reserve L for Abelian left_zeroed add-associative associative right_zeroed
              right_complementable distributive non empty doubleLoopStr;
reserve a,b,c for Element of L;
reserve R for non degenerated comRing;
reserve n,m,i,j,k for Nat;
 reserve D for Function of R, R;
 reserve x,y,z for Element of R;
reserve D for Derivation of R;
reserve s for FinSequence of the carrier of R;
reserve h for Function of R,R;

theorem Th19:
   Sum LBZ0(D,n,x,y) = Sum LBZ1(D,n,x,y) + Sum LBZ2(D,n,x,y)
   proof
      set p= LBZ1(D,n,x,y);
      set q= LBZ2(D,n,x,y);
      set r= LBZ0(D,n,x,y);
A1:   dom p = Seg len p by FINSEQ_1:def 3 .= Seg n by Def6;
A2:   dom q = Seg len q by FINSEQ_1:def 3 .= Seg n by Def7;
      Sum r = Sum(p + q) by Th18 .= Sum p + Sum q by A1,A2,BINOM:7;
      hence thesis;
   end;
