Proofwiki says the following about difference in natural numbers:
In the context of the natural numbers, the difference is defined as:
$n−m=p⟺m+p=n$
from which it can be seen that the above congruence can be understood as:
$(x_1,y_1)⊠(x_2,y_2)⟺x_1+y_2=x_2+y_1⟺x_1−y_1=x_2−y_2$
Thus this congruence defines an equivalence between pairs of elements which have the same difference.
What does that all mean? I don't understand it at all. Why did they call $n−m=p⟺m+p=n$ a congruence? How does it imply that $(x_1,y_1)(x_2,y_2)⟺x_1+y_2=x_2+y_1⟺x_1−y_1=x_2−y_2$?
Here is the definition of $⊠$.