reserve T for non empty TopSpace,
  X,Z for Subset of T;
reserve x,y for Element of OpenClosedSet(T);
reserve x,y,X for set;
reserve BL for non trivial B_Lattice,
  a,b,c,p,q for Element of BL,
  UF,F,F0,F1,F2 for Filter of BL;

theorem Th23:
  x in StoneR BL iff ex a st (UFilter BL).a =x
proof
A1: x in StoneR BL implies ex a st (UFilter BL).a =x
  proof
    assume x in StoneR BL;
    then consider y being object such that
A2: y in dom UFilter BL and
A3: x = UFilter BL.y by FUNCT_1:def 3;
    reconsider a=y as Element of BL by A2;
    take a;
    thus thesis by A3;
  end;
  ( ex a st UFilter BL.a =x ) implies x in StoneR BL
  proof
    given a such that
A4: x=UFilter BL.a;
    a in the carrier of BL;
    then a in dom UFilter BL by Def6;
    hence thesis by A4,FUNCT_1:def 3;
  end;
  hence thesis by A1;
end;
