reserve TS for 1-sorted,
  K, Q for Subset of TS;
reserve TS for TopSpace,
  GX for TopStruct,
  x for set,
  P, Q for Subset of TS,
  K , L for Subset of TS,
  R, S for Subset of GX,
  T, W for Subset of GX;

theorem Th12:
  for GX being non empty TopSpace, R being Subset of GX, p being
Point of GX holds p in Cl R iff for T being Subset of GX st T is open & p in T
  holds R meets T
proof
  let GX be non empty TopSpace, R be Subset of GX, p be Point of GX;
  hereby
    assume
A1: p in Cl R;
    given T being Subset of GX such that
A2: T is open and
A3: p in T and
A4: R misses T;
A5: (R`) \/ T` = (R /\ T)` by XBOOLE_1:54;
A6: R /\ T = {} GX by A4;
A7: R c= T`
    proof
      let x be object;
      assume
A8:   x in R;
      then x in R` or x in T` by A5,A6,XBOOLE_0:def 3;
      hence thesis by A8,XBOOLE_0:def 5;
    end;
    Cl(T`) = T` by A2,PRE_TOPC:22;
    then Cl R c= T` by A7,PRE_TOPC:19;
    hence contradiction by A1,A3,XBOOLE_0:def 5;
  end;
  assume
A9: for T being Subset of GX st T is open & p in T holds R meets T;
  assume not p in Cl R;
  then p in (Cl R)` by SUBSET_1:29;
  then
A10: R meets (Cl R)` by A9;
  R misses R` by XBOOLE_1:79;
  then
A11: R /\ R` = {};
  R c= Cl R by PRE_TOPC:18;
  then (Cl R)` c= R` by SUBSET_1:12;
  then R /\ (Cl R)` = {} by A11,XBOOLE_1:3,26;
  hence contradiction by A10;
end;
