
theorem Th6:
  for R being Ring, n being non zero Nat holds power(R).(0.R,n) = 0.R
   proof
     let R be Ring, n be non zero Nat;
     defpred P[Nat] means power(R).(0.R,$1) = 0.R;
     power(R).(0.R,0+1) = power(R).(0.R,0) * 0.R by GROUP_1:def 7 .= 0.R; then
A1:  P[1];
A2:  now let k be non zero Nat;
       assume P[k];
       power(R).(0.R,k+1) = power(R).(0.R,k) * 0.R by GROUP_1:def 7 .= 0.R;
       hence P[k + 1];
     end;
     for k being non zero Nat holds P[k] from NAT_1:sch 10(A1,A2);
     hence thesis;
   end;
