
theorem Th16:
  for F being Field for V being finite-dimensional VectSp of F for
W1,W2 being Subspace of V st W1 is Subspace of W2 for k being Nat st dim W1+1=k
  & dim W2=k+1 holds pencil(W1,W2,k) is non trivial
proof
  let F be Field;
  let V be finite-dimensional VectSp of F;
  let W1,W2 be Subspace of V;
  assume W1 is Subspace of W2;
  then reconsider U=W1 as Subspace of W2;
  let k be Nat such that
A1: dim W1+1=k & dim W2=k+1;
  set W = the Linear_Compl of U;
A2: W2 is_the_direct_sum_of W,U by VECTSP_5:def 5;
  then
A3: W /\ U = (0).W2 by VECTSP_5:def 4;
  dim W2 = dim U + dim W by A2,VECTSP_9:34;
  then consider u,v being Vector of W such that
A4: u <> v and
A5: {u,v} is linearly-independent and
  (Omega).W = Lin{u,v} by A1,VECTSP_9:31;
A6: now
    assume v in Lin{u};
    then ex a being Element of F st v = a*u by VECTSP10:3;
    hence contradiction by A4,A5,VECTSP_7:5;
  end;
  reconsider u,v as Vector of W2 by VECTSP_4:10;
  reconsider u1=u,v1=v as Vector of V by VECTSP_4:10;
A7: v in W;
A8: now
    assume v in W1;
    then v in (0).W2 by A3,A7,VECTSP_5:3;
    then v=0.W2 by VECTSP_4:35;
    then v=0.W by VECTSP_4:11;
    hence contradiction by A5,VECTSP_7:4;
  end;
A9: u in W;
A10: now
    assume u in W1;
    then u in (0).W2 by A3,A9,VECTSP_5:3;
    then u=0.W2 by VECTSP_4:35;
    then u=0.W by VECTSP_4:11;
    hence contradiction by A5,VECTSP_7:4;
  end;
  v in {v1} by TARSKI:def 1;
  then v in Lin{v1} by VECTSP_7:8;
  then
A11: v in W1+Lin{v1} by VECTSP_5:2;
A12: not v in Lin{u} by A6,VECTSP10:13;
A13: now
    reconsider Wx=W as Subspace of V by VECTSP_4:26;
    assume W1+Lin{v1} = W1+Lin{u1};
    then consider h,j being Vector of V such that
A14: h in W1 and
A15: j in Lin{u1} and
A16: v1 = h+j by A11,VECTSP_5:1;
    consider a being Element of F such that
A17: j=a*u1 by A15,VECTSP10:3;
    v1-a*u1=h+(a*u1-a*u1) by A16,A17,RLVECT_1:def 3;
    then v1-a*u1=h+0.V by RLVECT_1:15;
    then
A18: v1-a*u1=h by RLVECT_1:4;
    a*u = a*u1 by VECTSP_4:14;
    then
A19: v1-a*u1 = v-a*u by VECTSP_4:16;
    a*u in Wx by A9,VECTSP_4:21;
    then v-a*u in Wx by A7,VECTSP_4:23;
    then v-a*u in W/\U by A14,A18,A19,VECTSP_5:3;
    then v-a*u = 0.W2 by A3,VECTSP_4:35;
    then h = 0.V by A18,A19,VECTSP_4:11;
    then v1 = j by A16,RLVECT_1:4;
    hence contradiction by A12,A15,VECTSP10:13;
  end;
  v in W2;
  then
A20: W1+Lin{v1} in pencil(W1,W2,k) by A1,A8,Th15;
  u in W2;
  then W1+Lin{u1} in pencil(W1,W2,k) by A1,A10,Th15;
  then 2 c= card pencil(W1,W2,k) by A13,A20,PENCIL_1:2;
  hence thesis by PENCIL_1:4;
end;
