Ordering is Preserved on Integers by Addition

Theorem
The usual ordering on the integers is preserved by the operation of addition:


 * $\forall a, b, c, d, \in \Z: a \le b, c \le d \implies a + c \le b + d$

Proof
Recall that Integers form Ordered Integral Domain.

Then from Relation Induced by Strict Positivity Property is Compatible with Addition:


 * $\forall x, y, z \in \Z: x \le y \implies \paren {x + z} \le \paren {y + z}$
 * $\forall x, y, z \in \Z: x \le y \implies \paren {z + x} \le \paren {z + y}$

So:

Finally: