theorem Th35:
  v = Sum lmlt(v|--b1,b1)
proof
  consider KL be Linear_Combination of V1 such that
A1: v = Sum KL & Carrier KL c= rng b1 and
A2: for k st 1<=k & k<=len (v|--b1) holds (v|--b1)/.k=KL.(b1/.k) by Def7;
  len (v|--b1) = len b1 by Def7;
  then
A3: dom (v|--b1) = dom b1 by FINSEQ_3:29;
  then
A4: dom b1 = dom lmlt(v|--b1,b1) by Th12;
 len (KL (#) b1) = len b1 by VECTSP_6:def 5
    .= len lmlt(v|--b1,b1) by A4,FINSEQ_3:29; then
A5: dom (KL (#) b1) = dom lmlt(v|--b1,b1) by FINSEQ_3:29;
A6: now
    let t be Nat;
    assume
A7: t in dom lmlt(v|--b1,b1);
    then
A8: b1/.t = b1.t by A4,PARTFUN1:def 6;
    t in dom (v|--b1) by A3,A7,Th12;
    then
A9: t<=len (v|--b1) by FINSEQ_3:25;
A10: 1<=t by A7,FINSEQ_3:25;
    then
A11: (v|--b1)/.t = (v|--b1).t by A9,FINSEQ_4:15;
    t in dom (KL (#) b1) by A5,A7;
    hence (KL (#) b1).t = KL.(b1/.t) * (b1/.t) by VECTSP_6:def 5
      .= ((v|--b1)/.t) * (b1/.t) by A2,A10,A9
      .= (the lmult of V1).((v|--b1).t,b1.t) by A8,A11,VECTSP_1:def 12
      .= lmlt(v|--b1,b1).t by A7,FUNCOP_1:22;
  end;
  b1 is one-to-one by Def2;
  hence v = Sum(KL (#) b1) by A1,Th20
    .= Sum lmlt(v|--b1,b1) by A5,A6,FINSEQ_1:13;
end;
