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

theorem Th7:
  Union coprod A = {} implies Coprod A is empty-yielding
  proof
    assume
A1: Union coprod A = {};
    let i be object;
    assume i in I;
    then ex F being Function of A.i,Union coprod A st (Coprod A).i = F &
    for x being object st x in A.i holds F.x = [x,i] by Def10;
    hence thesis by A1;
  end;
