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 Th6:
  for l being Nat ex k being Nat st l = il.(T,k)
proof
  let l be Nat;
  consider f1 being sequence of NAT such that
A1: f1 is bijective and
A2: for m, n being Element of NAT holds m <= n iff f1.m <= f1.n, T and
  il.(T,0) = f1.0 by Def4;
  l in NAT & rng f1 = NAT by A1,FUNCT_2:def 3,ORDINAL1:def 12;
  then consider k being object such that
A3: k in dom f1 and
A4: f1.k = l by FUNCT_1:def 3;
  reconsider k as Nat by A3;
  take k;
  reconsider l as Element of NAT by ORDINAL1:def 12;
   l = il.(T,k) by A1,A2,A4,Def4;
  hence thesis;
end;
