
theorem Th34:
  for N being meet-continuous 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} = {inf J where J is Subset of N : x in
  J & J in lambda N}
proof
  let N be meet-continuous Lawson complete TopLattice, S be Scott
  TopAugmentation of N, x be Element of N;
  set l = {inf A where A is Subset of N : x in A & A in lambda N}, s = {inf J
  where J is Subset of S : x in J & J in sigma S};
  thus s c= l by Th33;
  let k be object;
  assume k in l;
  then consider A being Subset of N such that
A1: k = inf A and
A2: x in A and
A3: A in lambda N;
A4: the RelStr of N = the RelStr of S by YELLOW_9:def 4;
  then reconsider J = A as Subset of S;
  A is open by A3,Th12;
  then uparrow J is open by Th15;
  then
A5: uparrow J in sigma S by WAYBEL14:24;
A6: J c= uparrow J by WAYBEL_0:16;
  inf A = inf J by A4,YELLOW_0:17,27
    .= inf uparrow J by WAYBEL_0:38,YELLOW_0:17;
  hence thesis by A5,A1,A2,A6;
end;
