reserve r, s, t for Real;
reserve seq for Real_Sequence,
  X, Y for Subset of REAL;
reserve r3, r1, q3, p3 for Real;

theorem Th56:
  for Z being Subset-Family of REAL st Z is closed holds meet Z is closed
proof
  let Z be Subset-Family of REAL;
  assume
A1: Z is closed;
  let seq be Real_Sequence;
  set mZ = meet Z;
  assume that
A2: rng seq c= mZ and
A3: seq is convergent;
  per cases;
  suppose
    Z = {};
    then mZ = {} by SETFAM_1:def 1;
    hence thesis by A2;
  end;
  suppose
A4: Z <> {};
    now
      let X be set;
      assume
A5:   X in Z;
      then reconsider X9 = X as Subset of REAL;
A6:   rng seq c= X
      by A2,A5,SETFAM_1:def 1;
      X9 is closed by A1,A5;
      hence lim seq in X by A3,A6;
    end;
    hence thesis by A4,SETFAM_1:def 1;
  end;
end;
