reserve x, y, y1, y2 for object;
reserve V for Z_Module;
reserve W, W1, W2 for Submodule of V;
reserve u, v for VECTOR of V;
reserve i, j, k, n for Element of NAT;
reserve V,W for finite-rank free Z_Module;
reserve T for linear-transformation of V,W;

theorem LMTh441:
  for R being Ring
  for V, W being LeftMod of R, T being linear-transformation of V, W,
  A being Subset of V
  st A c= the carrier of (ker T)
  holds Lin(T .: A) = (0).W
  proof
    let R be Ring;
    let V, W be LeftMod of R,
    T be linear-transformation of V, W,
    A be Subset of V;
    assume
    A1: A c= the carrier of (ker T);
    per cases;
    suppose A = {};
      then T .: A = {}(the carrier of W);
      hence Lin(T .: A) = (0).W by MOD_3:6;
    end;
    suppose A <> {};
      then consider a be object such that
      A6: a in A by XBOOLE_0:def 1;
      A8: a in (ker T) by A1,A6;
      reconsider a as Vector of V by A6;
      A9: T.a = 0.W by A8,RANKNULL:10;
      for x being object holds x in T.:A iff x in {0.W}
      proof
        let x be object;
        hereby
          assume x in T.:A;
          then consider z be Element of V such that
          A4: z in A & x=T.z by FUNCT_2:65;
          z in ker T by A1,A4;
          then T.z = 0.W by RANKNULL:10;
          hence x in {0.W} by TARSKI:def 1,A4;
        end;
        assume x in {0.W};
        then x = T.a by A9,TARSKI:def 1;
        hence x in T.:A by FUNCT_2:35,A6;
      end;
      then T.:A = {0.W} by TARSKI:2;
      hence Lin(T .: A) = (0).W by ZMODUL06:22;
    end;
  end;
