reserve X for non empty TopSpace;
reserve Y for non empty TopStruct;
reserve x for Point of Y;
reserve Y for non empty TopStruct;
reserve X for non empty TopSpace;
reserve x,y for Point of X;

theorem Th51:
  for x, y being Point of X holds Cl {x} c= Cl {y} iff meet {G
where G is Subset of X : G is open & y in G} c= meet {G where G is Subset of X
  : G is open & x in G}
proof
  let x, y be Point of X;
  set FX = {G where G is Subset of X : G is open & x in G};
  set FY = {G where G is Subset of X : G is open & y in G};
A1: [#]X in FX;
  thus Cl {x} c= Cl {y} implies meet {G where G is Subset of X : G is open & y
  in G} c= meet {G where G is Subset of X : G is open & x in G}
  proof
    assume
A2: Cl {x} c= Cl {y};
    now
      let P be object;
      assume P in FX;
      then consider G being Subset of X such that
A3:   G = P and
A4:   G is open and
A5:   x in G;
      now
        assume not y in G;
        then {y} misses G by ZFMISC_1:50;
        then
A6:     (Cl {y}) misses G by A4,TSEP_1:36;
        {x} c= Cl {x} by PRE_TOPC:18;
        then x in Cl {x} by ZFMISC_1:31;
        hence contradiction by A2,A5,A6,XBOOLE_0:3;
      end;
      hence P in FY by A3,A4;
    end;
    then FX c= FY;
    hence thesis by A1,SETFAM_1:6;
  end;
  assume
A7: meet {G where G is Subset of X : G is open & y in G} c= meet {G
  where G is Subset of X : G is open & x in G};
  set G = (Cl {y})`;
  assume
A8: not Cl {x} c= Cl {y};
  not x in Cl {y} by A8,TOPS_1:5,ZFMISC_1:31;
  then x in G by SUBSET_1:29;
  then G in FX;
  then meet FX c= G by SETFAM_1:3;
  then
A9: meet FY c= G by A7;
  {y} c= Cl {y} by PRE_TOPC:18;
  then
A10: y in Cl {y} by ZFMISC_1:31;
  {y} c= MaxADSet(y) by Th12;
  then
A11: y in MaxADSet(y) by ZFMISC_1:31;
  MaxADSet(y) c= meet FY by Th46;
  then y in meet FY by A11;
  hence contradiction by A9,A10,XBOOLE_0:def 5;
end;
