reserve X for non empty set;

theorem Th13:
  for T being non empty TopSpace, ET being FMT_TopSpace st
  the carrier of T = the carrier of ET &
  Family_open_set(ET) = the topology of T holds
  for x being Element of ET holds
  U_FMT x = {V where V is Subset of ET:
  ex O being Subset of T st O in the topology of T & x in O & O c= V}
  proof
    let T be non empty TopSpace,
    ET be FMT_TopSpace;
    assume that
    the carrier of T = the carrier of ET and
A1: Family_open_set(ET)=the topology of T;
A2: for o be set st o in Family_open_set(ET) holds
    for x be Element of ET st x in o holds o in U_FMT x
    proof
      let o be set;
      assume o in Family_open_set(ET);
      then consider O be open Subset of ET such that
A3:   o=O;
      thus thesis by A3,Def1;
    end;
    for x be Element of ET holds
    U_FMT x = {V where V is Subset of ET:
    ex O be Subset of T st O in the topology of T & x in O & O c= V}
    proof
      let x be Element of ET;
A4:   U_FMT x c= {V where V is Subset of ET:
      ex O be Subset of T st O in the topology of T & x in O & O c= V}
      proof
        let t be object;
        assume
A5:     t in U_FMT x; then
A6:     t is a_neighborhood of x by Def5;
        reconsider t as Subset of ET by A5;
        consider OO be open Subset of ET such that
A7:     x in OO and
A8:     OO c= t by A6,Th10;
A9:     OO in Family_open_set(ET);
        then reconsider OO as Subset of T by A1;
        thus thesis by A7,A8,A9,A1;
      end;
      {V where V is Subset of ET:
      ex O be Subset of T st O in the topology of T & x in O & O c= V} c=
      U_FMT x
      proof
        let t be object;
        assume t in {V where V is Subset of ET:
        ex O be Subset of T st O in the topology of T & x in O & O c= V};
        then consider V be Subset of ET such that
A10:    t=V and
A11:    ex O be Subset of T st O in the topology of T & x in O & O c= V;
        consider O2 be Subset of T such that
A12:    O2 in the topology of T and
A13:    x in O2 and
A14:    O2 c= V by A11;
A15:    O2 in U_FMT x by A12,A1,A2,A13;
        U_FMT x is Filter of the carrier of ET by Def2;
        hence thesis by A14,CARD_FIL:def 1,A15,A10;
      end;
      hence thesis by A4;
    end;
    hence thesis;
  end;
