reserve T, T1 for non empty TopSpace;
reserve F,G,H for Subset-Family of T,
  A,B,C,D for Subset of T,
  O,U for open Subset of T,
  p,q for Point of T,
  x,y,X for set;
reserve Un for FamilySequence of T,
  r,r1,r2 for Real,
  n for Element of NAT;

theorem Th19:
  for T st T is regular for Bn being FamilySequence of T st (Union
Bn) is Basis of T holds for U being Subset of T,p being Point of T st U is open
  & p in U ex O being Subset of T st p in O & Cl O c= U & O in Union Bn
proof
  let T;
  assume
A1: T is regular;
  let Bn be FamilySequence of T;
  assume
A2: (Union Bn) is Basis of T;
  for U,p st U is open & p in U ex O being Subset of T st p in O & Cl O c=
  U & O in Union Bn
  proof
    let U,p;
    assume that
    U is open and
A3: p in U;
    now
      per cases;
      suppose
A4:     U=the carrier of T;
        consider O being Subset of T such that
A5:     O in Union Bn & p in O and
        O c= U by A2,A3,YELLOW_9:31;
        take O;
        Cl O c= U by A4;
        hence thesis by A5;
      end;
      suppose
        U<>the carrier of T;
        then U c< the carrier of T by XBOOLE_0:def 8;
        then
A6:     U`<>{} by XBOOLE_1:105;
        p in U`` by A3;
        then consider W,V being Subset of T such that
A7:     W is open and
A8:     V is open and
A9:     p in W and
A10:    U` c= V and
A11:    W misses V by A1,A6;
        consider O being Subset of T such that
A12:    O in Union Bn & p in O and
A13:    O c= W by A2,A7,A9,YELLOW_9:31;
        W c=V` by A11,SUBSET_1:23;
        then O c=V` by A13;
        then Cl O c= Cl V` by PRE_TOPC:19;
        then
A14:    Cl O c= V` by A8,PRE_TOPC:22;
        take O;
        V` c= U by A10,SUBSET_1:17;
        hence thesis by A12,A14,XBOOLE_1:1;
      end;
    end;
    hence thesis;
  end;
  hence thesis;
end;
