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
  ex F1 being FinSequence of bool X st for k st k in dom F1 holds F1.k = X
proof
  now
    let n be Element of NAT;
    set F1 = n |-> X;
A1: dom F1 = Seg n by FUNCOP_1:13;
    rng F1 c= bool X
    proof
      let x be object;
      assume x in rng F1;
      then ex i be Nat st i in dom F1 & F1.i = x by FINSEQ_2:10;
      then x = X by A1,FINSEQ_2:57;
      hence thesis by ZFMISC_1:def 1;
    end;
    then
A2: F1 is FinSequence of bool X by FINSEQ_1:def 4;
    for k being Nat st k in dom F1 holds F1.k = X by A1,FINSEQ_2:57;
    hence thesis by A2;
  end;
  hence thesis;
end;
