 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 Th18:
  for a being FinSequence of INT.Ring
  st len a = len b2 & g is additive homogeneous holds
  g.Sum(lmlt(a,b2)) = Sum(lmlt(a,g*b2))
  proof
    let a be FinSequence of INT.Ring such that
    A1: len a = len b2 and
    A2: g is additive homogeneous;
    thus g.Sum(lmlt(a,b2)) = Sum(g*lmlt(a,b2)) by A2,Th16
    .= Sum(lmlt(a,g*b2)) by A1,A2,Th17;
  end;
