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
  for F, G being non empty
   NAT-defined (the InstructionsF of T)-valued finite Function
    st F c= G
  holds LastLoc F <= LastLoc G, T
proof
  let F, G be non empty
   NAT-defined (the InstructionsF of T)-valued finite Function
  such that
A1: F c= G;
  consider N being finite non empty natural-membered set such that
A2: N = { locnum(l,T) where l is Element of NAT : l in dom G } and
A3: LastLoc G = il.(T, max N) by Def11;
  consider M being finite non empty natural-membered set such that
A4: M = { locnum(l,T) where l is Element of NAT : l in dom F } and
A5: LastLoc F = il.(T, max M) by Def11;
  reconsider MM = M, NN = N as non empty finite Subset of REAL by MEMBERED:3;
  M c= N
  proof
    let a be object;
    assume a in M;
    then
A6: ex l being Element of NAT st a = locnum(l,T) & l in dom F by A4;
    dom F c= dom G by A1,GRFUNC_1:2;
    hence thesis by A2,A6;
  end;
  then max MM <= max NN by XXREAL_2:59;
  hence thesis by A5,A3,Th8;
end;
