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;

theorem Th7:
     D.(x|^(m+1)) = (m+1)*((x|^m)*D.x)
     proof
       defpred P[Nat] means D.(x|^($1+1)) = ($1+1)*((x|^$1)*D.x);
A1:    for n be Nat st P[n] holds P[n+1]
       proof
         let n be Nat;
         assume
A2:      P[n];
         set t = n+1;
A3:      x|^((n+1)+1) = (x|^1)*(x|^t) by BINOM:10
         .= x*x|^(n+1) by BINOM:8;
         D.(x|^((n+1)+1))
          = x*((n+1)*((x|^n)*D.x)) + (x|^(n+1))*D.x by A2, Def1,A3
         .= (n+1)*(x*((x|^n)*D.x)) + (x|^(n+1))*D.x by BINOM:19
         .= (n+1)*((x*(x|^n))*D.x) + (x|^(n+1))*D.x by GROUP_1:def 3
         .= (n+1)*(((x|^1)*(x|^n))*D.x) + (x|^(n+1))*D.x by BINOM:8
         .= (n+1)*((x|^(n+1))*D.x) + (x|^(n+1))*D.x by BINOM:10
         .= (n+1)*((x|^(n+1))*D.x) + 1*((x|^(n+1))*D.x) by BINOM:13
         .= (n+1+1)*((x|^(n+1))*D.x) by BINOM:15;
         hence thesis;
       end;
       (0+1)*((x|^0)*D.x) = 1*((1_R)*D.x) by BINOM:8
       .= D.x by BINOM:13
       .= D.(x|^(0+1)) by BINOM:8; then
A4:    P[0];
       for n be Nat holds P[n] from NAT_1:sch 2(A4,A1);
       hence thesis;
     end;
