how exactly did he apply 1.39 to get
$$f(x-1)^2 =f(1-x)^2$$
It is not difficult; note that $$ f(x-1)^2=f((x-1)^2)=f((1-x)^2)=f(1-x)^2.$$
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It is not difficult; note that $$ f(x-1)^2=f((x-1)^2)=f((1-x)^2)=f(1-x)^2.$$