
theorem Th15:
  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 for v being Vector of V st v in W2 & not v in W1 holds W1+Lin{v}
  in pencil(W1,W2,k)
proof
  let F be Field;
  let V be finite-dimensional VectSp of F;
  let W1,W2 be Subspace of V such that
A1: W1 is Subspace of W2;
  let k be Nat such that
A2: dim W1+1=k and
A3: dim W2=k+1;
  let v be Vector of V such that
A4: v in W2 and
A5: not v in W1;
  set W=W1+Lin{v};
A6: dim W = k by A2,A5,Th13;
  then
A7: W in k Subspaces_of V by VECTSP_9:def 2;
  v in the carrier of W2 by A4;
  then {v} c= the carrier of W2 by ZFMISC_1:31;
  then Lin{v} is Subspace of W2 by VECTSP_9:16;
  then W1 is Subspace of W & W is Subspace of W2 by A1,VECTSP_5:7,34;
  then
A8: W in segment(W1,W2) by A1,Def1;
  dim (Omega).W2 = k+1 by A3,VECTSP_9:27;
  then
A9: W <> (Omega).W2 by A6;
  dim (Omega).W1 + 1 = k by A2,VECTSP_9:27;
  then W <> (Omega).W1 by A6;
  then not W in {(Omega).W1,(Omega).W2} by A9,TARSKI:def 2;
  then W in pencil(W1,W2) by A8,XBOOLE_0:def 5;
  hence thesis by A7,XBOOLE_0:def 4;
end;
