reserve V,W for Z_Module;
reserve T for linear-transformation of V,W;
reserve T for linear-transformation of V,W;
reserve l for Linear_Combination of V;
reserve V,W for Z_Module;
reserve l for Linear_Combination of V;
reserve T for linear-transformation of V,W;

theorem Th38:
  for R being Ring
  for V,W being LeftMod of R
  for l being Linear_Combination of V,
      T being linear-transformation of V,W
  for G being FinSequence of V st rng G = Carrier l &
    T | (Carrier l) is one-to-one holds T*(l (#) G) = (T@*l) (#) (T*G)
  proof
    let R be Ring;
    let V,W be LeftMod of R;
    let l be Linear_Combination of V,
        T be linear-transformation of V,W;
    let G be FinSequence of V such that
    A1: rng G = Carrier l and
    A2: T | (Carrier l) is one-to-one;
    reconsider R = (T@*l) (#) (T*G) as FinSequence of W;
    reconsider L = T*(l (#) G) as FinSequence of W;
    A3: len R = len (T*G) by VECTSP_6:def 5
    .= len G by FINSEQ_2:33;
    A4: len L = len(l (#) G) by FINSEQ_2:33
    .= len G by VECTSP_6:def 5;
    for k being Nat st 1 <= k & k <= len L holds L.k = R.k
    proof
      A5: dom R = Seg len G by A3,FINSEQ_1:def 3;
      let k be Nat such that
      A6: 1 <= k & k <= len L;
      reconsider gk = G/.k as Element of V;
      len (l (#) G) = len G by VECTSP_6:def 5; then
      A7: dom (l (#) G) = Seg len G by FINSEQ_1:def 3;
      A8: k in dom (l (#) G) by A4,A6,A7; then
      A9: k in dom G by A7,FINSEQ_1:def 3; then
      A10: G.k = G/.k by PARTFUN1:def 6;
      then reconsider Gk = G.k as Element of V;
      (T*G).k = T.Gk by A9,FUNCT_1:13;
      then reconsider TGk = (T*G).k as Element of W;
      (l (#) G).k = (l.gk)*gk by A8,VECTSP_6:def 5; then
      A11: L.k = T.((l.gk)*gk) by A8,FUNCT_1:13
      .= (l.gk)*(T.gk) by MOD_2:def 2
      .= (l.Gk)*TGk by A9,A10,FUNCT_1:13;
      G.k in rng G & (T*G).k = T.(G.k) by A9,FUNCT_1:3,13; then
      A12: (T@*l).((T*G).k) = l.(G.k) by A1,A2,Th37;
      dom T = [#]V by RANKNULL:7;
      then dom (T*G) = dom G by A1,RELAT_1:27;
      then (T*G)/.k = (T*G).k by A9,PARTFUN1:def 6;
      hence thesis by A5,A7,A8,A11,A12,VECTSP_6:def 5;
    end;
    hence thesis by A3,A4;
  end;
