
theorem Th33:
  for N being Lawson complete TopLattice for S being Scott
TopAugmentation of N for x being Element of N holds {inf A where A is Subset of
S: x in A & A in sigma S} c= {inf J where J is Subset of N : x in J & J in
  lambda N}
proof
  let N be Lawson complete TopLattice, S be Scott TopAugmentation of N, x be
  Element of N;
  set s = {inf A where A is Subset of S : x in A & A in sigma S};
  let k be object;
  assume k in s;
  then consider J being Subset of S such that
A1: k = inf J and
A2: x in J and
A3: J in sigma S;
A4: the RelStr of N = the RelStr of S by YELLOW_9:def 4;
  then reconsider A = J as Subset of N;
  sigma N c= lambda N by Th10;
  then
A5: sigma S c= lambda N by A4,YELLOW_9:52;
  inf A = inf J by A4,YELLOW_0:17,27;
  hence thesis by A5,A1,A2,A3;
end;
