
theorem Th20:
  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
  identifying_close_blocks
proof
  let F be Field;
  let V be finite-dimensional VectSp of F;
  let k be Nat such that
A1: 1 <= k and
A2: k < dim V;
  set S=PencilSpace(V,k);
  thus S is identifying_close_blocks
  proof
    let X,Y be Block of S;
    assume 2 c= card(X/\Y);
    then consider P,Q being object such that
A3: P in X/\Y & Q in X/\Y and
A4: P<>Q by PENCIL_1:2;
A5: P in X & Q in X by A3,XBOOLE_0:def 4;
A6: P in Y & Q in Y by A3,XBOOLE_0:def 4;
A7: S is non void by A1,A2,Th17;
    then consider W1,W2 being Subspace of V such that
A8: W1 is Subspace of W2 and
A9: dim W1+1=k & dim W2=k+1 and
A10: X=pencil(W1,W2,k) by Def4;
    the topology of S is non empty by A7;
    then X in the topology of S;
    then reconsider P,Q as Point of S by A5;
A11: S is non empty by A2,VECTSP_9:36;
    then ex P1 being strict Subspace of V st P1=P & dim P1 = k by
VECTSP_9:def 2;
    then reconsider P as strict Subspace of V;
    ex Q1 being strict Subspace of V st Q1=Q & dim Q1 = k by A11,VECTSP_9:def 2
;
    then reconsider Q as strict Subspace of V;
    consider U1,U2 being Subspace of V such that
A12: U1 is Subspace of U2 and
A13: dim U1+1=k & dim U2=k+1 and
A14: Y=pencil(U1,U2,k) by A7,Def4;
A15: (Omega).W2=P+Q by A4,A5,A9,A10,Th11
      .= (Omega).U2 by A4,A6,A13,A14,Th11;
    (Omega).W1=P/\Q by A4,A5,A9,A10,Th11
      .= (Omega).U1 by A4,A6,A13,A14,Th11;
    hence thesis by A8,A10,A12,A14,A15,Th6;
  end;
end;
