theorem Th46:
  FB is being_ultrafilter implies for a holds a in FB iff not a` in FB
proof
  assume
A1: FB is being_ultrafilter;
  let a;
  thus a in FB implies not a` in FB
  proof
    assume a in FB & a` in FB;
    then a"/\"a` in FB by Th8;
    then Bottom B in FB by LATTICES:20;
    then FB = the carrier of B by Th26;
    hence contradiction by A1;
  end;
  thus thesis by A1,Th44;
end;
