 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 Th10:
  for a being Element of V1, F being FinSequence of INT.Ring,
  G being FinSequence of V1 st len F = len G &
  for k st k in dom F holds G.k = (F/.k) * a
  holds Sum(G) = Sum(F) * a
  proof
    let a be Element of V1;
    let F be FinSequence of INT.Ring;
    let G be FinSequence of V1;
    assume that
    A1: len F = len G and
    A2: for k st k in dom F holds G.k = (F/.k) * a;
    now
      let k;
      let v be Element of INT.Ring;
      assume that
      A3: k in dom G and
      A4: v = F.k;
      A5: k in dom F by A1,A3,FINSEQ_3:29;
      then v = F/.k by A4,PARTFUN1:def 6;
      hence G.k = v * a by A2,A5;
    end;
    hence thesis by A1,Th9;
  end;
