Square of Element Less than Unity in Ordered Integral Domain
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Theorem
Let $\struct {D, +, \times, \le}$ be an ordered integral domain.
Let $x \in D$ such that $0 < x < 1$.
Then:
- $0 < x \times x < x$
Proof
We have that $0 < x < 1$.
From Relation Induced by Strict Positivity Property is Compatible with Multiplication:
- $0 \times x < x \times x < 1 \times x$
Hence the result.
$\blacksquare$