Real Number Divided by Itself

Theorem

$\forall x \in \R_{\ne 0}: \dfrac x x = 1$

Proof

\(\ds \forall x \ne 0: \, \)

\(\ds \frac x x\)

\(=\)

\(\ds x \times \frac 1 x\)

Definition of Real Division

\(\ds \)

\(=\)

\(\ds 1\)

Real Number Axiom $\R \text M4$: Inverses for Multiplication

$\blacksquare$

Sources

• 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers: Exercise $1 \ \text{(j)}$