reserve
  I for set,
  E for non empty set;
reserve A for ObjectsFamily of I,EnsCat E;

theorem Th7:
  product A in E & product A = {} implies EnsCatProduct A = I --> {}
  proof
    assume that
A1: product A in E and
A2: product A = {};
    now
      let i be object;
      assume
   i in I;
      hence (EnsCatProduct A).i = proj(A,i) by A1,Def11
      .= {} by A2
      .= (I --> {}).i;
    end;
    hence thesis by PBOOLE:3;
  end;
