reserve Omega for set;
reserve X, Y, Z, p,x,y,z for set;
reserve D, E for Subset of Omega;
reserve f for Function;
reserve m,n for Nat;
reserve r,r1 for Real;
reserve seq for Real_Sequence;
reserve F for Field_Subset of X;
reserve ASeq,BSeq for SetSequence of Omega;
reserve A1 for SetSequence of X;
reserve Sigma for SigmaField of Omega;
reserve Si for SigmaField of X;

theorem Th17:
  for ASeq being SetSequence of Si holds Union ASeq in Si
proof
  let ASeq be SetSequence of Si;
A1: rng A1 c= Si implies for n being Nat holds (Complement A1).n in Si
  proof
    assume
A2: rng A1 c= Si;
    let n be Nat;
    A1.n in rng A1 by NAT_1:51;
    then n in NAT & (A1.n)` in Si by A2,Def1,ORDINAL1:def 12;
    hence thesis by Def2;
  end;
A3: rng A1 c= Si implies Union Complement Complement A1 in Si
  proof
    assume rng A1 c= Si;
    then for n being Nat holds (Complement A1).n in Si by A1;
    then rng Complement A1 c= Si by NAT_1:52;
    then Intersection Complement A1 in Si by Def6;
    then (Union Complement Complement A1)`` in Si by Def1;
    hence thesis;
  end;
A4: for A1 st rng A1 c= Si holds Union A1 in Si
  proof
    let A1;
    assume rng A1 c= Si;
    then Union Complement Complement A1 in Si by A3;
    hence thesis;
  end;
  rng ASeq c= Si by RELAT_1:def 19;
  hence thesis by A4;
end;
