<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] $$x^2+y^2+(x-1)(y-1) = x^2 + y^2 + xy - x -y+1 = \dfrac{(x+y)^2}2 + \dfrac{(x-1)^2}2 + \dfrac{(y-1)^2}2$$
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