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;
