The primes in the statement of prop 4.2.2 are not operators; a' and b' are simply names of variables which are different from $a$ and $b$, although suggestively named.
It is, however, not very clearly written. It appears that the author is using $[a]$ and [a'] denote the residue classes of the integers $a$ and a', and so forth. But it is strictly speaking nonsense to write "if [a]=[a'] and [b]=[b'] modulo $m$ ...". For then [a]=[a'] simply asserts that the residue classes are the same, and this identity is just an identity between sets of numbers; there is nothing modular about the way these sets are equal.
But if the author does mean the premises to be [a]=[a'] and [b]=[b'], then the conclusions [a]+[b]=[a']+[b'] and [a][b]=[a'][b'] are completely vacuous, because of course we're allowed to substitute equals for equals.
What the proposition ought to have been, in order to be meaningful, is
If a\equiv a'\pmod m and b\equiv b'\pmod m, then (a+b)\equiv(a'+b')\pmod m and ab\equiv a'b' \pmod m.
or, equivalently,
If [a]=[a'] and [b]=[b'], then [a+b]=[a'+b'] and [ab]=[a'b'].
... and because of this fact it is possible and meaningful to define the sum and product of residue classes by $[a]+[b]=[a+b]$ and $[a]\cdot[b]=[ab]$.