reserve X,Y,Z,Z1,Z2,D for set,x,y for object;
reserve SFX,SFY,SFZ for set;

theorem
  SFX is_finer_than {} implies SFX = {}
proof
  assume
A1: for X st X in SFX ex Y st Y in {} & X c= Y;
  set x = the Element of SFX;
  assume not thesis;
  then ex Y st Y in {} & x c= Y by A1;
  hence contradiction;
end;
