
theorem Th70:
  for C being with_exponential_objects with_binary_products non empty category,
      a,b being Object of C holds Hom(b|^a [x] a, b)<>{} &
      b|^a,eval(a,b) is_exponent_of a,b
  proof
    let C be with_exponential_objects with_binary_products non empty category;
    let a,b be Object of C;
    set T = the categorical_exponent of a,b;
    consider c be Object of C, e be Morphism of c [x] a,b such that
A1: T = [c,e] & Hom(c [x] a, b)<>{} & c,e is_exponent_of a,b by Def31;
    thus thesis by A1;
  end;
