Proposition 1.8.1.17.  Let a, b, c ∈  such that a < b ≤ c. Then:
Proof.  We show that ∃ x ∈ + : a + x = c:
a < bdef∃ y ∈ + : a + y = b
b ≤ cdef∃ z ∈  : b + z = c
Take x:= y + z.
References.