 reserve x,y,z for object,
   i,j,k,l,n,m for Nat,
   D,E for non empty set;
 reserve M for Matrix of D;
 reserve L for Matrix of E;
 reserve k,t,i,j,m,n for Nat,
   D for non empty set;
 reserve V for free Z_Module;
 reserve a for Element of INT.Ring,
   W for Element of V;
 reserve KL1,KL2,KL3 for Linear_Combination of V,
   X for Subset of V;
 reserve V for finite-rank free Z_Module,
   W for Element of V;
 reserve KL1,KL2,KL3 for Linear_Combination of V,
   X for Subset of V;
 reserve s for FinSequence,
   V1,V2,V3 for finite-rank free Z_Module,
   f,f1,f2 for Function of V1,V2,
   g for Function of V2,V3,
   b1 for OrdBasis of V1,
   b2 for OrdBasis of V2,
   b3 for OrdBasis of V3,
   v1,v2 for Vector of V2,
   v,w for Element of V1;
 reserve p2,F for FinSequence of V1,
   p1,d for FinSequence of INT.Ring,
   KL for Linear_Combination of V1;

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;
    A51: 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 A51,A7,FINSEQ_3:29;
      hence (KL (#) b1).t = KL.(b1/.t) * (b1/.t) by VECTSP_6:def 5
      .= ((v|--b1)/.t) * (b1/.t) by A2,A10,A9
      .= lmlt(v|--b1,b1).t by A7,A8,A11,FUNCOP_1:22;
    end;
    thus v = Sum(KL (#) b1) by A1,defOrdBasis,Th20
    .= Sum lmlt(v|--b1,b1) by A5,A6,FINSEQ_1:13;
  end;
