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 Th20:
    D*LBZ0(D,n,x,y) = Del(LBZ2(D,n+1,x,y),n+1) + Del(LBZ1(D,n+1,x,y),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:   dom p = Seg len p by FINSEQ_1:def 3 .= Seg (n+1) by Def7;
A4:   dom q = Seg len q by FINSEQ_1:def 3 .= Seg (n+1) by Def6;
A5:   dom r = Seg len r by FINSEQ_1:def 3 .= Seg n by Def5;
A6:   1 <= n+1 by NAT_1:12; then
A7:   n+1 in dom p by A1,FINSEQ_3:25;
A8:   1 in dom q by A2,A6,FINSEQ_3:25;
      reconsider p1 = Del(p,n+1) as FinSequence of the carrier of R;
      reconsider q1 = Del(q,1) as FinSequence of the carrier of R;
A9:   dom Del(p,n+1) = Seg len p1 by FINSEQ_1:def 3
      .= Seg n by A1,FINSEQ_3:109,A7;
A10:  dom Del(q,1) = Seg len q1 by FINSEQ_1:def 3
      .= Seg n by A2,FINSEQ_3:109,A8;
A11:  dom (p1+q1) = Seg n by A9,BINOM:def 1;
A12:  Seg len(p1+q1) = dom (p1+q1) by FINSEQ_1:def 3
      .= Seg n by A9,BINOM:def 1;
A14:  len (D*r) = len r by FINSEQ_2:33 .= n by Def5;
      for i st 1 <= i & i <= len (D*r) holds (D*r).i = (p1 + q1).i
       proof
        let i;
        assume
A16:    1 <= i & i <= len (D*r); then
A17:    i in dom r by A5,A14;
A18:    i in dom (D*r) by A16,FINSEQ_3:25;
A19:    i in dom (p1+q1) by A14,A16,A11;
A20:    1 <= i & i <= len (p1+q1) by A12,FINSEQ_1:6,A14,A16;
A21:    i < n+1 by A14,A16,XREAL_1:39;
A22:    1 <=i<=n+1 by A14,A16,XREAL_1:39; then
A23:    i in dom p by A3;
        i in dom p1 by A14,A16,A9; then
A24:    p1/.i = p1.i by PARTFUN1:def 6 .= p.i by A21,FINSEQ_3:110
        .= ((n+1) choose i)*((D|^((n+1) +1 -'i)).x)*((D|^i).y) by A23,Def7
        .= ((n+1) choose i)*((D|^((n+2)-'i)).x)*((D|^i).y);
A25:    1 <= i+1 by NAT_1:12;
A26:    i+1 <= n+1 by A21,INT_1:7; then
A28:    (n+1)+1 -'(i+1) = (n+2) -(i+1) by XREAL_1:38,233
         .= n+1-i
         .= n+1-'i by A22,XREAL_1:233;
A27:    (i+1) in dom q by A25,A26,A4;
A29:    i in dom q1 by A14,A16,A10;
A30:    1 in dom q by A4,A6;
        reconsider qq1 = q1 as (the carrier of R)-valued Function;
        reconsider ii = i as object;
A31:    q1/.i = qq1.ii by A29,PARTFUN1:def 6
        .= q.(i+1) by A14,A16,A2,A30,FINSEQ_3:111;
A32:    q1/.i
        = ((n+1) choose ((i+1)-'1))*((D|^((n+1)+1 -'(i+1))).x)*((D|^(i+1)).y)
         by A27,A31,Def6
        .=((n+1) choose i)*((D|^(n+1-'i)).x)*((D|^(i+1)).y) by A28,NAT_D:34;
A33:    i<=n+2 by A14,A16,XREAL_1:39;
A34:    n+1 -'i+1 = (n+1) -i +1 by A22,XREAL_1:233 .= n+2 - i
        .= (n+2) -'i by A33,XREAL_1:233;
        i in dom r by A14,A16,A5; then
A35:    r.i = ((n choose i)+(n choose (i-'1)))*((D|^(n+1 -'i)).x)*((D|^i).y)
          by Def5
        .= ((n choose ((i-'1)+1))+(n choose (i-'1)))*((D|^(n+1 -'i)).x)
            *((D|^i).y) by A16,XREAL_1:235
        .= ((n+1) choose ((i-'1)+1))*((D|^(n+1 -'i)).x)*((D|^i).y) by NEWTON:22
        .= ((n+1) choose i)*((D|^(n+1 -'i)).x)*((D|^i).y)
            by A16,XREAL_1:235;
       (D*r)/.i = (D*r).i by A18,PARTFUN1:def 6
        .= D.((((n+1) choose i)*((D|^(n+1 -'i)).x))*((D|^i).y) )
           by A35, A17,FUNCT_1:13
        .= D.((((n+1) choose i)*((D|^(n+1 -'i)).x)))*((D|^i).y)
           + (((n+1) choose i)*((D|^(n+1 -'i)).x))*D.((D|^i).y) by Def1
        .= ((n+1) choose i)*(D.((D|^(n+1 -'i)).x))*((D|^i).y)
           + (((n+1) choose i)*((D|^(n+1 -'i)).x))*D.((D|^i).y) by Th6
        .= ((n+1) choose i)*(((D|^(n+1 -'i+1)).x))*((D|^i).y)
           + (((n+1) choose i)*((D|^(n+1 -'i)).x))*D.((D|^i).y) by Th9
        .= p1/.i + q1/.i by A32,A24,A34,Th9
        .= (p1+q1)/.i by A20,BINOM:def 1; then
        (D*r).i = (p1+q1)/.i by A18,PARTFUN1:def 6
        .= (p1+q1).i by A19,PARTFUN1:def 6;
        hence thesis;
       end;
       hence thesis by A14,A12,FINSEQ_1:6;
     end;
