reserve N for with_zero set;
reserve N for with_zero set;
reserve x,y,z,A,B for set,
  f,g,h for Function,
  i,j,k for Nat;
reserve S for IC-Ins-separated non empty
     with_non-empty_values AMI-Struct over N,
  s for State of S;
reserve N for non empty with_zero set,
 S for IC-Ins-separated non empty with_non-empty_values AMI-Struct over N,
  s for State of S;
reserve n for Nat;
reserve N for with_zero non empty set;
reserve N for with_zero non empty set,
  S for IC-Ins-separated non empty AMI-Struct over N;
reserve m,n for Nat;

theorem
  for S being halting IC-Ins-separated non empty with_non-empty_values
       AMI-Struct over N,
  P being Instruction-Sequence of S,
  s being State of S st s = Following(P,s)
  holds for n holds Comput(P,s,n) = s
proof
  let S be halting IC-Ins-separated non empty
  with_non-empty_values AMI-Struct over N,
  P be Instruction-Sequence of S,
  s be State of S;
  defpred X[Nat] means Comput(P,s,$1) = s;
  assume s = Following(P,s);
  then
A1: for n st X[n] holds X[n+1] by Th3;
A2: X[ 0];
  thus for n holds X[n] from NAT_1:sch 2(A2, A1);
end;
