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;

theorem Th18:
  for S being halting IC-Ins-separated
  non empty with_non-empty_values AMI-Struct over N
 for P being (the InstructionsF of S)-valued NAT-defined Function,
  s being State of S, k st P halts_at IC Comput(P,s,k)
   holds Result(P,s) = Comput(P,s,k)
proof
  let S be halting IC-Ins-separated non empty
  with_non-empty_values AMI-Struct over N,
  P be (the InstructionsF of S)-valued NAT-defined Function,
  s be State of S, k;
  assume
A1: P halts_at IC Comput(P,s,k);
  then P halts_on s by Th16;
  hence thesis by A1,Th17;
end;
