reserve x for set,
  D for non empty set,
  k, n for Element of NAT,
  z for Nat;
reserve N for with_zero set,
  S for
    IC-Ins-separated non empty with_non-empty_values AMI-Struct over N,
  i for Element of the InstructionsF of S,
  l, l1, l2, l3 for Element of NAT,
  s for State of S;
reserve ss for Element of product the_Values_of S;

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;
