 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 Th30:
  for P1,P2 being FinSequence of V1 st len P1 = len P2 holds
  Sum(P1 + P2) = (Sum P1) + (Sum P2)
  proof
    let P1, P2 be FinSequence of V1;
    assume len P1 = len P2;
    then reconsider R1 = P1, R2 = P2
    as Element of (len P1) -tuples_on the carrier of V1 by FINSEQ_2:92;
    thus Sum (P1 + P2) = Sum (R1 + R2)
    .= (Sum P1) + (Sum P2) by FVSUM_1:76;
  end;
