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 Th52:
  F1 <> {} implies (x in meet rng F1 iff for n st n in dom F1 holds x in F1.n)
proof
  assume F1 <> {};
  then
A1: rng F1 <> {} by RELAT_1:41;
A2: now
    let x;
    assume
A3: for n st n in dom F1 holds x in F1.n;
    now
      let Y;
      assume Y in rng F1;
      then consider n be object such that
A4:   n in dom F1 and
A5:   Y = F1.n by FUNCT_1:def 3;
      thus x in Y by A3,A4,A5;
    end;
    hence x in meet rng F1 by A1,SETFAM_1:def 1;
  end;
  now
    let x;
    assume
A6: x in meet rng F1;
    now
      let k;
      assume k in dom F1;
      then F1.k in rng F1 by FUNCT_1:3;
      hence x in F1.k by A6,SETFAM_1:def 1;
    end;
    hence for n st n in dom F1 holds x in F1.n;
  end;
  hence thesis by A2;
end;
