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;
reserve T for weakly_standard
 IC-Ins-separated non empty
  with_non-empty_values AMI-Struct over N;

theorem Th17:
  il.(STC N, z) = z
proof
  set M = STC N;
  reconsider f2 = id NAT as sequence of NAT;
  consider f being sequence of NAT such that
A1: f is bijective & for m, n being Element of NAT holds m <= n iff f.m
  <= f.n, STC N and
A2: il.(M,z) = f.z by Def4;
  now
    let k be Element of NAT;
      reconsider fk = f2.k as Element of NAT;
A3: SUCC(fk,STC N) = {k,k+1} by AMISTD_1:8;
    thus f2.(k+1) in SUCC(f2.k,STC N) by A3,TARSKI:def 2;
    let j be Element of NAT;
    assume f2.j in SUCC(f2.k,STC N);
    then j = k or j = k+1 by A3,TARSKI:def 2;
    hence k <= j by NAT_1:11;
  end;
  then for m, n being Element of NAT holds m <= n iff f2.m <= f2.n, M by Th3;
  then z is Element of NAT & f = f2 by A1,Th2,ORDINAL1:def 12;
  hence thesis by A2;
end;
