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
  A1 is SetSequence of Si iff for n holds A1.n is Event of Si
proof
  thus A1 is SetSequence of Si implies for n holds A1.n is Event of
  Si
  proof
    assume A1 is SetSequence of Si;
    then
A1: rng A1 c= Si by RELAT_1:def 19;
    for n holds A1.n is Event of Si
    by NAT_1:51,A1;
    hence thesis;
  end;
  assume
A2: for n holds A1.n is Event of Si;
  for n be Nat holds A1.n in Si
  proof
    let n be Nat;
    A1.n is Event of Si by A2;
    hence thesis;
  end;
  then rng A1 c= Si by NAT_1:52;
  hence thesis by RELAT_1:def 19;
end;
