theorem Th59:
  Sum(L * a) = Sum(L) * a
proof
  per cases;
  suppose
A1: a <> 0.R;
    set l = L * a;
A2: Carrier(l) = Carrier(L) by A1,Th43;
    consider G being FinSequence of V such that
A3: G is one-to-one and
A4: rng G = Carrier(L) and
A5: Sum(L) = Sum(L (#) G) by Def7;
    set g = L (#) G;
    consider F being FinSequence of V such that
A6: F is one-to-one and
A7: rng F = Carrier(L * a) and
A8: Sum(L * a) = Sum((L * a) (#) F) by Def7;
A9: len G = len F by A1,A6,A7,A3,A4,Th43,FINSEQ_1:48;
    set f = (L * a) (#) F;
    deffunc Q(Nat)= F <- (G.$1);
    consider P being FinSequence such that
A10: len P = len F and
A11: for k be Nat st k in dom P holds P.k = Q(k) from FINSEQ_1:sch 2;
A12: dom P = Seg len F by A10,FINSEQ_1:def 3;
A13: now
      let x be object;
      assume
A14:  x in dom G;
      then reconsider n = x as Nat by FINSEQ_3:23;
      G.n in rng F by A7,A4,A2,A14,FUNCT_1:def 3;
      then
A15:  F just_once_values G.n by A6,FINSEQ_4:8;
      n in Seg len F by A9,A14,FINSEQ_1:def 3;
      then F.(P.n) = F.(F <- (G.n)) by A11,A12
        .= G.n by A15,FINSEQ_4:def 3;
      hence G.x = F.(P.x);
    end;
A16: rng P c= dom F
    proof
      let x be object;
      assume x in rng P;
      then consider y being object such that
A17:  y in dom P and
A18:  P.y = x by FUNCT_1:def 3;
      reconsider y as Nat by A17,FINSEQ_3:23;
      y in dom G by A10,A9,A17,FINSEQ_3:29;
      then G.y in rng F by A7,A4,A2,FUNCT_1:def 3;
      then
A19:  F just_once_values G.y by A6,FINSEQ_4:8;
      P.y = F <- (G.y) by A11,A17;
      hence thesis by A18,A19,FINSEQ_4:def 3;
    end;
    now
      let x be object;
      thus x in dom G implies x in dom P & P.x in dom F
      proof
        assume x in dom G;
        then x in Seg(len P) by A10,A9,FINSEQ_1:def 3;
        hence x in dom P by FINSEQ_1:def 3;
        then P.x in rng P by FUNCT_1:def 3;
        hence thesis by A16;
      end;
      assume that
A20:  x in dom P and
      P.x in dom F;
      x in Seg(len P) by A20,FINSEQ_1:def 3;
      hence x in dom G by A10,A9,FINSEQ_1:def 3;
    end;
    then
A21: G = F * P by A13,FUNCT_1:10;
    dom F c= rng P
    proof
      set f = F" * G;
      let x be object;
      assume
A22:  x in dom F;
      dom(F") = rng G by A6,A7,A4,A2,FUNCT_1:33;
      then
A23:  rng f = rng(F") by RELAT_1:28
        .= dom F by A6,FUNCT_1:33;
      f = F " * F * P by A21,RELAT_1:36
        .= id(dom F) * P by A6,FUNCT_1:39
        .= P by A16,RELAT_1:53;
      hence thesis by A22,A23;
    end;
    then
A24: dom F = rng P by A16;
A25: dom P = dom F by A10,FINSEQ_3:29;
    then
A26: P is one-to-one by A24,FINSEQ_4:60;
    reconsider P as Function of dom F, dom F by A16,A25,FUNCT_2:2;
A27: len f = len F by Def6;
    then
A28: dom f = dom F by FINSEQ_3:29;
    then reconsider P as Function of dom f, dom f;
    reconsider Fp = f * P as FinSequence of V by FINSEQ_2:47;
    reconsider P as Permutation of dom f by A24,A26,A28,FUNCT_2:57;
A29: Fp = f * P;
    then
A30: len Fp = len f by FINSEQ_2:44;
    then
A31: len Fp = len g by A9,A27,Def6;
A32: now
      let k;
      let v be Vector of V;
      assume that
A33:  k in dom Fp and
A34:  v = g.k;
A35:  k in Seg(len g) by A31,A33,FINSEQ_1:def 3;
      then
A36:  k in dom P by A10,A27,A30,A31,FINSEQ_1:def 3;
A37:  k in dom G by A9,A27,A30,A31,A35,FINSEQ_1:def 3;
      then G.k in rng G by FUNCT_1:def 3;
      then F just_once_values G.k by A6,A7,A4,A2,FINSEQ_4:8;
      then
A38:  (F <- (G.k)) in dom F by FINSEQ_4:def 3;
      then reconsider i = F <- (G.k) as Nat by FINSEQ_3:23;
A39:  G/.k = G.k by A37,PARTFUN1:def 6
        .= F.(P.k) by A21,A36,FUNCT_1:13
        .= F.i by A11,A12,A27,A30,A31,A35
        .= F/.i by A38,PARTFUN1:def 6;
A40:  k in dom g by A35,FINSEQ_1:def 3;
      i in Seg(len f) by A27,A38,FINSEQ_1:def 3;
      then
A41:  i in dom f by FINSEQ_1:def 3;
      thus Fp.k = f.(P.k) by A36,FUNCT_1:13
        .= f.(F <- (G.k)) by A11,A12,A27,A30,A31,A35
        .= (F/.i) * l.(F/.i) by A41,Def6
        .= (F/.i) * (L.(F/.i) * a) by Def10
        .= (F/.i) * L.(F/.i) * a by VECTSP_2:def 9
        .= v * a by A34,A40,A39,Def6;
    end;
    Sum(Fp) = Sum(f) by A29,RLVECT_2:7;
    hence thesis by A8,A5,A31,A32,Th1;
  end;
  suppose
A42: a = 0.R;
    hence Sum(L * a) = Sum(ZeroLC(V)) by Th44
      .= 0.V by Lm3
      .= Sum(L) * a by A42,VECTSP_2:32;
  end;
end;
