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

theorem Th8:
  Union coprod A = {} implies A is empty-yielding
  proof
    assume
A1: Union coprod A = {};
    let i be object;
    assume i in I;
    then consider F being Function of A.i,Union coprod A such that
    (Coprod A).i = F and
A2: for x being object st x in A.i holds F.x = [x,i] by Def10;
    assume A.i is non empty;
    then consider x being object such that
A3: x in A.i by XBOOLE_0:7;
    F.x = [x,i] by A2,A3;
    hence thesis by A1;
  end;
