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 ThTF3C3:
  for V,W be non empty set,
  f,g be V-valued FinSequence,
  l be Function of V,W
  holds l*(f^g) = (l*f)^(l*g)
  proof
    let V,W be non empty set,
        f,g be V-valued FinSequence,
        l be Function of V,W;
    A1: dom l = V by FUNCT_2:def 1;
    A2: rng f c= V;
    A3: rng g c= V;
    A4: rng(f^g) = rng f \/ rng g by FINSEQ_1:31;
    A5: dom(l*f) = dom f by A1,A2,RELAT_1:27
    .= Seg(len f) by FINSEQ_1:def 3; then
    A6: len(l*f) = len f by FINSEQ_1:def 3;
    dom(l*g) = dom g by A1,A3,RELAT_1:27
    .= Seg(len g) by FINSEQ_1:def 3; then
    A7: len(l*g) = len g by FINSEQ_1:def 3;
    A8: dom(f^g) = Seg(len(f^g)) by FINSEQ_1:def 3
    .= Seg(len f + len g) by FINSEQ_1:22;
    A9: len((l*f)^(l*g)) = len(l*f) + len(l*g) by FINSEQ_1:22
    .= len f + len g by A5,A7,FINSEQ_1:def 3;
    A10: dom((l*f)^(l*g)) = Seg(len f + len g) by A9,FINSEQ_1:def 3;
    now let k be object;
      assume
      A11: k in dom(l*(f^g));
      then reconsider i = k as Nat;
      A12: i in dom(f^g) by A1,A4,A11,RELAT_1:27;
      per cases by A12,FINSEQ_1:25;
      suppose
        A13: i in dom f; then
        A14: i in dom(l*f) by A1,A2,RELAT_1:27;
        thus (l*(f^g)).k = l.((f^g).i) by A11,FUNCT_1:12
        .= l.(f.i) by A13,FINSEQ_1:def 7
        .= (l*f).i by A13,FUNCT_1:13
        .= ((l*f)^(l*g)).k by A14,FINSEQ_1:def 7;
      end;
      suppose ex n be Nat st n in dom g & i = len f + n;
        then consider n be Nat such that
        A15: n in dom g & i = len f + n;
        A16: n in dom(l*g) by A1,A3,A15,RELAT_1:27;
        thus (l*(f^g)).k = l.((f^g).i) by A11,FUNCT_1:12
        .= l.(g.n) by A15,FINSEQ_1:def 7
        .= (l*g).n by A15,FUNCT_1:13
        .= ((l*f)^(l*g)).k by A6,A15,A16,FINSEQ_1:def 7;
      end;
    end;
    hence thesis by A1,A4,A8,A10,FUNCT_1:2,RELAT_1:27;
  end;
