Proposition 1.8.1.18.  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.