
theorem Th29:
  for R being complete LATTICE for LL being lower correct
TopAugmentation of R for S being Scott TopAugmentation of R for T being correct
  TopAugmentation of R holds T is Lawson iff T is Refinement of S,LL
proof
  let R be complete LATTICE;
  let LL be lower correct TopAugmentation of R;
  let S be Scott TopAugmentation of R;
  let T be correct TopAugmentation of R;
A1: the topology of S = sigma R by YELLOW_9:51;
A2: (the carrier of R) \/ the carrier of R = the carrier of R;
A3: the topology of LL = omega R by Def2;
A4: the RelStr of LL = the RelStr of R by YELLOW_9:def 4;
A5: the RelStr of S = the RelStr of R by YELLOW_9:def 4;
A6: the RelStr of T = the RelStr of R by YELLOW_9:def 4;
  then
A7: sigma T = sigma R by YELLOW_9:52;
  omega T = omega R by A6,Th3;
  hence thesis by A7,A3,A1,A2,A4,A5,A6,YELLOW_9:def 6;
end;
