Congruence of Sum with Constant

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Theorem

Let $a, b, z \in \R$.

Let $a$ be congruent to $b$ modulo $z$:

$a \equiv b \pmod z$


Then:

$\forall c \in \R: a + c \equiv b + c \pmod z$


Proof

Follows directly from the definition of Modulo Addition:

\(\ds a\) \(\equiv\) \(\ds b\) \(\ds \pmod z\) given
\(\ds c\) \(\equiv\) \(\ds c\) \(\ds \pmod z\) Congruence Modulo Real Number is Equivalence Relation
\(\ds \leadsto \ \ \) \(\ds a + c\) \(\equiv\) \(\ds b + c\) \(\ds \pmod z\) Definition of Modulo Addition

$\blacksquare$