reserve PM for MetrStruct;
reserve x,y for Element of PM;
reserve r,p,q,s,t for Real;
reserve T for TopSpace;
reserve A for Subset of T;
reserve T for non empty TopSpace;
reserve x for Point of T;
reserve Z,X,V,W,Y,Q for Subset of T;
reserve FX for Subset-Family of T;
reserve a for set;
reserve x,y for Point of T;
reserve A,B for Subset of T;
reserve FX,GX for Subset-Family of T;

theorem Th17:
  FX is_finer_than clf FX
proof
  set GX = clf FX;
  for X be set st X in FX ex Y be set st Y in GX & X c= Y
  proof
    let X be set;
    assume
A1: X in FX;
    then reconsider X as Subset of T;
    thus thesis
    proof
      reconsider Y = Cl X as set;
      take Y;
      thus Y in GX by A1,Def2;
      thus thesis by PRE_TOPC:18;
    end;
  end;
  hence thesis by SETFAM_1:def 2;
end;
