
theorem
  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
  for k1 being Nat, k2 being Nat holds A1.k1 /\ A1.k2 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}};
  let k1,k2 be Nat;
  D1: dom A1=NAT by FUNCT_2:def 1; then
   k1 in dom A1 by ORDINAL1:def 12; then
  B1: A1.k1 in MySigmaField by FUNCT_1:3, A0;
   k2 in dom A1 by D1,ORDINAL1:def 12; then
  B2: A1.k2 in MySigmaField by FUNCT_1:3,A0;
  A1.k1 /\ A1.k2 in MySigmaField
  proof
    (A1.k1 = {} or A1.k1 = {1,2,3,4}) &
      (A1.k2 = {} or A1.k2 = {1,2,3,4}) by B2,B1,A0,TARSKI:def 2;
    hence thesis by A0,TARSKI:def 2;
  end;
  hence thesis;
 end;
