reserve R for non empty RelStr,
  N for net of R,
  i for Element of N;

theorem Th43:
  for L being complete continuous Scott TopLattice,
  p be Element of L, S be Subset of L st S is open & p in S
  ex q being Element of L st q << p & q in S
proof
  let L be complete continuous Scott TopLattice, p be Element of L;
  let S be Subset of L such that
A1: S is open and
A2: p in S;
A3: S is inaccessible by A1,Def4;
  sup waybelow p = p by WAYBEL_3:def 5;
  then waybelow p meets S by A2,A3;
  then (waybelow p) /\ S <> {};
  then consider u being Element of L such that
A4: u in (waybelow p) /\ S by SUBSET_1:4;
  reconsider u as Element of L;
  take u;
  u in waybelow p by A4,XBOOLE_0:def 4;
  hence u << p by WAYBEL_3:7;
  thus thesis by A4,XBOOLE_0:def 4;
end;
