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

theorem
  Union coprod A in E & Union coprod A = {} implies
  EnsCatCoproduct A = I --> {}
  proof
    assume that
A1: Union coprod A in E and
A2: Union coprod A = {};
    let i be object;
    assume
 i in I;
A4: Coprod A is empty-yielding by A2,Th7;
    thus (EnsCatCoproduct A).i = (Coprod A).i by A1,Def11
      .= {} by A4
      .= (I --> {}).i;
  end;
