
theorem Th14:
  for F being Field for V being finite-dimensional VectSp of F for
  W being Subspace of V for v,u being Vector of V st v<>u & {v,u} is
  linearly-independent & W/\Lin{v,u}=(0).V holds dim(W+Lin{v,u})=dim W + 2
proof
  let F be Field;
  let V be finite-dimensional VectSp of F;
  let W be Subspace of V;
  let v,u be Vector of V such that
A1: v<>u and
A2: {v,u} is linearly-independent and
A3: W/\Lin{v,u}=(0).V;
  u in {v,u} by TARSKI:def 2;
  then
A4: u in Lin{v,u} by VECTSP_7:8;
  v in {v,u} by TARSKI:def 2;
  then v in Lin{v,u} by VECTSP_7:8;
  then reconsider v1=v,u1=u as Vector of Lin{v,u} by A4;
  reconsider L={v1,u1} as linearly-independent Subset of Lin{v,u} by A2,
VECTSP_9:12;
  (Omega).Lin{v,u}=Lin L by VECTSP_9:17;
  then
A5: dim Lin{v,u}=2 by A1,VECTSP_9:31;
  (Omega).(W/\Lin{v,u})=(0).(W/\Lin{v,u}) by A3,VECTSP_4:36;
  then dim(W/\Lin{v,u}) = 0 by VECTSP_9:29;
  hence dim(W+Lin{v,u}) = dim(W+Lin{v,u}) + dim(W/\Lin{v,u})
    .= dim W + 2 by A5,VECTSP_9:32;
end;
