
theorem Th13:
  for S, T being TopSpace st the TopStruct of S = the TopStruct of
  T holds Omega S = Omega T
proof
  let S, T be TopSpace such that
A1: the TopStruct of S = the TopStruct of T;
A2: the TopStruct of Omega S = the TopStruct of S by Def2
    .= the TopStruct of Omega T by A1,Def2;
  the InternalRel of Omega S = the InternalRel of Omega T
  proof
    let a,b be object;
    thus [a,b] in the InternalRel of Omega S implies [a,b] in the InternalRel
    of Omega T
    proof
      assume
A3:   [a,b] in the InternalRel of Omega S;
      then reconsider s1 = a, s2 = b as Element of Omega S by ZFMISC_1:87;
      reconsider t1 = s1, t2 = s2 as Element of Omega T by A2;
      s1 <= s2 by A3;
      then consider Y being Subset of S such that
A4:   Y = {s2} and
A5:   s1 in Cl Y by Def2;
      reconsider Z = Y as Subset of T by A1;
      t1 in Cl Z by A1,A5,TOPS_3:80;
      then t1 <= t2 by A4,Def2;
      hence thesis;
    end;
    assume
A6: [a,b] in the InternalRel of Omega T;
    then reconsider s1 = a, s2 = b as Element of Omega T by ZFMISC_1:87;
    reconsider t1 = s1, t2 = s2 as Element of Omega S by A2;
    s1 <= s2 by A6;
    then consider Y being Subset of T such that
A7: Y = {s2} and
A8: s1 in Cl Y by Def2;
    reconsider Z = Y as Subset of S by A1;
    t1 in Cl Z by A1,A8,TOPS_3:80;
    then t1 <= t2 by A7,Def2;
    hence thesis;
  end;
  hence thesis by A2;
end;
