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y-x=√(1-x^2)+√(1-y^2)

The equation look pretty simple, and according to Mathematica the solution is simple too: y=√(1-x^2)

The (proposed) solution looks so simple that I would guess there is some fairly straightforward way to get there. But I can't see it.

Is the only way squaring both sides until all roots are eradicated, and then solving whatever mess comes out of that? Or is there perhaps some nicer way?