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
     (D|^n).(x*y) = Sum(LBZ(D,n,x,y))
     proof
       per cases;
         suppose
A1:        n <> 0;
           defpred P[Nat] means (D|^$1).(x*y) = Sum(LBZ(D,$1,x,y));
           Sum(LBZ(D,1,x,y)) = Sum <* y*D.x, x*D.y *> by Th16
           .= Sum(<* y*D.x *>) + x*D.y by FVSUM_1:71
           .= y*D.x + x*D.y  by BINOM:3
           .= D.(x*y) by Def1; then
A2:        P[1] by VECTSP11:19;
A3:        for n being non zero Nat st P[n] holds P[n+1]
           proof
             let n be non zero Nat;
             assume
A4:          P[n];
             reconsider k = n-1 as Nat;
             (D|^(n+1)).(x*y) = D.(Sum(LBZ(D,k+1,x,y))) by A4,Th9
             .= Sum(LBZ(D,k+2,x,y)) by Th24
             .= Sum(LBZ(D,n+1,x,y));
             hence thesis;
           end;
           for n be non zero Nat holds P[n] from NAT_1:sch 10(A2,A3);
           hence thesis by A1;
         end;
         suppose
A5:        n = 0;
           Sum(LBZ(D,0,x,y)) = Sum<* x*y *> by Th15
           .= (id R).(x*y) by BINOM:3
           .= (D|^0).(x*y) by VECTSP11:18;
           hence thesis by A5;
         end;
       end;
