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;

theorem Th3:
  FX is Cover of T implies for x ex W st x in W & W in FX
proof
  assume FX is Cover of T;
  then
A1: union FX = [#](T) by SETFAM_1:45;
  thus thesis
  proof
    let x;
    thus ex W st x in W & W in FX
    proof
      consider W being set such that
A2:   x in W and
A3:   W in FX by A1,TARSKI:def 4;
      reconsider W as Subset of T by A3;
      take W;
      thus thesis by A2,A3;
    end;
  end;
end;
