
theorem Lm9:
  for MySigmaField, MySet being set,
      A1 being SetSequence of MySet st MySet={1,2,3,4} &
    rng A1 c= MySigmaField & MySigmaField = {{},{1,2,3,4}} holds
  Intersection A1 <> {} implies Intersection A1 in MySigmaField
proof
 let MySigmaField, MySet be set;
 let A1 be SetSequence of MySet;
 assume A0: MySet = {1,2,3,4} &
        rng A1 c= MySigmaField & MySigmaField = {{},{1,2,3,4}};
 assume A5: Intersection A1 <> {};
 D1: dom A1=NAT by FUNCT_2:def 1;
 A4: for n being Nat holds A1.n={} or A1.n={1,2,3,4}
 proof
  let n be Nat;
   A1.n in MySigmaField
   proof
    n in dom A1 by D1,ORDINAL1:def 12;
   hence thesis by FUNCT_1:3, A0;
  end;
 hence thesis by A0,TARSKI:def 2;
 end;
   H1: (ex n being Nat st A1.n={}) implies Intersection A1={}
   proof
     assume ex n being Nat st A1.n={}; then
     for x being object holds x in Intersection A1 iff x in {}
       by PROB_1:13;
     hence thesis by TARSKI:def 3;
   end;
   Intersection A1 = {1,2,3,4}
   proof
      for x being object holds
       x in Intersection A1 iff x in {1,2,3,4}
      proof
       let x be object;
       x in Intersection A1 iff for n being Nat holds x in {1,2,3,4}
       proof
        thus x in Intersection A1 implies
             for n being Nat holds x in {1,2,3,4}
        proof
         assume G1: x in Intersection A1;
         for n being Nat holds x in {1,2,3,4}
         proof
          let n be Nat;
          for x being object holds x in A1.n iff x in {1,2,3,4} by H1,A4,A5;
         hence thesis by PROB_1:13,G1;
         end;
        hence thesis;
        end;
         assume G1: for n being Nat holds x in {1,2,3,4};
         for n being Nat holds x in A1.n
         proof
          let n be Nat;
          x in {1,2,3,4} by G1;
          hence thesis by H1,A4,A5;
         end;
        hence thesis by PROB_1:13;
       end;
      hence thesis;
      end;
   hence thesis by TARSKI:def 3;
   end;
  hence thesis by A0,TARSKI:def 2;
end;
