Is there any way to prove that $x = y \Rightarrow x + z = y + z$?

Question

Terence Tao, Analysis I, 3e,

A.7 Equality

(...) How equality is defined depends on the class T of objects under consideration, and to some extent is just a matter of definition. However, for the purposes of logic we require that equality obeys the following four axioms of equality:

• (Reflexive axiom) (...)
• (Symmetry axiom) (...)
• (Transitive axiom) (...)
• (Substitution axiom). Given any two objects $x$ and $y$ of the same type, if $x = y$, then $f(x) = f(y)$ for all functions or operations $f$.

Concerning the substitution axiom, I keep wondering if there is really no way one could prove that

$$ x = y \Rightarrow x + z = y + z, $$

where $x, y, z$ are natural numbers?

Answer 1

$y+z = (y+0) + z = y+(x-x) + z = (y-x) + x+ z$

if $y = x$ then $y-x = 0$ and $y+z = x+z$