Changes http://wiki.axiom-developer.org/285EqualInAlgebraicNumberFails/diff
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If you write:
\begin{axiom}
-a=(a^2)^(1/2)
coerce(%%(7))$Equation(AlgebraicNumber)
\end{axiom}

you are asking about equality in the domain of
AlgebraicNumber (whatever that happens to be). But
if you write:
\begin{axiom}
%%(7)::Equation(Float)::Boolean
\end{axiom}

then you are referring to equality in the domain
Float. These need not be the same thing. But I agree
that Axiom does not have an entirely consistent
treatment of 'sqrt' even within a given domain. :(

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