
theorem SuperLemma1:
  for A1 being SetSequence of {1,2,3,4},
      w being Real st w = 1 or w = 3 holds
    (for n being Nat holds
      A1.n = {} or A1.n = {1,2} or A1.n = {3,4} or A1.n = {1,2,3,4})
  implies {w}<>Intersection A1
proof
  let A1 be SetSequence of {1,2,3,4};
  let w be Real;
  assume KX: w = 1 or w = 3;
  assume 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};
  per cases;
  suppose Intersection A1={};
    hence thesis;
  end;
  suppose SUPP1: Intersection A1<>{};
        ex x being object st
         (for n being Nat holds x in A1.n) & not x in {w}
        proof
         per cases;
         suppose ex j being Nat st A1.j={};
         then consider j being Nat such that J0: A1.j={};
         thus thesis by The1,J0,SUPP1;
         end;
         suppose USUPP1: for n being Nat holds A1.n<>{};
          per cases;
          suppose P1: for j being Nat holds {1,2} c= A1.j;
           set x=2;
           Z1: not x in {w} by TARSKI:def 1,KX;
           for n being Nat holds x in A1.n
           proof
            let n be Nat;
            per cases by S2;
            suppose A1.n={};
              hence thesis by USUPP1;
            end;
            suppose A1.n = {1,2};
              hence thesis by TARSKI:def 2;
            end;
            suppose J0: A1.n = {3,4};
              x in {1,2} by TARSKI:def 2; then
              not {1,2} c= A1.n by J0,TARSKI:def 2;
              hence thesis by P1;
            end;
            suppose A1.n = {1,2,3,4};
              hence thesis by ENUMSET1:def 2;
            end;
           end;
          hence thesis by Z1;
          end;
          suppose ex j being Nat st not {1,2} c= A1.j; then
           consider j being Nat such that BSUPP1: not {1,2} c= A1.j;
           per cases;
           suppose P1: for n being Nat holds {3,4} c= A1.n;
           set x=4;
           Z1: not x in {w} by TARSKI:def 1,KX;
           for n being Nat holds x in A1.n
           proof
            let n be Nat;
            per cases by S2;
            suppose A1.n={};
              hence thesis by USUPP1;
            end;
            suppose A1.n = {3,4};
              hence thesis by TARSKI:def 2;
            end;
            suppose J0: A1.n = {1,2};
              x in {3,4} by TARSKI:def 2; then
              not {3,4} c= A1.n by J0,TARSKI:def 2;
              hence thesis by P1;
            end;
            suppose A1.n = {1,2,3,4};
              hence thesis by ENUMSET1:def 2;
            end;
           end;
           hence thesis by Z1;
           end;
           suppose ex k being Nat st not {3,4} c= A1.k;
           then consider k being Nat such that CSUPP1: not {3,4} c= A1.k;
           ZW1: A1.k /\ A1.j={}
           proof
            per cases by S2,Lm2,BSUPP1;
            suppose DSUPP1: A1.j = {3,4};
              A1.k = {1,2} or A1.k = {} by S2,CSUPP1,Lm1;
              hence thesis by Lm3,DSUPP1;
            end;
            suppose A1.j = {};
              hence thesis;
            end;
           end;
           Intersection A1={}
           proof
            Intersection A1 c= A1.k /\ A1.j
            proof
             for x being object st x in Intersection A1 holds x in A1.k /\ A1.j
             proof
              let x be object;
              assume x in Intersection A1; then
              x in A1.k & x in A1.j by PROB_1:13;
             hence thesis by XBOOLE_0:def 4;
             end;
              hence thesis;
            end;
           hence thesis by ZW1;
           end;
           hence thesis by SUPP1;
           end;
          end;
         end;
      end;
     hence thesis by PROB_1:13;
    end;
end;
