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 Th8:
    (D|^(n+1)) = (D*D|^n) & dom D = the carrier of R &
    dom (D|^n) = the carrier of R & D|^n is (the carrier of R)-valued Function
    proof
      D|^1 = D by VECTSP11:19;
      hence thesis by FUNCT_2:def 1,VECTSP11:20;
    end;
