
theorem Th10:
  for F being Field for V being finite-dimensional VectSp of F for
  k being Nat st 1 <= k & k < dim V holds k Pencils_of V is non empty
proof
  let F be Field;
  let V be finite-dimensional VectSp of F;
  let k be Nat;
  assume that
A1: 1 <= k and
A2: k < dim V;
  k+1 <= dim V by A2,NAT_1:13;
  then consider W2 being strict Subspace of V such that
A3: dim W2 = k+1 by VECTSP_9:35;
  k-'1<=k by NAT_D:35;
  then k-'1 <= dim W2 by A3,NAT_1:13;
  then consider W1 being strict Subspace of W2 such that
A4: dim W1 = k-'1 by VECTSP_9:35;
  reconsider W19=W1 as Subspace of V by VECTSP_4:26;
  dim W1+1=k by A1,A4,XREAL_1:235;
  then pencil(W19,W2,k) in k Pencils_of V by A3,Def4;
  hence thesis;
end;
