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;
