
theorem
  for L being complete LATTICE, x,y being Element of L
  st for X being Subset of L st y <= sup X
  ex A being finite Subset of L st A c= X & x <= sup A holds x << y
proof
  let L be complete LATTICE, x,y be Element of L such that
A1: for X being Subset of L st y <= sup X
  ex A being finite Subset of L st A c= X & x <= sup A;
  let D be non empty directed Subset of L;
  assume y <= sup D;
  then consider A being finite Subset of L such that
A2: A c= D and
A3: x <= sup A by A1;
  reconsider B = A as finite Subset of D by A2;
  consider a being Element of L such that
A4: a in D and
A5: a is_>=_than B by WAYBEL_0:1;
  take a;
  a >= sup A by A5,YELLOW_0:32;
  hence thesis by A3,A4,YELLOW_0:def 2;
end;
