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 Th30:
  for F being non empty
  NAT-defined (the InstructionsF of T)-valued finite Function,
      l being
  Element of NAT st l in dom F holds l <= LastLoc F, T
proof
  let F be non empty
  NAT-defined (the InstructionsF of T)-valued finite Function,
  l be Element of NAT
   such that
A1: l in dom F;
  consider M being finite non empty natural-membered set such that
A2: M = { locnum(w,T) where w is Element of NAT : w in dom F } and
A3: LastLoc F = il.(T, max M) by Def11;
  locnum(l,T) in M by A1,A2;
  then
A4: locnum(l,T) <= max M by XXREAL_2:def 8;
  max M is Nat by TARSKI:1;
  then locnum(LastLoc F,T) = max M by A3,Def5;
  hence thesis by A4,Th9;
end;
