 reserve V for Z_Module;
 reserve W for Subspace of V;
 reserve v, u for Vector of V;
 reserve i for Element of INT.Ring;

theorem HM7:
  for R being Ring
  for X, Y be LeftMod of R, A be Subset of X,
      T be linear-transformation of X, Y
  st T is bijective holds
  T.:(the carrier of Lin(A)) = the carrier of Lin(T.: A)
  proof
    let R be Ring;
    let X, Y be LeftMod of R,
    A be Subset of X,
    T be linear-transformation of X, Y;
    assume AS1: T is bijective;
    X1: T.:(the carrier of Lin(A)) c= the carrier of Lin(T.: A) by ThTF3C;
    reconsider B = T.: A as Subset of Y;
    D1: dom T = the carrier of X by FUNCT_2:def 1;
    R1: rng T = the carrier of Y by FUNCT_2:def 3,AS1;
    consider K be linear-transformation of Y, X such that
    AS3: K= T" & K is bijective by HM1,AS1;
    K.: B = A by D1,AS1,AS3,FUNCT_1:107; then
    X3: T.: (K.:(the carrier of Lin(B))) c= T.: the carrier of Lin(A)
    by RELAT_1:123,ThTF3C;
    X4: the carrier of Lin(B) c= rng T by R1,VECTSP_4:def 2;
    T.:(K.: the carrier of Lin(B)) = T.:(T" the carrier of Lin(B))
    by FUNCT_1:85,AS1,AS3
    .= the carrier of Lin(B) by X4,FUNCT_1:77;
    hence thesis by X1,XBOOLE_0:def 10,X3;
  end;
