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 Th28:
  for GX being non empty TopSpace, R being Subset of GX, p being
Point of GX holds p in Fr R iff for S being Subset of GX st S is open & p in S
  holds R meets S & R` meets S
proof
  let GX be non empty TopSpace, R be Subset of GX, p be Point of GX;
  hereby
    assume
A1: p in Fr R;
    then
A2: p in Cl R` by XBOOLE_0:def 4;
    let S be Subset of GX;
    assume that
A3: S is open and
A4: p in S;
    p in Cl R by A1,XBOOLE_0:def 4;
    hence R meets S & R` meets S by A3,A4,A2,Th12;
  end;
  assume
A5: for S being Subset of GX st S is open & p in S holds R meets S & R`
  meets S;
  then for S being Subset of GX st S is open & p in S holds R` meets S;
  then
A6: p in Cl R` by Th12;
  for S being Subset of GX st S is open & p in S holds R meets S by A5;
  then p in Cl R by Th12;
  hence thesis by A6,XBOOLE_0:def 4;
end;
