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
  for R being Ring
  for V, W being LeftMod of R,
      T being linear-transformation of V, W holds
  T = Zdecom(T) * ZQMorph(V, ker T)
  proof
    let R be Ring;
    let V, W be LeftMod of R;
    let T be linear-transformation of V, W;
    set g = Zdecom(T) * ZQMorph(V,ker T);
    A1: dom g = the carrier of V by FUNCT_2:def 1;
    the carrier of (im T) c= the carrier of W by VECTSP_4:def 2;
    then rng g c= the carrier of W;
    then reconsider g as Function of V, W by FUNCT_2:2,A1;
    for z being Element of V holds T.z = g.z
    proof
      let z be Element of V;
      thus T.z = (Zdecom(T)).((ZQMorph(V,ker T)).z) by defdecom
      .= g.z by FUNCT_2:15;
    end;
    hence thesis by FUNCT_2:def 8;
  end;
