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

theorem
  SFX is_finer_than {Y} implies for X st X in SFX holds X c= Y
proof
  assume
A1: for X st X in SFX ex Z st Z in {Y} & X c= Z;
  let X;
  assume X in SFX;
  then ex Z st Z in {Y} & X c= Z by A1;
  hence thesis by TARSKI:def 1;
end;
