reserve K for Ring,
  V1,W1 for VectSp of K;
reserve F for Field,
  V,W for VectSp of F;
reserve T for linear-transformation of V,W;
reserve l for Linear_Combination of V;

theorem Th38:
  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 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 Th7;
    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 A7,A8,A11,A5,A12,VECTSP_6:def 5;
  end;
  hence thesis by A4,A3;
end;
