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;
