reserve K for Ring,
  V1,W1 for VectSp of K;
reserve F for Field,
  V,W for VectSp of F;
reserve T for linear-transformation of V,W;

theorem Th18:
  for F being Ring, V, W being VectSp of F
  for A being Subset of V, x,y being Element of V st x - y in Lin
  A holds x in Lin (A \/ {y})
proof
  let F be Ring, V, W be VectSp of F;
  let A be Subset of V, x,y be Element of V such that
A1: x - y in Lin A;
A2: Lin (A \/ {y}) = (Lin A) + (Lin {y}) by VECTSP_7:15;
  y in {y} by TARSKI:def 1;
  then
A3: y in Lin ({y}) by VECTSP_7:8;
  (x - y) + y = x - (y - y) by RLVECT_1:29
    .= x - 0.V by VECTSP_1:19
    .= x by RLVECT_1:13;
  hence thesis by A1,A3,A2,VECTSP_5:1;
end;
