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let x be a natural number

i want to prove that f(x)=[tex]x^2[/tex]

By induction

suppose that f(x)=[tex]x^2[/tex]

f(0)=0 holds (from the functional equation)

i want to prove that f(x)= [tex](x+1)^2[/tex]

when putting (x,1)(we have f(x+y)+f(x-y) in the equation) I need to know the value of f(x-1) ,does the assumption that f(x)=[tex]x^2[/tex], imply that f(x-1)=[tex](x-1)^2[/tex]

(this is not a homework i just saw this excercise online and decided to do it).