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 Th23:
   Sum(<* ((D|^(n+1)).x)*y *>^(LBZ0(D,n,x,y))^<* x*((D|^(n+1)).y) *>)
   = ((D|^(n+1)).x)*y + Sum(LBZ0(D,n,x,y))+ x*((D|^(n+1)).y)
   proof
     set p1 = <* ((D|^(n+1)).x)*y *>;
     set p2 = <* x*((D|^(n+1)).y) *>;
     set q = (LBZ0(D,n,x,y));
     set r = <*((D|^(n+1)).x)*y *>^(LBZ0(D,n,x,y))^<* x*((D|^(n+1)).y)*>;
     set r1 = p1^q;
     set r2 = r1^p2;
     set r3 = q^p2;
A1:  Sum(p1) = ((D|^(n+1)).x)*y by RLVECT_1:44;
     r = p1^r3 by FINSEQ_1:32; then
     Sum(r) = Sum(p1)+ Sum(q^p2) by RLVECT_1:41
     .= ((D|^(n+1)).x)*y + (Sum(q)+Sum(p2)) by A1,RLVECT_1:41
     .= ((D|^(n+1)).x)*y + (Sum(q)+ x*((D|^(n+1)).y) ) by RLVECT_1:44
     .= ((D|^(n+1)).x)*y + Sum(q)+ x*((D|^(n+1)).y) by RLVECT_1:def 3;
     hence thesis;
   end;
