The equivalence relation only needs to be proved to satisfy reflexivity, symmetry and transitivity.

The reflexivity is obvious (in the relationship, X and Y are replaced by U and V respectively, which can be proved)

Symmetry:

because

x+v=u+y

& ltx，y & gtR & ltu，v & gt

Transitivity:

& ltu，v & gtR & ltx，y & gt？ u+y=x+v

& ltx，y & gtR & lta，b & gt？ x+b=a+y

rule

(u+y)+(x+b)=(x+v)+(a+y)

u+b=a+v

& ltu，v & gtR & lta，b & gt