theorem Th153:
  for R being add-associative right_zeroed right_complementable
  non empty addLoopStr,
  i be Element of NAT holds
  (Nat-mult-left(R)).(i,0.R) = 0.R
  proof
    let R be add-associative right_zeroed right_complementable
    non empty addLoopStr,
    i be Element of NAT;
    defpred P[Nat] means (Nat-mult-left(R)).($1,0.R) = 0.R;
    A1: P[0] by BINOM:def 3;
    A2: for n be Nat st P[n] holds P[n+1]
    proof
      let n be Nat;
      assume A3:P[n];
      (Nat-mult-left(R)).(n+1,0.R) = 0.R + 0.R by A3,BINOM:def 3
      .= 0.R by RLVECT_1:4;
      hence P[n+1];
    end;
    for n be Nat holds P[n] from NAT_1:sch 2(A1,A2);
    hence thesis;
  end;
