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 LMFirst3:
  for R being Ring
  for V, U, W being LeftMod of R,
  f being linear-transformation of V, U,
  g being linear-transformation of U, W holds
  the carrier of ker (g*f) = f"(the carrier of ker g)
  proof
    let R be Ring;
    let V, U, W be LeftMod of R,
    f be linear-transformation of V, U,
    g be linear-transformation of U, W;
    thus the carrier of ker (g*f) = (g*f)"{0.W} by LMFirst2
    .= f"(g"{0.W}) by RELAT_1:146
    .= f"(the carrier of ker g) by LMFirst2;
  end;
