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 Th20:
  for A being Subset of V st A is linearly-independent holds A is
  Basis of Lin A
proof
  let A be Subset of V such that
A1: A is linearly-independent;
  A c= [#](Lin A)
  proof
    let x be object such that
A2: x in A;
    reconsider x as Element of V by A2;
    x in Lin A by A2,VECTSP_7:8;
    hence thesis;
  end;
  then reconsider B = A as Subset of Lin A;
A3: Lin B = Lin A by VECTSP_9:17;
  B is linearly-independent by A1,VECTSP_9:12;
  hence thesis by A3,VECTSP_7:def 3;
end;
