theorem Th162:
  for R being Abelian right_zeroed add-associative
  right_complementable non empty addLoopStr,
  a being Element of R, i, j being Element of NAT holds
  (Nat-mult-left(R)).(i*j,a) = (Nat-mult-left(R)).(i,(Nat-mult-left(R)).(j,a))
  proof
    let R be Abelian right_zeroed add-associative right_complementable
    non empty addLoopStr,
    a be Element of R, i, j be Element of NAT;
    defpred P[Nat] means (Nat-mult-left(R)).($1*j,a)
    =(Nat-mult-left(R)).($1,(Nat-mult-left(R)).(j,a));
    A1: P[0]
    proof
      (Nat-mult-left(R)).(0*j,a) = 0.R by BINOM:def 3
      .= (Nat-mult-left(R)).(0,(Nat-mult-left(R)).(j,a)) by BINOM:def 3;
      hence thesis;
    end;
    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)*j,a) = (j+(n*j))*a
      .= j*a+(n*j)*a by BINOM:15
      .= (Nat-mult-left(R)).(n+1,j*a) by A3,BINOM:def 3;
      hence P[n+1];
    end;
    for n be Nat holds P[n] from NAT_1:sch 2(A1,A2);
    hence thesis;
  end;
