theorem Th4:
  S is weakly_standard iff ex f being sequence of NAT st f is
  bijective & for k being Element of NAT holds f.(k+1) in SUCC(f.k,S) & for j
  being Element of NAT st f.j in SUCC(f.k,S) holds k <= j
proof
  hereby
    assume S is weakly_standard;
    then consider f being sequence of NAT such that
A1: f is bijective and
A2: for m, n being Element of NAT holds m <= n iff f.m <= f.n, S;
    thus ex f being sequence of NAT st f is bijective & for k being
Element of NAT holds f.(k+1) in SUCC(f.k,S) & for j being Element of NAT st f.j
    in SUCC(f.k,S) holds k <= j
    proof
      take f;
      thus f is bijective by A1;
      thus thesis by A1,A2,Th3;
    end;
  end;
  given f be sequence of NAT such that
A3: f is bijective and
A4: for k being Element of NAT holds f.(k+1) in SUCC(f.k,S) & for j being
  Element of NAT st f.j in SUCC(f.k,S) holds k <= j;
  take f;
  thus f is bijective by A3;
  thus thesis by A3,A4,Th3;
end;
