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**need help understanding "Superposition Principle"..!!**

hello everyone..

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.