Definition 2.3.1 (Multiplication of natural numbers). Let \(m\) be a natural number. To multiply zero to \(m\), we define \(0 \times m := 0\). Now suppose inductively that we have defined how to multiply \(n\) to \(m\). Then we can multiply \(n\pp\) to \(m\) by defining \((n\pp) \times m := (n \times m) + m\).
Definition 2.3.1
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Exercise 2.3.5 (Proposition 2.3.9)
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Exercise 2.3.4
\(=\) { Definition of multiplication: \((n\pp)\times m:=n\times m+m\);
\(\hspace{1cm} 2m=(1\pp)m=1m+m=m+m\) } -
Exercise 2.3.3 (Proporsition 2.3.5 )
\(=\) { Definition 2.3.1: \((n\pp)\times m:=(n\times m)+m\) }
\(\equiv\) { Definition 2.3.1: \(\forall m.~0\times m:=0\) }