reserve a for set;
reserve L for lower-bounded sup-Semilattice;
reserve x for Element of L;
reserve L for complete LATTICE;
reserve AR for Relation of L;
reserve x, y, z for Element of L;

theorem
  for L being lower-bounded continuous LATTICE holds
  for x,y being Element of L holds ( x << y iff
  for D being non empty directed Subset of L st y <= sup D
  ex d being Element of L st d in D & x << d )
proof
  let L be lower-bounded continuous LATTICE;
  let x,y be Element of L;
  hereby
    assume
A1: x << y;
    let D be non empty directed Subset of L;
    assume
A2: y <= sup D;
    consider x9 be Element of L such that
A3: x << x9 and
A4: x9 << y by A1,Th52;
    ex d be Element of L st ( d in D)&( x9 <= d) by A2,A4,WAYBEL_3:def 1;
    hence ex d be Element of L st d in D & x << d by A3,WAYBEL_3:2;
  end;
  assume
A5: for D be non empty directed Subset of L st y <= sup D
  ex d be Element of L st d in D & x << d;
  for D being non empty directed Subset of L st y <= sup D
  ex d being Element of L st d in D & x <= d
  proof
    let D be non empty directed Subset of L;
    assume y <= sup D;
    then ex d be Element of L st ( d in D)&( x << d) by A5;
    hence thesis by WAYBEL_3:1;
  end;
  hence thesis by WAYBEL_3:def 1;
end;
