
theorem Th15:
  for S being Scott complete TopLattice holds Omega S = the TopRelStr of S
proof
  let S be Scott complete TopLattice;

A1: the TopStruct of Omega S = the TopStruct of S by Def2;
  the InternalRel of Omega S = the InternalRel of S
  proof
    let x, y be object;
    hereby
      assume
A2:   [x,y] in the InternalRel of Omega S;
      then
A3:   y in the carrier of Omega S by ZFMISC_1:87;
      x in the carrier of Omega S by A2,ZFMISC_1:87;
      then reconsider t1 = x, t2 = y as Element of S by A3,Lm1;
      reconsider o1 = x, o2 = y as Element of Omega S by A2,ZFMISC_1:87;
      o1 <= o2 by A2;
      then ex Y2 being Subset of S st Y2 = {o2} & o1 in Cl Y2 by Def2;
      then t1 in downarrow t2 by WAYBEL11:9;
      then ex t3 being Element of S st t3 >= t1 & t3 in {t2} by WAYBEL_0:def 15
;
      then t1 <= t2 by TARSKI:def 1;
      hence [x,y] in the InternalRel of S;
    end;
    assume
A4: [x,y] in the InternalRel of S;
    then reconsider t1 = x, t2 = y as Element of S by ZFMISC_1:87;
    reconsider o1 = x, o2 = y as Element of Omega S by A1,A4,ZFMISC_1:87;
A5: t2 in {t2} by TARSKI:def 1;
    t1 <= t2 by A4;
    then t1 in downarrow t2 by A5,WAYBEL_0:def 15;
    then t1 in Cl {t2} by WAYBEL11:9;
    then o1 <= o2 by Def2;
    hence thesis;
  end;
  hence thesis by A1;
end;
