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 Th9:
    for n,x holds (D|^(n+1)).x = D.((D|^n).x)
    proof
      let n,x;
A1:   dom (D|^n) = the carrier of R by Th8;
      (D|^(n+1)).x = ((D*D|^n)).x by Th8
      .= D.((D|^n).x) by A1,FUNCT_1:13;
      hence thesis;
    end;
