reserve X for non empty set;

theorem Th9:
  for ET being FMT_TopSpace holds
  {} in Family_open_set(ET) &
  the carrier of ET in Family_open_set(ET) &
  (for a being Subset-Family of ET st a c= Family_open_set(ET) holds
    union a in Family_open_set(ET)) &
  (for a, b being Subset of ET st a in Family_open_set(ET) &
    b in Family_open_set(ET) holds a /\ b in Family_open_set(ET))
  proof
    let ET be FMT_TopSpace;
A1: ET is U_FMT_filter;
    thus {} in Family_open_set(ET)
    proof
      set S={};
      S c= the carrier of ET;
      then reconsider S as Subset of ET;
      S is open;
      then reconsider S as open Subset of ET;
      S in the set of all O where O is open Subset of ET;
      hence thesis;
    end;
    thus the carrier of ET in Family_open_set(ET)
    proof
      set S=the carrier of ET;
      S c= the carrier of ET;
      then reconsider S as Subset of ET;
      for x be Element of ET st x in S holds S in U_FMT x
      proof
        let x be Element of ET;
        assume x in S;
        U_FMT x is Filter of the carrier of ET by A1;
        hence S in U_FMT x by CARD_FIL:5;
      end;
      then S is open;
      then reconsider S as open Subset of ET;
      S in the set of all O where O is open Subset of ET;
      hence thesis;
    end;
    thus ( for a being Subset-Family of ET st a c= Family_open_set(ET) holds
    union a in Family_open_set(ET) )
    proof
      let a being Subset-Family of ET such that
A2:   a c= Family_open_set(ET);
      reconsider UA=union a as Subset of ET;
      now
        let x be Element of ET;
        assume x in UA;
        then consider Y be set such that
A3:     x in Y and
A4:     Y in a by TARSKI:def 4;
        Y in the set of all O where O is open Subset of ET by A2,A4;
        then consider Y0 be open Subset of ET such that
A5:     Y=Y0;
A6:     Y in U_FMT x by A3,A5,Def1;
A7:     Y c= UA by A4,ZFMISC_1:74;
        U_FMT x is Filter of the carrier of ET by A1;
        hence UA in U_FMT x by A6,A7,CARD_FIL:def 1;
      end;
      then UA is open;
      hence thesis;
    end;
    thus for a, b being Subset of ET st a in Family_open_set(ET) &
      b in Family_open_set(ET)
      holds
    a /\ b in Family_open_set(ET)
    proof
      let a,b being Subset of ET such that
A9:   a in Family_open_set(ET) and
A10:  b in Family_open_set(ET);
      now
        let x be Element of ET such that
A11:    x in a/\b;
A12:    x in a & x in b by A11,XBOOLE_0:def 4;
        consider a0 be open Subset of ET such that
A13:    a = a0 by A9;
A14:    a in U_FMT x by A12,A13,Def1;
        consider b0 be open Subset of ET such that
A15:    b = b0 by A10;
A16:    b in U_FMT x by A12,A15,Def1;
        U_FMT x is Filter of the carrier of ET by A1;
        hence a/\b in U_FMT x by A14,A16,CARD_FIL:def 1;
      end;
      then a/\b is open;
      then reconsider AB=a/\b as open Subset of ET;
      AB in the set of all O where O is open Subset of ET;
      hence thesis;
    end;
  end;
