[_ouml_]

The equality test is equivalent to deciding if something is 0. You can probably ask the app to calculate sqrt(2) * sqrt(2). If that doesn't show a 2 as answer, you've lost. Same goes for sin(arcsin(1/2)). You want the exact answer.

[H8crilA]

Equality is also needed if you want to know whether "x<y".

Proof: 1) suppose you can decide whether "x<y" for all x,y. 2) Then, for all x,y you can decide whether x==y because that's just "!(x<y) && !(y<x)"