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 Th21:
    Sum(D*LBZ0(D,n,x,y)) =
    - (LBZ1(D,n+1,x,y)/.1) + Sum LBZ0(D,n+1,x,y) - (LBZ2(D,n+1,x,y)/.(n+1))
    proof
      set p= LBZ2(D,n+1,x,y);
      set q= LBZ1(D,n+1,x,y);
      set r= LBZ0(D,n,x,y);
A1:   len p = n+1 by Def7;
A2:   len q = n+1 by Def6;
A3:   1 <= n+1 by NAT_1:12; then
A4:   n+1 in dom p by A1,FINSEQ_3:25;
A5:   1 in dom q by A2,A3,FINSEQ_3:25;
      set p1 = Del(p,n+1);
      set q1 = Del(q,1);
      set r1 = D*r;
A6:   dom p1 = Seg len p1 by FINSEQ_1:def 3 .= Seg n by A1,A4,FINSEQ_3:109;
A7:   dom q1 = Seg len q1 by FINSEQ_1:def 3 .= Seg n by A2,A5,FINSEQ_3:109;
      reconsider p1,p,q1,q as FinSequence of R;
      len p >= 1 by NAT_1:12,A1; then
A8:   len p in dom p by FINSEQ_3:25;
      reconsider a = p.(len p) as Element of R by A8,FINSEQ_2:11;
A9:   a = (LBZ2(D,n+1,x,y)/.(n+1)) by A1,A8,PARTFUN1:def 6;
      len q >= 1 by NAT_1:12,A2; then
A10:  1 in dom q by FINSEQ_3:25; then
      reconsider b = q.1 as Element of R by FINSEQ_2:11;
A11:  b = (LBZ1(D,n+1,x,y)/.1) by A10,PARTFUN1:def 6;
      p <> {} by Def7; then
A13:  Sum p = Sum( p1^<*a*> ) by A1,PRE_POLY:13
      .= Sum p1 + Sum <*a*> by RLVECT_1:41
      .= Sum p1 + a by BINOM:3;
      q <> {} by Def6; then
A15:  Sum q = Sum( <*b*>^q1 ) by FINSEQ_5:86
      .= Sum <*b*> + Sum q1 by RLVECT_1:41
      .= b + Sum q1 by BINOM:3;
      Sum(D*LBZ0(D,n,x,y)) = Sum( p1 + q1) by Th20
      .= Sum p1 + Sum q1 by A6,A7,BINOM:7
      .= (Sum p - a) + Sum q1 by A13,VECTSP_2:2
      .= (Sum p + (-a) + (Sum q - b)) by A15,VECTSP_2:2
      .= (Sum p + (-a) + Sum q - b) by RLVECT_1:def 3
      .= -b + ((Sum p + Sum q) -a) by RLVECT_1:def 3
      .= -LBZ1(D,n+1,x,y)/.1 +(Sum p + Sum q) -LBZ2(D,n+1,x,y)/.(n+1)
      by A9,A11,RLVECT_1:def 3;
      hence thesis by Th19;
    end;
