reserve N for Cardinal;
reserve M for Aleph;
reserve X for non empty set;
reserve Y,Z,Z1,Z2,Y1,Y2,Y3,Y4 for Subset of X;
reserve S for Subset-Family of X;
reserve x for set;
reserve F,Uf for Filter of X;
reserve S for non empty Subset-Family of X;
reserve I for Ideal of X;

theorem Th14:
  (for Y1 st Y1 in F holds Y1 meets Z) implies Z in Extend_Filter(
  F,Z) & Extend_Filter(F,Z) is Filter of X & F c= Extend_Filter(F,Z)
proof
  assume
A1: for Y1 st Y1 in F holds Y1 meets Z;
  X in F & X /\ Z c= Z by Th5,XBOOLE_1:17;
  hence Z in Extend_Filter(F,Z) by Th13;
  thus Extend_Filter(F,Z) is Filter of X
  proof
    thus not {} in Extend_Filter(F,Z)
    proof
      assume {} in Extend_Filter(F,Z);
      then consider Z2 such that
A2:   Z2 in F and
A3:   Z2 /\ Z c= {} by Th13;
      Z2 meets Z by A1,A2;
      then Z2 /\ Z <> {};
      hence contradiction by A3;
    end;
    let Y1,Y2;
    thus Y1 in Extend_Filter(F,Z) & Y2 in Extend_Filter(F,Z) implies (Y1 /\ Y2
    ) in Extend_Filter(F,Z)
    proof
      assume that
A4:   Y1 in Extend_Filter(F,Z) and
A5:   Y2 in Extend_Filter(F,Z);
      consider Y3 such that
A6:   Y3 in F and
A7:   Y3 /\ Z c= Y1 by A4,Th13;
      consider Y4 such that
A8:   Y4 in F and
A9:   Y4 /\ Z c= Y2 by A5,Th13;
      (Y3 /\ Z) /\ (Y4 /\ Z) = Y3 /\ (Z /\ (Y4 /\ Z)) by XBOOLE_1:16
        .= Y3 /\ (Y4 /\ (Z /\ Z)) by XBOOLE_1:16
        .= (Y3 /\ Y4) /\ Z by XBOOLE_1:16;
      then
A10:  (Y3 /\ Y4) /\ Z c= Y1 /\ Y2 by A7,A9,XBOOLE_1:27;
      Y3 /\ Y4 in F by A6,A8,Def1;
      hence thesis by A10,Th13;
    end;
    thus Y1 in Extend_Filter(F,Z) & Y1 c= Y2 implies Y2 in Extend_Filter(F,Z )
    proof
A11:  (X \ Z) misses Z by XBOOLE_1:79;
      (Y2 \/ (X \ Z)) /\ Z = (Y2 /\ Z) \/ ((X \ Z) /\ Z) by XBOOLE_1:23
        .= (Y2 /\ Z) \/ {} by A11
        .= Y2 /\ Z;
      then
A12:  (Y2 \/ (X \ Z)) /\ Z c= Y2 by XBOOLE_1:17;
      assume that
A13:  Y1 in Extend_Filter(F,Z) and
A14:  Y1 c= Y2;
      consider Y3 such that
A15:  Y3 in F and
A16:  Y3 /\ Z c= Y1 by A13,Th13;
      Y3 = (Y3 /\ Z) \/ (Y3 \ Z) & Y3 \ Z c= X \ Z by XBOOLE_1:33,51;
      then
A17:  Y3 c= (Y3 /\ Z) \/ (X \ Z) by XBOOLE_1:9;
      Y3 /\ Z c= Y2 by A14,A16;
      then (Y3 /\ Z) \/ (X \ Z) c= Y2 \/ (X \ Z) by XBOOLE_1:9;
      then Y3 c= Y2 \/ (X \ Z) by A17;
      then Y2 \/ (X \ Z) in F by A15,Def1;
      hence thesis by A12,Th13;
    end;
  end;
  thus F c= Extend_Filter(F,Z)
  proof
    let Y be object;
    assume
A18: Y in F;
    then reconsider Y as Subset of X;
    Y /\ Z c= Y by XBOOLE_1:17;
    hence thesis by A18,Th13;
  end;
end;
