
theorem Th29:
  for S being Scott complete TopLattice holds S is monotone-convergence
proof
  let S be Scott complete TopLattice;
  let D be non empty directed Subset of Omega S;
  thus ex_sup_of D,Omega S by YELLOW_0:17;
A1: Omega S = the TopRelStr of S by Th15;
  then
A2: the RelStr of Omega S = the RelStr of S;
  reconsider E = D as Subset of S by A1;
  let V be open Subset of S;
  assume sup D in V;
  then
A3: sup E in V by A2,YELLOW_0:17,26;
  E is non empty directed Subset of S by A2,WAYBEL_0:3;
  hence thesis by A3,WAYBEL11:def 1;
end;
