theorem Th15:
  Suc(f) is_tail_of Ant(f) implies Ant(f) |= Suc(f)
proof
  assume Suc(f) is_tail_of Ant(f);
  then ex i st i in dom Ant(f) & (Ant(f)).i = Suc(f) by Lm1;
  then
A1: Suc(f) in rng(Ant(f)) by FUNCT_1:3;
  let A,J,v;
  assume J,v |= rng(Ant(f));
  hence thesis by A1;
end;
