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 LMFirst2:
  for R being Ring
  for V, W being LeftMod of R,
  f being linear-transformation of V, W holds
  the carrier of ker f = f"{0.W}
  proof
    let R be Ring;
    let V, W be LeftMod of R,
    f be linear-transformation of V, W;
    A0: [#](ker f) = { u where u is Element of V : f.u = 0.W }
    by RANKNULL:def 1;
    for x being object holds
    x in the carrier of ker f iff x in f"{0.W}
    proof
      let x be object;
      hereby
        assume x in the carrier of ker f;
        then consider v be Vector of V such that
        A2: x = v & f.v = 0.W by A0;
        f.x in {0.W} by A2,TARSKI:def 1;
        hence x in f"{0.W} by FUNCT_2:38,A2;
      end;
      assume A11: x in f"{0.W};
      then
      A1: x in the carrier of V & f.x in {0.W} by FUNCT_2:38;
      reconsider v = x as Vector of V by A11;
      f.v = 0.W by A1,TARSKI:def 1;
      hence x in the carrier of ker f by A0;
    end;
    hence thesis by TARSKI:2;
  end;
