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 being lower non empty
   NAT-defined (the InstructionsF of T)-valued finite Function,
  G being non empty NAT-defined
   NAT-defined (the InstructionsF of T)-valued finite Function
    holds F c= G & LastLoc F = LastLoc G
  implies F = G
proof
  let F be lower non empty
   NAT-defined (the InstructionsF of T)-valued finite Function,
     G be non empty
     NAT-defined (the InstructionsF of T)-valued finite Function
      such that
A1: F c= G and
A2: LastLoc F = LastLoc G;
  dom F = dom G
  proof
    thus dom F c= dom G by A1,GRFUNC_1:2;
    let x be object;
    assume
A3: x in dom G;
    reconsider x as Element of NAT by A3;
A4: LastLoc F in dom F by Th28;
    x <= LastLoc F, T by A2,A3,Th30;
    hence thesis by A4,Def10;
  end;
  hence thesis by A1,GRFUNC_1:3;
end;
