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 Th21:
  for A being Subset of V, x being Element of V st x in Lin A &
  not x in A holds A \/ {x} is linearly-dependent
proof
  let A be Subset of V, x be Element of V such that
A1: x in Lin A and
A2: not x in A;
  per cases;
  suppose
A3: A is linearly-independent;
    x in [#](Lin A) by A1;
    then reconsider X = {x} as Subset of Lin A by SUBSET_1:41;
    reconsider A9 = A as Basis of Lin A by A3,Th20;
    reconsider B = A9 \/ X as Subset of Lin A;
    X misses A9
    proof
      assume X meets A9;
      then ex y being object st y in X & y in A9 by XBOOLE_0:3;
      hence contradiction by A2,TARSKI:def 1;
    end;
    then B is linearly-dependent by VECTSP_9:15;
    hence thesis by VECTSP_9:12;
  end;
  suppose
    A is linearly-dependent;
    hence thesis by VECTSP_7:1,XBOOLE_1:7;
  end;
end;
