x 1=√2a+b，a，b∈Z

x2=√2c+d，c，d∈Z

rule

x 1+x2=√2(a+c)+(b+d)，a+c，b+d∈Z

Therefore, x = x1+x2 ∈ a.

Similarly:

x 1x 2 =√2ad+√2bc+2ac+BD =√2(ad+BC)+(2ac+BD)，ad+bc∈Z，2ac+bd∈Z

Therefore x = x1x2 ∈ a.

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