
theorem Th13:
  for F being Field for V being finite-dimensional VectSp of F for
W being Subspace of V for v being Vector of V st not v in W holds dim(W+Lin{v})
  =dim W + 1
proof
  let F be Field;
  let V be finite-dimensional VectSp of F;
  let W be Subspace of V;
  let v be Vector of V such that
A1: not v in W;
  the carrier of (Omega).(W/\Lin{v}) = {0.(W/\Lin{v})}
  proof
    thus the carrier of (Omega).(W/\Lin{v}) c= {0.(W/\Lin{v})}
    proof
      let a be object;
      assume a in the carrier of (Omega).(W/\Lin{v});
      then
A2:   a in (the carrier of W)/\the carrier of Lin{v} by VECTSP_5:def 2;
      then a in the carrier of Lin{v} by XBOOLE_0:def 4;
      then a in Lin{v};
      then consider e being Element of F such that
A3:   a=e*v by VECTSP10:3;
      a in the carrier of W by A2,XBOOLE_0:def 4;
      then
A4:   a in W;
      now
        assume e<>0.F;
        then v=e"*(e*v) by VECTSP_1:20;
        hence contradiction by A1,A4,A3,VECTSP_4:21;
      end;
      then a=0.V by A3,VECTSP_1:14;
      then a = 0.(W/\Lin{v}) by VECTSP_4:11;
      hence thesis by TARSKI:def 1;
    end;
    let a be object;
    assume a in {0.(W/\Lin{v})};
    then a=0.(W/\Lin{v}) by TARSKI:def 1;
    then a=0.V by VECTSP_4:11;
    then a in W/\Lin{v} by VECTSP_4:17;
    hence thesis;
  end;
  then (Omega).(W/\Lin{v})=(0).(W/\Lin{v}) by VECTSP_4:def 3;
  then
A5: dim(W/\Lin{v})=0 by VECTSP_9:29;
A6: dim(W+Lin{v}) + dim(W/\Lin{v}) = dim W + dim Lin{v} by VECTSP_9:32;
  v <> 0.V by A1,VECTSP_4:17;
  hence thesis by A5,A6,Th12;
end;
