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

theorem
  SFX is_finer_than {X,Y} implies for Z st Z in SFX holds Z c= X or Z c= Y
proof
  assume
A1: for Z1 st Z1 in SFX ex Z2 st Z2 in {X,Y} & Z1 c= Z2;
  let Z;
  assume Z in SFX;
  then consider Z2 such that
A2: Z2 in {X,Y} and
A3: Z c= Z2 by A1;
  {X,Y} = {X} \/ {Y} by ENUMSET1:1;
  then Z2 in {X} or Z2 in {Y} by A2,XBOOLE_0:def 3;
  hence thesis by A3,TARSKI:def 1;
end;
