
theorem Lm700000:
  for A1 being SetSequence of {1,2,3,4} st
  (for n being Nat,k being Nat holds not (A1.n /\ A1.k = {})) &
  rng A1 c= {{},{1,2},{3,4},{1,2,3,4}} holds
  (Intersection A1 = {} or
   Intersection A1 = {1,2} or Intersection A1 = {3,4} or
   Intersection A1 = {1,2,3,4})
proof
 let A1 be SetSequence of {1,2,3,4};
 assume GENASS0: (for n being Nat,k being Nat holds not A1.n /\ A1.k = {}) &
 rng A1 c= {{},{1,2},{3,4},{1,2,3,4}};
 set MyOmega={{},{1,2},{3,4},{1,2,3,4}};
 D1: dom A1=NAT by FUNCT_2:def 1;
 S20: for n being Nat holds A1.n in {{},{1,2},{3,4},{1,2,3,4}}
 proof
   let n be Nat;
   n in dom A1 by D1,ORDINAL1:def 12;
   hence thesis by FUNCT_1:3,GENASS0;
 end;
 S2: for n being Nat holds
     A1.n = {} or A1.n = {1,2} or A1.n = {3,4} or A1.n = {1,2,3,4}
 proof
  let n be Nat;
  A1.n in {{},{1,2},{3,4},{1,2,3,4}} by S20;
  hence thesis by ENUMSET1:def 2;
 end;
 Fin1: for n being Nat holds Intersection A1 c= A1.n by PROB_1:13;
   per cases;
   suppose Intersection A1 = {1,2,3,4};
     hence thesis;
   end;
   suppose MYSUPP0: Intersection A1 <> {1,2,3,4};
   W0: Intersection A1 c= {} or Intersection A1 c= {1,2} or
   Intersection A1 c= {3,4}
   proof
    Intersection A1 = {} or Intersection A1 = {1,2} or
    Intersection A1 = {3,4}
    proof
     per cases;
     suppose ASS1: for n being Nat holds A1.n={1,2,3,4};
      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;
        (for n being Nat holds x in A1.n) iff x in {1,2,3,4}
        proof
         (for n being Nat holds x in A1.n) implies x in {1,2,3,4}
         proof
          assume for n being Nat holds x in A1.n; then
          x in A1.0; then
          consider k being Nat such that N10: x in A1.k;
          thus thesis by N10;
         end;
         hence thesis by ASS1;
        end;
       hence thesis by PROB_1:13;
       end;
      hence thesis by TARSKI:2;
      end;
     hence thesis by MYSUPP0;
     end;
     suppose ex n being Nat st A1.n <> {1,2,3,4}; then
      consider n being Nat such that KK: A1.n <> {1,2,3,4};
      per cases by S2;
      suppose A1.n = {};
      hence thesis by The1;
      end;
      suppose JSUPP1: A1.n = {1,2};
  Intersection A1 = {} or Intersection A1 = {1,2}
  proof
   per cases;
   suppose Intersection A1 = {1,2};
     hence thesis;
   end;
   suppose Intersection A1 <> {1,2}; then
    Intersection A1 = {1} or Intersection A1 = {2} or Intersection A1 = {}
      by ZFMISC_1:36,JSUPP1,Fin1;
    hence thesis by SuperLemma2,SuperLemma1,S2;
   end;
  end;
  hence thesis;
    end;
    suppose SUPP1: A1.n = {3,4};
  Intersection A1 = {} or Intersection A1 = {3,4}
  proof
   per cases;
   suppose Intersection A1 = {3,4};
     hence thesis;
   end;
   suppose Intersection A1 <> {3,4}; then
    Intersection A1 = {3} or Intersection A1 = {4} or Intersection A1 = {}
      by SUPP1,Fin1,ZFMISC_1:36;
    hence thesis by SuperLemma1,S2,SuperLemma2;
   end;
  end;
  hence thesis;
      end;
      suppose A1.n = {1,2,3,4};
        hence thesis by KK;
      end;
     end;
    end;
   hence thesis;
   end;
   per cases by W0;
   suppose Intersection A1 c= {};
     hence thesis;
   end;
   suppose ZW10: Intersection A1 c= {1,2};
   per cases;
   suppose Intersection A1 = {1,2};
     hence thesis;
   end;
   suppose Intersection A1 <> {1,2}; then
    Intersection A1 = {1} or Intersection A1 = {2} or Intersection A1 = {}
      by ZW10,ZFMISC_1:36;
    hence thesis by SuperLemma1,S2,SuperLemma2;
   end;
   end;
   suppose ZW10: Intersection A1 c= {3,4};
   per cases;
   suppose Intersection A1={3,4};
     hence thesis;
   end;
   suppose Intersection A1 <> {3,4}; then
    Intersection A1 = {3} or Intersection A1 = {4} or Intersection A1 = {}
      by ZW10,ZFMISC_1:36;
    hence thesis by SuperLemma1,S2,SuperLemma2;
   end;
   end;
   end;
end;
