
theorem Th1:
  for LL being non empty RelStr ex T being strict correct
  TopAugmentation of LL st T is lower
proof
  let LL be non empty RelStr;
  set A = the set of all (uparrow x)` where x is Element of LL;
  A c= bool the carrier of LL
  proof
    let a be object;
    assume a in A;
    then ex x being Element of LL st a = (uparrow x)`;
    hence thesis;
  end;
  then reconsider A as Subset-Family of LL;
  set T = TopRelStr(#the carrier of LL, the InternalRel of LL, UniCl FinMeetCl
    A#);
  reconsider S = TopStruct(#the carrier of LL, UniCl FinMeetCl A#) as non
  empty TopSpace by CANTOR_1:15;
A1: the TopStruct of T = S;
  T is TopSpace-like
  by A1,PRE_TOPC:def 1;
  then reconsider T as strict non empty TopSpace-like TopRelStr;
  take T;
  set BB = the set of all (uparrow x)` where x is Element of T;
  the RelStr of T = the RelStr of LL;
  hence T is strict correct TopAugmentation of LL by YELLOW_9:def 4;
A2: A is prebasis of S by CANTOR_1:18;
  then consider F being Basis of S such that
A3: F c= FinMeetCl A by CANTOR_1:def 4;
A4: the topology of T c= UniCl F by CANTOR_1:def 2;
  F c= the topology of T by TOPS_2:64;
  then
A5: F is Basis of T by A4,CANTOR_1:def 2,TOPS_2:64;
  the RelStr of T = the RelStr of LL;
  then
A6: A = BB by Lm1;
  A c= the topology of S by A2,TOPS_2:64;
  hence BB is prebasis of T by A5,A3,A6,CANTOR_1:def 4,TOPS_2:64;
end;
