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;

theorem
  for F being Instruction-Sequence of S
  for s being State of S, k being Nat st IC
  Comput(F,s,k) <> IC Comput(F,s,k+1) & F.(IC Comput(F,s,k+1)) = halt S
   holds LifeSpan(F,s) = k+1
proof
  let F be Instruction-Sequence of S;
  let s be State of S, k be Nat;
  assume that
A1: IC Comput(F,s,k) <> IC Comput(F,s,k+1) and
A2: F.(IC Comput(F,s,k+1)) = halt S;
A3: dom F = NAT by PARTFUN1:def 2;
  now
    assume F.(IC Comput(F,s,k)) = halt S;
    then CurInstr(F,Comput(F,s,k)) = halt S by A3,PARTFUN1:def 6;
    hence contradiction by A1,Th5,NAT_1:11;
  end;
  hence thesis by A2,Th31;
end;
