Exercise 4.1.3

Exercise 4.1.3 Show that \((−1) × a = −a\) for every integer \(a\).

• \(\bullet\) Show that \((−1) × a = −a\) for every integer \(a\).

• \(\Vdash\) \((−1) × a\)

• \(=\) { By Definition 4.1.1, \(a = b — c\) for some natural number \(b\) and \(c\); Definition 4.1.4 }

  • \((0 — 1) × (b — c)\)

• \(=\) { Definition 4.1.2 }

  • \((0b + 1c)—(0c + 1b)\)

• \(=\) { Multiplication on \(\mathbb N\) }

  • \((c — b)\)

• \(=\) { Definition 4.1.4 }

  • \(−(b — c)\)

• \(=\) { \(a = b — c\) }

  • \(−a\)

• \(\square\)