lin{u1,v1} by A37,VECTSP_5:2;
              hence thesis;
            end;
            suppose
              a in {i};
              then a=i by TARSKI:def 1;
              hence thesis by A34;
            end;
          end;
          then Lin IV is Subspace of (Lin J)+Lin{u1,v1} by VECTSP_9:16;
          then Lin I is Subspace of (Lin J)+Lin{u1,v1} by VECTSP_9:17;
          then
A38:      W1 is Subspace of (Lin J)+Lin{u1,v1} by A31,Th5;
          the carrier of (W1/\Lin{u1,v1}) c= the carrier of (0).V
          proof
            let a be object;
            assume a in the carrier of (W1/\Lin{u1,v1});
            then
A39:        a in W1/\Lin{u1,v1};
            then a in Lin {u1,v1} by VECTSP_5:3;
            then a in Lin {u,v} by VECTSP_9:17;
            then a in the carrier of the ModuleStr of T by A24;
            then
A40:        a in T;
            a in W1 by A39,VECTSP_5:3;
            then a in W1 /\ T by A40,VECTSP_5:3;
            hence thesis by A21;
          end;
          then (0).V is Subspace of W1/\Lin{u1,v1} & W1/\Lin{u1,v1} is
          Subspace of (0).V by VECTSP_4:27,39;
          then
A41:      dim (W1+Lin{u1,v1}) = dim W1 + 2 by A22,A25,Th14,VECTSP_4:25;
          Lin{u1,v1} is Subspace of (Lin J)+Lin{u1,v1} by VECTSP_5:7;
          then W1+Lin{u1,v1} is Subspace of (Lin J)+Lin{u1,v1} by A38,
VECTSP_5:34;
          then dim W1 + 1+1 <= dim W1 + 1 by A30,A41,VECTSP_9:25;
          hence contradiction by NAT_1:13;
        end;
        assume (Lin J)+Lin{u1,v1} in B;
        then W1 is Subspace of (Lin J)+Lin{u1,v1} by A7,Th8;
        hence contradiction by A32,A33,VECTSP_4:8;
      end;
      hence the carrier of S <> B by A5,A30,VECTSP_9:def 2;
    end;
  end;
  then not the carrier of S is Block of S;
  hence S is non degenerated;
end;
