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 N be non empty with_zero set,
  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, k being Nat
   st CurInstr(P,Comput(P,s,k)) = halt S
 holds Comput(P,s,LifeSpan(P,s)) = Comput(P,s,k)
proof
  let N;
  let S be halting
IC-Ins-separated
 non empty with_non-empty_values AMI-Struct over N,
     P being Instruction-Sequence of S,
     s be State of S, k be Nat such that
A1: CurInstr(P,Comput(P,s,k)) = halt S;
A2: dom P = NAT by PARTFUN1:def 2;
A3: P halts_on s by A2,A1;
  set Ls = LifeSpan(P,s);
A4: CurInstr(P,Comput(P,s,Ls)) = halt S by A3,Def15;
  Ls <= k by A1,A3,Def15;
  hence thesis by A4,Th5;
end;
