= Solution
For $E:y^2=x^3-9x+9$, the addition formulas use
$$
m=\frac{y_2-y_1}{x_2-x_1},
\qquad
x(P_1+P_2)=m^2-x_1-x_2,
\qquad
y(P_1+P_2)=m(x_1-x(P_1+P_2))-y_1,
$$
with $m=(3x_1^2-9)/(2y_1)$ for doubling. They give
$$
2P=\left(\frac94,\frac38\right),
\qquad
2Q=(3,-3)=-Q,
$$
and
$$
P+Q=(-3,-3),
\qquad
P-Q=(1,-1).
$$
Let $(x,y)\ne O$ be rational. If the <P-adic valuation>[3-adic valuation] satisfies $v_3(y)\geq2$, first $v_3(x)$ cannot be negative: otherwise $x^3$ is the unique term of least valuation in $x^3-9x+9$, giving $2v_3(y)=3v_3(x)<0$. If $v_3(x)=0$, the right side has valuation zero. If $v_3(x)\geq1$, its three terms have valuations at least $3$, at least $3$, and exactly $2$, so $2v_3(y)=2$. In every case $v_3(y)\leq1$.
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