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 Th6:
    for n,x holds D.(n*x) = n*D.x
    proof
      let n,x;
      defpred P[Nat] means D.($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];
        D.((n+1)*x) = D.(n*x + 1*x) by BINOM:15
        .= D.(n*x) + D.(1*x) by Def1
        .= D.(n*x) + D.x by BINOM:13
        .= n*D.x + 1*D.x by A2,BINOM:13
        .= (n+1)*D.x by BINOM:15;
        hence thesis;
      end;
      0*D.x = 0.R by BINOM:12 .= D.(0.R) by Th5 .= D.(0*x) by BINOM:12;
      then
A3:   P[0];
      for n be Nat holds P[n] from NAT_1:sch 2(A3,A1);
      hence thesis;
    end;
