<s>[INST] Proof that $ \forall x,y \in \mathbb{R} \qquad x^2+y^2+(x-1)(y-1)>0 $ How to proof simply that 
$$
\forall x,y \in \mathbb{R} \qquad x^2+y^2+(x-1)(y-1)>0
$$
 [/INST] $x^2+y^2+(x-1)(y-1)>0\iff x^2+y^2+xy-x-y+1>0\iff2x^2+2y^2+2xy-2x-2y+2>0\iff(x-1)^2+(y-1)^2+(x+y)^2>0$ which is obviously true.
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