Zeros and poles of a nonzero rational function form a finite formal sum whose total degree is zero. I will separate the object being defined from the consequence being claimed.
Definitions first
On a smooth projective curve \(C\), each closed point \(p\) defines a valuation \(\operatorname{ord}_p\). Divisors combine these local orders into a global bookkeeping device.
There are two layers here: the object \(\mathsf D\) and the law \(\mathsf C\). Writing them separately makes the direction of \(\Longrightarrow\) visible and keeps an accidental converse from slipping in.
A small case in full
The computation below is not a second theorem. It is a checksum for the definitions and a place to inspect the difficult LaTeX at full size.
The reusable statement
The aligned summary deliberately puts the datum and conclusion on different rows. Mathematically, this is the distinction between specifying an object and proving a property of it.
A nearby false statement
A divisor being degree zero does not make it principal. The difference is measured by the Picard group and, in degree zero, by the Jacobian.
The notation is dense, but it is doing honest work: every delimiter records scope and every index records dependence. Removing one should require a mathematical reason.