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 LMFirst4:
  for R being Ring
  for V, W being LeftMod of R, f being linear-transformation of V, W
  st f is onto holds
  im f = (Omega).W
  proof
    let R be Ring;
    let V, W be LeftMod of R, f be linear-transformation of V, W;
    assume f is onto; then
    B1: rng f = the carrier of W by FUNCT_2:def 3;
    B2: dom f = the carrier of V by FUNCT_2:def 1;
    the carrier of im f = [#]im f
    .= f .: [#]V by RANKNULL:def 2
    .= the carrier of (Omega).W by B1,B2,RELAT_1:113;
    hence thesis by VECTSP_4:29;
  end;
