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 Th5:
    D.(1.R) = 0.R & D.(0.R) = 0.R
    proof
A1:   D.(0.R) = D.(0.R + 0.R) .= D.(0.R) + D.(0.R) by Def1;
      D.(1.R) = D.((1.R)*(1.R))
      .= (1.R)*D.(1.R) + (1.R)*D.(1.R) by Def1 .= D.(1.R) + D.(1.R);
      hence thesis by A1,RLVECT_1:9;
    end;
