theorem Th10:
  for F being unique-halt Program of S,
  I being Nat st I in dom CutLastLoc F
  holds (CutLastLoc F).I <> halt S
proof
  let F be unique-halt Program of S,
  I be Nat such that
A1: I in dom CutLastLoc F and
A2: (CutLastLoc F).I = halt S;
A3: dom CutLastLoc F c= dom F by GRFUNC_1:2;
  F.I = halt S by A1,A2,GRFUNC_1:2;
  then
A4: I = LastLoc F by A1,A3,Def7;
  dom CutLastLoc F = (dom F) \ {LastLoc F} by VALUED_1:36;
  then not I in {LastLoc F} by A1,XBOOLE_0:def 5;
  hence thesis by A4,TARSKI:def 1;
end;
