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 $ab\equiv a'b' \pmod m$.
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 $[a]\cdot[b]=[ab]$.