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In his Analysis I book, Terence Tao defines two functions $f,g:X\to Y$ to be equal if $f(x)=g(x)$, for all $x\in X$. After giving some examples of this concept, he then says:

This notion of equality obeys the usual axioms (Exercise 3.3.1).

It is quite easy to show that this relation defined on the class of functions is reflexive, symmetric, and transitive. But how do one verify that the substitution property is satisfied too?

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    $\begingroup$ Am I right that you need to show that, if $f:X\to Y = g:X\to , $ then $h(f) = h(g)$ for all functions/operations $h$ ? $\endgroup$
    – Adam Rubinson
    Mar 23 at 23:26
  • $\begingroup$ @AdamRubinson I think so. $\endgroup$
    – Gleison Stanlley
    Mar 23 at 23:32
  • $\begingroup$ If $f=g$ and $x=y$ then $f(x)=f(y)=g(x)=g(y)$. $\endgroup$
    – CyclotomicField
    Mar 24 at 3:35
  • $\begingroup$ For properties $P$, use induction on complexity of formulas. $\endgroup$
    – Mauro ALLEGRANZA
    Mar 24 at 6:44
  • $\begingroup$ See page 329: "How equality is defined depends on the class T of objects under consideration". Thus, you have to consider functions that have functions as their argument. Example: the "sum" of two functions. $\endgroup$
    – Mauro ALLEGRANZA
    Mar 24 at 6:47

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