
theorem Th48:
  for S being non empty reflexive RelStr, T being TopAugmentation of S
  st the topology of T = sigma S holds T is correct
proof
  let R be non empty reflexive RelStr;
  let T be TopAugmentation of R such that
A1: the topology of T = sigma R;
A2: the RelStr of T = the RelStr of R by Def4;
  set IT = ConvergenceSpace Scott-Convergence R;
A3: the carrier of IT = the carrier of R by YELLOW_6:def 24;
  then
A4: the carrier of T in sigma R by A2,PRE_TOPC:def 1;
A5: for a being Subset-Family of T st a c= the topology of T holds union a
  in the topology of T by A1,A2,A3,PRE_TOPC:def 1;
  for a,b being Subset of T st a in the topology of T & b in the topology
  of T holds a /\ b in the topology of T by A1,PRE_TOPC:def 1;
  hence thesis by A1,A4,A5;
end;
