
theorem Lm8:
  for MyOmega being set,
      A1 being SetSequence of {1,2,3,4} st
     rng A1 c= MyOmega &
     MyOmega = {{},{1,2},{3,4},{1,2,3,4}}
  holds Intersection A1 in MyOmega
 proof
  let MyOmega be set;
  let A1 be SetSequence of {1,2,3,4};
  assume A0: rng A1 c= MyOmega & MyOmega = {{},{1,2},{3,4},{1,2,3,4}};
   Intersection A1 in {{},{1,2},{3,4},{1,2,3,4}}
   proof
    per cases;
    suppose CASE000: ex n being Nat,k being Nat st A1.n /\ A1.k = {};
    CASE0: ex n being Nat,k being Nat st A1.n misses A1.k
    proof
     consider n being Nat,k being Nat such that
      BCASE0: A1.n /\ A1.k = {} by CASE000;
      A1.n misses A1.k by BCASE0;
      hence thesis;
    end;
    CASE1: for y being object holds
     (for n being Nat,k being Nat holds y in A1.k & y in A1.n) iff y in {}
    proof
     let y be object;
     (for n being Nat,k being Nat holds y in A1.k & y in A1.n) implies
         y in {}
     proof
      assume G1: for n being Nat,k being Nat holds y in A1.k & y in A1.n;
      ex n being Nat,k being Nat st y in A1.k & y in A1.n & y in {}
      proof
       consider n being Nat,k being Nat such that
        H1: A1.n misses A1.k by CASE0;
       (y in A1.k & y in A1.n) iff y in {} by H1,XBOOLE_0:3;
      hence thesis by G1;
      end;
     hence thesis;
     end;
    hence thesis;
    end;
    Intersection A1 c= {}
    proof
     let y be object;
     y in Intersection A1 iff
      (for n being Nat,k being Nat holds y in A1.k & y in A1.n) by PROB_1:13;
     hence thesis by CASE1;
    end; then
    Intersection A1={};
    hence thesis by ENUMSET1:def 2;
    end;
    suppose CASE0: not ex n being Nat,k being Nat st A1.n /\ A1.k = {};
    Intersection A1 in {{},{1,2},{3,4},{1,2,3,4}}
     proof
      Intersection A1 = {} or
       Intersection A1 = {1,2} or Intersection A1 = {3,4} or
       Intersection A1 = {1,2,3,4} by CASE0,A0,Lm700000;
      hence thesis by ENUMSET1:def 2;
     end;
   hence thesis;
  end;
 end;
 hence thesis by A0;
end;
