
theorem
  for R being complete LATTICE for T being lower correct TopAugmentation
of R for S being Scott correct TopAugmentation of R for M being Refinement of S
  ,T holds lambda R = the topology of M
proof
  let R be complete LATTICE;
  let T be lower correct TopAugmentation of R;
  let S be Scott correct TopAugmentation of R;
  let M be Refinement of S,T;
A1: the RelStr of T = the RelStr of R by YELLOW_9:def 4;
A2: (the carrier of R) \/ the carrier of R = the carrier of R;
  the RelStr of S = the RelStr of R by YELLOW_9:def 4;
  then
A3: the carrier of M = the carrier of R by A2,A1,YELLOW_9:def 6;
A4: sigma R = the topology of S by YELLOW_9:51;
  omega R = the topology of T by Def2;
  then (sigma R) \/ omega R is prebasis of M by A4,YELLOW_9:def 6;
  then
A5: FinMeetCl ((sigma R) \/ omega R) is Basis of M by A3,YELLOW_9:23;
  thus lambda R = UniCl FinMeetCl ((sigma R) \/ (omega R)) by Th33
    .= the topology of M by A3,A5,YELLOW_9:22;
end;
