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 Th13:
  FX = { V } implies clf FX = { Cl V }
proof
  reconsider CFX = clf FX as set;
  assume
A1: FX = { V };
  for W be object holds W in CFX iff W = Cl V
  proof
    let W be object;
A2: W = Cl V implies W in CFX
    proof
      assume
A3:   W = Cl V;
      ex X st W = Cl X & X in FX
      proof
        take V;
        thus thesis by A1,A3,TARSKI:def 1;
      end;
      hence thesis by Def2;
    end;
    W in CFX implies W = Cl V
    proof
      assume
A4:   W in CFX;
      then reconsider W as Subset of T;
      ex X st W = Cl X & X in FX by A4,Def2;
      hence thesis by A1,TARSKI:def 1;
    end;
    hence thesis by A2;
  end;
  hence thesis by TARSKI:def 1;
end;
