
theorem Th12:
  for X being set holds Fin X is c=directed c=filtered
proof
  let X be set;
  now
    let a,b be set;
    assume
A1: a in Fin X & b in Fin X;
    take c = a \/ b;
    thus a \/ b c= c;
    a c= X & b c= X by A1,FINSUB_1:def 5;
    then a \/ b c= X by XBOOLE_1:8;
    hence c in Fin X by A1,FINSUB_1:def 5;
  end;
  hence Fin X is c=directed by Th6;
  now
    reconsider c = {} as set;
    let a,b be set;
    assume that
    a in Fin X and
    b in Fin X;
    take c;
    thus c c= a /\ b;
    thus c in Fin X by FINSUB_1:7;
  end;
  hence thesis by Th8;
end;
