need help understanding superposition principle..!

need help understanding superposition principle..!

if we have a function y=f(x) then in-order to prove linearity we try to
justify according to superposition principle as : let x1 and x2 be two
inputs then f(x1+x2)=f(x1)+f(x2) please correct me if i am wrong upto
here.. now what if we have more than two variables..let's say we have
three variables two independent and one dependent now we have function
z=g(x,y)..now in-order to prove linearity for function involving more than
two variables can i say this that for g(x,y) to be linear
g(x1+x2,y1+y2)=g(x1,y1)+g(x2,y2)..?? and if this isn't the correct way for
proving linearity in functions involving more than two variables..then
please justify the correct method along with examples.