
theorem Th8:
  for F being Field for V being finite-dimensional VectSp of F for
k being Nat for W1,W2,W being Subspace of V st W in pencil(W1,W2,k) holds W1 is
  Subspace of W & W is Subspace of W2
proof
  let F be Field;
  let V be finite-dimensional VectSp of F;
  let k be Nat;
  let W1,W2,W be Subspace of V;
  assume
A1: W in pencil(W1,W2,k);
  then
A2: ex X being strict Subspace of V st W=X & dim X=k by VECTSP_9:def 2;
  W in pencil(W1,W2) by A1,XBOOLE_0:def 4;
  then
A3: W in segment(W1,W2) by XBOOLE_0:def 5;
  then W1 is Subspace of W2 by Def1;
  hence thesis by A2,A3,Def1;
end;
