
theorem Th22:
  for N being meet-continuous Lawson complete TopLattice st
  InclPoset sigma N is continuous holds N is topological_semilattice
proof
  let N be meet-continuous Lawson complete TopLattice such that
A1: InclPoset sigma N is continuous;
  set NN = the Lawson correct TopAugmentation of [:N,N:];
  N is TopAugmentation of N by YELLOW_9:44;
  then
A2: the TopStruct of NN = [:N,N qua TopSpace:] by A1,Th20;
A3: the RelStr of NN = the RelStr of [:N,N:] by YELLOW_9:def 4;
  then reconsider h = inf_op N as Function of NN, N;
A4: the RelStr of N = the RelStr of N;
A5: h is directed-sups-preserving
  proof
    let X be Subset of NN;
    assume X is non empty directed;
    then reconsider X1 = X as non empty directed Subset of [:N,N:] by A3,
WAYBEL_0:3;
    inf_op N preserves_sup_of X1 by WAYBEL_0:def 37;
    hence thesis by A3,A4,WAYBEL_0:65;
  end;
A6: h is infs-preserving
  by A3,WAYBEL_0:def 32,A4,WAYBEL_0:65;
  then h is SemilatticeHomomorphism of NN, N by WAYBEL21:5;
  then
A7: h is continuous by A5,A6,WAYBEL21:46;
A8: the TopStruct of N = the TopStruct of N;
  let g be Function of [:N,N qua TopSpace:], N;
  assume g = inf_op N;
  hence thesis by A2,A7,A8,YELLOW12:36;
end;
