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 Th13:
  for Z1 holds ( Z1 in Extend_Filter(F,Z) iff ex Z2 st Z2 in F & Z2 /\ Z c= Z1)
proof
  let Z1;
  thus Z1 in Extend_Filter(F,Z) implies ex Z2 st Z2 in F & Z2 /\ Z c= Z1
  proof
    defpred P[set] means ex Y2 st (Y2 in {Y1 /\ Z : Y1 in F} & Y2 c= $1);
    assume Z1 in Extend_Filter(F,Z);
    then
A1: Z1 in {Y: P[Y]};
    P[Z1] from ElemProp(A1);
    then consider Y2 such that
A2: Y2 in {Y1 /\ Z: Y1 in F} and
A3: Y2 c= Z1;
    consider Y3 such that
A4: Y2 = Y3 /\ Z and
A5: Y3 in F by A2;
    take Y3;
    thus Y3 in F by A5;
    thus thesis by A3,A4;
  end;
  given Z2 such that
A6: Z2 in F and
A7: Z2 /\ Z c= Z1;
  ex Y2 st Y2 in {Y1 /\ Z : Y1 in F} & Y2 c= Z1
  proof
    take Z2 /\ Z;
    thus Z2 /\ Z in {Y1 /\ Z : Y1 in F} by A6;
    thus thesis by A7;
  end;
  hence thesis;
end;
