TLA + "Specifying Systems" draft translation --Section 6.5 Using Functions (function uses)

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defined function with the best $f'=[i \in Nat \mapsto i+1]$ In this way

Consider the following two formulas:
(6.6) $f'=[i \in Nat \mapsto i+1]$
(6.7) $\forall i \in Nat: f'[i]=i+1$
both for each of a number of natural $i$ , have $f'[i]=i+1$ , but they are not equivalent. Formula (6.6) uniquely determine $f'$ , It declares it is a domain of $Nat$ function, but the formula (6.7), there are many different $f'$ Satisfies the condition, for example the following functions: $[i \in Real \mapsto \text{IF } i \in Nat \text{ THEN } i + 1 \text{ ELSE } i^2]$In fact, from the formula (6.7), we can not even deduce $f'$ Is a function. Equation (6.6) contains the formula (6.7), not vice versa.
When writing the statute, we generally like to variables $f$ simple assignment once on the line, not all elements of the collection $i$ Ode again $f[i]$ values. Therefore, we usually recommend to write formula (6.6) instead of (6.7).