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 LMTh44:
  for R being Ring
  for V, W being LeftMod of R,
  T being linear-transformation of V, W,
  A, B, X being Subset of V
  st A c= the carrier of (ker T) & X = B \/ A
  holds Lin(T .: X) = Lin(T.: B)
  proof
    let R be Ring;
    let V, W be LeftMod of R,
    T be linear-transformation of V, W,
    A, B, X be Subset of V;
    assume that
    A1: A c= the carrier of (ker T) and
    A2: X = B \/ A;
    P1: T .: X = (T.:B)  \/ (T.:A) by A2,RELAT_1:120;
    thus Lin(T .: X) = Lin(T.:B) + Lin(T.:A) by P1,MOD_3:12
    .= Lin(T.:B) + (0).W by LMTh441,A1
    .= Lin(T.:B) by VECTSP_5:9;
  end;
