reserve k,t,i,j,m,n for Nat,
  x,y,y1,y2 for object,
  D for non empty set;
reserve K for Field,
  V for VectSp of K,
  a for Element of K,
  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-dimensional VectSp of K,
  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 K,
  KL for Linear_Combination of V1;

theorem Th18:
  for a be FinSequence of K 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 K 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;
