theorem
  (for I,A holds Union coprod A in E) implies EnsCat E is with_coproducts
  proof
    assume
A1: for I,A holds Union coprod A in E;
    let I,A;
    take EnsCatCoproductObj A, EnsCatCoproduct A;
    Union coprod A in E by A1;
    hence thesis by Th10;
  end;
