theorem Th55:
  for F1 being FinSequence of bool X for A1 being SetSequence of X
  st (for k st k in dom F1 holds A1.k = F1.k) &
   (for k st not k in dom F1 holds A1.k = {})
  holds A1.0={} & Union A1 = Union F1
proof
  let F1 be FinSequence of bool X;
  let A1 be SetSequence of X such that
A1: for k st k in dom F1 holds A1.k = F1.k and
A2: for k st not k in dom F1 holds A1.k = {};
  thus A1.0 = {} by A2,Th1;
  thus Union A1 = Union F1
  proof
    thus Union A1 c= Union F1
    proof
      let x be object;
      assume x in Union A1;
      then consider n being Nat such that
A3:   x in A1.n by PROB_1:12;
      n in dom F1 & x in F1.n by A1,A2,A3;
      hence thesis by Th49;
    end;
    let x be object;
    assume x in Union F1;
    then consider n such that
A4: n in dom F1 & x in F1.n by Th49;
    n in NAT & x in A1.n by A1,A4;
    hence thesis by PROB_1:12;
  end;
end;
