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;

theorem Th9:
  dual S = {Y`:Y in S}
proof
  thus dual S c= {Y`:Y in S}
  proof
    let X1 be object such that
A1: X1 in dual S;
    reconsider Y1=X1 as Subset of X by A1;
    Y1` in S by A1,SETFAM_1:def 7;
    then Y1`` in {Y`:Y in S};
    hence thesis;
  end;
  let X1 be object;
  assume X1 in {Y`:Y in S};
  then consider Y such that
A2: Y`=X1 and
A3: Y in S;
  Y`` in S by A3;
  hence thesis by A2,SETFAM_1:def 7;
end;
