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 Th24:
  for S being weakly_standard halting IC-Ins-separated
 non empty with_non-empty_values AMI-Struct over N holds il.(S,0) .-->
  halt S qua NAT-defined (the InstructionsF of S)-valued
   finite Function is really-closed
proof
  let S be weakly_standard halting IC-Ins-separated
    non empty
  with_non-empty_values AMI-Struct over N;
  reconsider F = il.(S,0) .--> halt S as
   NAT-defined (the InstructionsF of S)-valued finite Function;
  let l be Nat;
  assume
A1: l in dom(il.(S,0) .--> halt S);
A3: l = il.(S,0) by A1,TARSKI:def 1;
  F/.l = F.l by A1,PARTFUN1:def 6
    .= halt S by A3,FUNCOP_1:72;
  hence thesis by A3,AMISTD_1:2;
end;
