
theorem
  for L be complete LATTICE for x,y,k be Element of L holds x <= k & k
  <= y & k in the carrier of CompactSublatt L implies x << y
proof
  let L be complete LATTICE;
  let x,y,k be Element of L;
  assume that
A1: x <= k & k <= y and
A2: k in the carrier of CompactSublatt L;
  k is compact by A2,Def1;
  then k << k by WAYBEL_3:def 2;
  hence thesis by A1,WAYBEL_3:2;
end;
