Equal Numbers are Congruent

Theorem

$\forall x, y, z \in \R: x = y \implies x \equiv y \pmod z$

where $x \equiv y \pmod z$ denotes congruence modulo $z$.

$\ds x$

$=$

$\ds y$

$\ds \leadsto \ \ $

$\ds x - y$

$=$

$\ds 0$

$\ds \leadsto \ \ $

$\ds x - y$

$=$

$\ds 0 \cdot z$

$\ds \leadsto \ \ $

$\ds x$

$\equiv$

$\ds y$

$\ds \pmod z$

Definition of Congruence Modulo $z$

$\blacksquare$

1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {4-1}$ Basic Properties of Congruences: Example $\text {4-3}$