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 Th12:
  for T being TopSpace, FX being Subset-Family of T st FX = {}
  holds clf FX = {}
proof
  let T be TopSpace, FX be Subset-Family of T;
  reconsider CFX = clf FX as set;
  set X = the Element of CFX;
  assume
A1: FX = {};
A2: for X be set holds not X in CFX
  proof
    let X be set;
    assume
A3: X in CFX;
    then reconsider X as Subset of T;
    ex V being Subset of T st X = Cl V & V in FX by A3,Def2;
    hence thesis by A1;
  end;
  assume not thesis;
  then X in CFX;
  hence contradiction by A2;
end;
