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 Th22:
   LBZ(D,n+1,x,y) = <*((D|^(n+1)).x)*y *>^(LBZ0(D,n,x,y))^<* x*((D|^(n+1)).y)*>
   proof
     set p= LBZ(D,n+1,x,y);
     set q= LBZ0(D,n,x,y);
     set r = <* ((D|^(n+1)).x)*y *>^q^<* x*((D|^(n+1)).y) *>;
     set r1 = <* ((D|^(n+1)).x)*y *>^q;
     set r2 = r1^<* x*((D|^(n+1)).y) *>;
     set r3 = q^<* x*((D|^(n+1)).y) *>;
     r1 = Ins(q,0,((D|^(n+1)).x)*y) by FINSEQ_5:67; then
A2:  len r1 = len q + 1 by FINSEQ_5:69 .= n + 1 by Def5;
A3:  len p = (n+1)+1 by Def4 .= n + 2;
     r = Ins(r1,n+1,x*((D|^(n+1)).y)) by A2,FINSEQ_5:68; then
A4:  len r = n + 1 +1 by A2,FINSEQ_5:69 .= len p by A3;
     for k st 1 <= k & k <= len p holds p.k = r.k
     proof
       let k;
       assume
A5:    1 <= k & k <= len p;
A6:    k in dom p by A5,FINSEQ_3:25;
       n+1+1 > 0+1 by XREAL_1:8; then
A7:    n+2 -'1 = n+2 - 1 by XREAL_1:233 .= n+1;
       1 < k+1 & k < n+2 +1 by A5,A3,NAT_1:13; then
       per cases by NAT_1:22;
         suppose
A9:       k = 1;
           n+1+1 > 0+1 by XREAL_1:8; then
A10:       n+2 -'1 = n+2 - 1 by XREAL_1:233 .= n+1;
A11:       p.k = ((n+1) choose (k-'1))*((D|^((n+1)+1-'k)).x)*((D|^(k -' 1)).y)
           by A6,Def4
           .= ((n+1) choose 0)*((D|^(n+2-'1)).x)*((D|^(1-'1)).y)
           by A9,XREAL_1:232
           .= (1*((D|^(n+2-'1)).x))*((D|^(1-'1)).y) by NEWTON:19
           .= ((D|^(n+2-'1)).x)*((D|^(1-'1)).y) by BINOM:13
           .= ((D|^(n+1)).x)*((D|^0).y) by A10,XREAL_1:232
           .= ((D|^(n+1)).x)*((id R).y) by VECTSP11:18
           .= ((D|^(n+1)).x)*y;
           r.k = (Ins(r1,n+1,x*((D|^(n+1)).y))).1 by A9,A2,FINSEQ_5:68
           .= (<* ((D|^(n+1)).x)*y *>^q).1 by NAT_1:11,FINSEQ_5:75
           .= p.k by A11,FINSEQ_1:41;
           hence thesis;
         end;
         suppose
A12:       1 < k & k < n+2;
A13:       dom r1 = Seg (n + 1) by A2,FINSEQ_1:def 3;
           1+1 <= k by A12,INT_1:7; then
A15:       2 -1 <= k -1 by XREAL_1:9;
           k + 1 <= n+2 by A12,INT_1:7; then
A17:       k+1-1 <= n+2-1 by XREAL_1:9; then
A18:       k in dom r1 by A13,A12;
           reconsider s = k - 1 as Nat by A12;
           reconsider q1 = <* ((D|^(n+1)).x)*y *> as FinSequence of R;
A19:       dom q = Seg len q  by FINSEQ_1:def 3 .= Seg n by Def5;
           k - 1 <= n + 1 -1 by A17,XREAL_1:9; then
A21:       s in dom q by A19,A15;
           reconsider t = s -1 as Nat by A15;
A22:       k-'1 = s by A12,XREAL_1:233;
A23:       (n+1)+1-'k = (n+1)+1-k by A12,XREAL_1:233 .= (n+1) - s
           .= (n+1) -'s by A12,A17,XREAL_1:9,233;
           r.k = (q1^q ).(s+1) by A18, FINSEQ_1:def 7
           .= (q1^q ).(len q1 + s) by FINSEQ_1:39
            .= (LBZ0(D,n,x,y)).s by A21,FINSEQ_1:def 7
           .= ((n choose (s-'1))+(n choose s))*((D|^(n+1 -'s)).x)*((D|^s).y)
              by A21,Def5
           .= ((n choose t)+(n choose (t+1)))*((D|^(n+1 -'s)).x)*((D|^s).y)
              by A15,XREAL_1:233
           .= ((n+1) choose (k-'1))*((D|^((n+1)+1-'k)).x)*((D|^(k-'1)).y)
              by A23,A22,NEWTON:22
           .= p.k by A6,Def4;
           hence thesis;
         end;
         suppose
A24:       k = n+2; then
A25:       r.k = (Ins(r1,n+1,x*((D|^(n+1)).y))).(n+1+1) by A2,FINSEQ_5:68
           .= x*((D|^(n+1)).y) by FINSEQ_5:73,A2;
           p.k = ((n+1) choose (n+1))*
              ((D|^((n+1)+1-'(n+2))).x)*((D|^(n+2 -' 1)).y) by A7,A24,A6,Def4
           .= 1*((D|^((n+1)+1-'(n+2))).x)*((D|^(n+2 -' 1)).y) by NEWTON:21
           .= ((D|^((n+2 -'(n+2)))).x)*((D|^(n+2 -' 1)).y) by BINOM:13
           .= ((D|^0).x)*((D|^(n+1)).y) by A7,XREAL_1:232
           .= ((id R).x)*((D|^(n+1)).y) by VECTSP11:18
           .= r.k by A25;
           hence thesis;
         end;
       end;
       hence thesis by A4;
     end;
