reserve n,m,k,i for Nat,
  g,s,t,p for Real,
  x,y,z for object, X,Y,Z for set,
  A1 for SetSequence of X,
  F1 for FinSequence of bool X,
  RFin for real-valued FinSequence,
  Si for SigmaField of X,
  XSeq,YSeq for SetSequence of Si,
  Omega for non empty set,
  Sigma for SigmaField of Omega,
  ASeq,BSeq for SetSequence of Sigma,
  P for Probability of Sigma;

theorem Th51:
 for x being object holds
  F1 <> {} implies (x in Intersection F1 iff for k st k in dom F1
  holds x in F1.k )
proof let x be object;
A1: for n st n in dom F1 holds X \ (Complement F1).n = F1.n
  proof
    let n;
    assume n in dom F1;
    then
A2: n in dom Complement F1 by Th50;
    X \ (Complement F1).n = ((Complement F1).n)` by SUBSET_1:def 4
      .= ((F1.n)`)` by A2,Def5
      .= F1.n;
    hence thesis;
  end;
  assume
A3: F1 <> {};
  then
A4: dom F1 <> {} by RELAT_1:41;
A5: x in X & (for n st n in dom F1 holds not x in (Complement F1).n) iff
  for n st n in dom F1 holds x in F1.n
  proof
    hereby
      assume that
A6:   x in X and
A7:   for n st n in dom F1 holds not x in (Complement F1).n;
      let n such that
A8:   n in dom F1;
      not x in (Complement F1).n by A7,A8;
      then x in X \ (Complement F1).n by A6,XBOOLE_0:def 5;
      hence x in F1.n by A1,A8;
    end;
    assume
A9: for n st n in dom F1 holds x in F1.n;
A10: now
      let n be Element of NAT such that
A11:  n in dom F1;
      x in F1.n by A9,A11;
      then x in X \ (Complement F1).n by A1,A11;
      hence x in X & not x in (Complement F1).n by XBOOLE_0:def 5;
    end;
    ex a being object st a in dom F1 by A4,XBOOLE_0:def 1;
    hence thesis by A10;
  end;
  dom Complement F1 = dom F1 by Th50;
  then
A12: x in X & (not x in Union (Complement F1)) iff x in X & for n st n in
  dom F1 holds not x in (Complement F1).n by Th49;
  x in (Union Complement F1)` iff x in X \ Union Complement F1 by
SUBSET_1:def 4;
  hence thesis by A3,A12,A5,Def6,XBOOLE_0:def 5;
end;
