
theorem Th19:
  for F being Field for V being finite-dimensional VectSp of F for
  k being Nat st 1 <= k & k < dim V holds PencilSpace(V,k) is
  with_non_trivial_blocks
proof
  let F be Field;
  let V be finite-dimensional VectSp of F;
  let k be Nat such that
A1: 1 <= k & k < dim V;
  set S=PencilSpace(V,k);
  thus S is with_non_trivial_blocks
  proof
    let X be Block of S;
    S is non void by A1,Th17;
    then
    ex W1,W2 being Subspace of V st W1 is Subspace of W2 & dim W1+1=k & dim
    W2=k+1 & X=pencil(W1,W2,k) by Def4;
    then X is non trivial by Th16;
    hence thesis by PENCIL_1:4;
  end;
end;
