
theorem Th11:
  for F being Field for V being finite-dimensional VectSp of F for
W1,W2,P,Q being Subspace of V for k being Nat st dim W1+1=k & dim W2=k+1 & P in
  pencil(W1,W2,k) & Q in pencil(W1,W2,k) & P<>Q holds P/\Q = (Omega).W1 & P+Q =
  (Omega).W2
proof
  let F be Field;
  let V be finite-dimensional VectSp of F;
  let W1,W2,P0,Q0 be Subspace of V;
  let k be Nat such that
A1: dim W1+1=k and
A2: dim W2=k+1 and
A3: P0 in pencil(W1,W2,k) and
A4: Q0 in pencil(W1,W2,k) and
A5: P0<>Q0;
  consider Q being strict Subspace of V such that
A6: Q=Q0 and
A7: dim Q = k by A4,VECTSP_9:def 2;
A8: W1 is Subspace of Q by A4,A6,Th8;
  consider P being strict Subspace of V such that
A9: P=P0 and
A10: dim P = k by A3,VECTSP_9:def 2;
  W1 is Subspace of P by A3,A9,Th8;
  then
A11: W1 is Subspace of (P/\Q) by A8,VECTSP_5:19;
  P/\Q is Subspace of P by VECTSP_5:15;
  then
A12: dim(P/\Q)<=k by A10,VECTSP_9:25;
  per cases by A1,A12,A11,NAT_1:9,VECTSP_9:25;
  suppose
A13: dim W1 = dim (P/\Q);
    then (Omega).W1 = (Omega).(P/\Q) by A11,VECTSP_9:28;
    hence P0/\Q0 = (Omega).W1 by A9,A6;
    P is Subspace of W2 & Q is Subspace of W2 by A3,A4,A9,A6,Th8;
    then
A14: P+Q is Subspace of W2 by VECTSP_5:34;
    dim(P+Q)+dim W1-dim W1 = dim P + dim Q - dim W1 by A13,VECTSP_9:32;
    then (Omega).(W2) = (Omega).(P+Q) by A1,A2,A10,A7,A14,VECTSP_9:28;
    hence thesis by A9,A6;
  end;
  suppose
A15: dim (P/\Q)=k;
    P/\Q is Subspace of Q by VECTSP_5:15;
    then
A16: (Omega).(P/\Q) = (Omega).Q by A7,A15,VECTSP_9:28;
    P/\Q is Subspace of P by VECTSP_5:15;
    then (Omega).(P/\Q) = (Omega).P by A10,A15,VECTSP_9:28;
    hence thesis by A5,A9,A6,A16;
  end;
end;
