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;
reserve S for IC-Ins-separated halting
  non empty with_non-empty_values AMI-Struct over N;
reserve S for halting IC-Ins-separated non empty
   with_non-empty_values AMI-Struct over N;
reserve P for Instruction-Sequence of S;

theorem
  for s being State of S st P halts_on s for k being Nat
    st LifeSpan(P,s) <= k
   holds CurInstr(P, Comput(P,s,k)) = halt S
proof
  let s be State of S;
  assume
  P halts_on s;
  then
A1: CurInstr(P,Comput(P,s,LifeSpan(P,s))) = halt S by Def15;
  let k be Nat;
  assume
   LifeSpan(P,s) <= k;
  hence thesis by A1,Th5;
end;
