reserve V,W for Z_Module;
reserve T for linear-transformation of V,W;

theorem ThLin4:
  for R being Ring
  for V being LeftMod of R,
      A being Subset of V holds A is Subset of Lin(A)
  proof
    let R be Ring;
    let V be LeftMod of R;
    let A be Subset of V;
    for x being object st x in A holds x in the carrier of Lin(A)
    proof
      let x be object such that
      A1: x in A;
      x in Lin(A) by A1,MOD_3:5;
      hence thesis;
    end;
    then A c= the carrier of Lin(A);
    hence thesis;
  end;
