This classic from euclid's elements, is it accepted everywhere? I was reading linear vector spaces. When doing some exercise to prove some statements based on the properties defined for linear vector spaces, i suddenly noticed, outside the things defined, i'm using a common notion without proof. This notion also surfaced when i was studying Group theory. After giving a thought i come to the conclusion that i've used it in all systems which are modeled after euclid. I mean first i give some postulates. Then i derive statements from those postulates. Euclid also has it as "common notion" :
If equals are added to equals, the wholes are equal.
So i started wondering that if anybody challenged this or attempted to build a system without this.

If this were not the case, how could we say that the things were equal in a sense meaningful for our purposes? 
In different areas of mathematics, we have different notions of equality that generally mean 'the same as far as everything we're currently concerned with'. For example if we have two Riemann surfaces, if we were studying them from the point of view of topology we might call them equal if they were homeomorphic (or maybe homotopy equivalent), from the point of view of Differential Geometry we would consider them equal if they were diffeomorphic to each other, whereas from the point of view of complex geometry they would need to be biholomorphic. We may not be interested in the structure at all, and consider them equal if their underlying sets had the same cardinality.
If we had a notion of equality that did not guarantee equivalence for all things we were interested in, it would not be a good notion of equality to study those phenomenon.
