reserve A,B for limit_ordinal infinite Ordinal;
reserve B1,B2,B3,B5,B6,D, C for Ordinal;
reserve X for set;
reserve X for Subset of A;
reserve M for non countable Aleph;
reserve X for Subset of M;
reserve N,N1 for cardinal infinite Element of M;

theorem Th21:
  for S being Subset-Family of M st for X being Element of S holds
  X is closed holds meet S is closed
proof
  let S be Subset-Family of M such that
A1: for X being Element of S holds X is closed;
  let C be limit_ordinal infinite Ordinal;
  assume
A2: C in M;
  per cases;
  suppose
A3: S = {};
    not sup (meet S /\ C) = C
    proof
      assume
A4:   sup (meet S /\ C) = C;
      meet S = {} by A3,SETFAM_1:def 1;
      hence contradiction by A4,ORDINAL2:18;
    end;
    hence thesis;
  end;
  suppose
A5: S <> {};
    assume
A6: sup (meet S /\ C) = C;
    for X being set holds X in S implies C in X
    proof
      let X be set such that
A7:   X in S;
      reconsider X1=X as Element of S by A7;
A8:   X1 is closed by A1;
      sup (X1 /\ C) c= sup C by ORDINAL2:22,XBOOLE_1:17;
      then
A9:   sup (X1 /\ C) c= C by ORDINAL2:18;
      (meet S /\ C) c= (X1 /\ C) by A7,SETFAM_1:3,XBOOLE_1:26;
      then sup (meet S /\ C) c= sup (X1 /\ C) by ORDINAL2:22;
      then sup (X1 /\ C) = C by A6,A9,XBOOLE_0:def 10;
      hence thesis by A2,A8;
    end;
    hence thesis by A5,SETFAM_1:def 1;
  end;
end;
