reserve X for non empty set;

theorem Th10:
  for ET being FMT_TopSpace,a being Element of ET,
  V being a_neighborhood of a holds
  ex O being open Subset of ET st a in O & O c= V
  proof
    let ET be FMT_TopSpace,
    a be Element of ET, V be a_neighborhood of a;
    set O={x where x is Element of ET : V is a_neighborhood of x};
    O is Subset of ET
    proof
      O c= the carrier of ET
      proof
        let x be object;
        assume  x in O;
        then consider t be Element of ET such that
A1:     x=t and
        V is a_neighborhood of t;
        thus thesis by A1;
      end;
      hence thesis;
    end;
    then reconsider O as Subset of ET;
A2: O is open Subset of ET
    proof
      for x be Element of ET st x in O holds O in U_FMT x
      proof
        let x be Element of ET;
        assume  x in O;
        then consider t be Element of ET such that
A3:     x=t and
A4:     V is a_neighborhood of t;
        x is Element of ET & V is Element of U_FMT x by A3,A4,Def5;
        then consider W be Element of U_FMT x such that
A5:     for y be Element of ET st y is Element of W holds
        V is Element of U_FMT y by Def4;
A6:     W c= O
        proof
          let v be object such that E1:v in W;
          U_FMT x is non empty by Th4;
          then W in U_FMT x;
          then reconsider v as Element of ET by E1;
A7:       v is Element of ET & V is Element of U_FMT v by E1,A5;
          U_FMT v is non empty by Th4;
          then V is a_neighborhood of v by A7,Def5;
          hence thesis;
        end;
        W in U_FMT x & U_FMT x is Filter of the carrier of ET
        proof
          hereby
            U_FMT x is non empty by Th4;
            hence W in U_FMT x;
          end;
          thus U_FMT x is Filter of the carrier of ET by Def2;
        end;
        hence thesis by A6,CARD_FIL:def 1;
      end;
      hence thesis by Def1;
    end;
A8: a in O;
    O c= V
    proof
      let x be object;
      assume x in O;
      then consider x0 be Element of ET such that
A9:   x=x0 and
A10:  V is a_neighborhood of x0;
      V in U_FMT x0 by A10,Def5;
      hence x in V by A9,Def3;
    end;
    hence thesis by A2,A8;
  end;
