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