Sufficient statistic Let $\mathbf{X}=(X_1,\ldots,X_n)$ with joint frequency function $f(\mathbf{x};\theta_1,\theta_2)$ where $\theta_1,\theta_2$ vary independently.
The set $S=\{\mathbf{x}:f(\mathbf{x};\theta_1,\theta_2)>0\}$ doesn't depend on $\theta_1,\theta_2$.
Suppose $T_1$ is sufficient for $\theta_1$ when $\theta_2$ is known, and $T_2$ is sufficient for $\theta_2$ when $\theta_1$ is known. 
I need to show that $(T_1,T_2)$ is sufficient for $(\theta_1,\theta_2)$, if $T_1$ doesn't depend on $\theta_2$ and $T_2$ doesn't depend on $\theta_1$. 
So, I'm having some difficulty in interpreting the problem and expressing it with mathematical expressions.
I was thinking of using the Factorization Criterion.
«Suppose $T_1$ is sufficient for $\theta_1$ when $\theta_2$ is known.» This sentence I'm writing it as : With $\theta_2$ known, $f(\mathbf{x};\theta_1,\theta_2)=g_1(T_1(x),\theta_1)h_1(x)$
Similarly, for the other sentence: $f(\mathbf{x};\theta_1,\theta_2)=g_2(T_2(x),\theta_2)h_2(x)$
and so, $f(\mathbf{x};\theta_1,\theta_2)=\left(g_2(T_2(x),\theta_2)\cdot g_1(T_1(x),\theta_1)\right)^{1/2}(h_1(x)\cdot h_2(x))^{1/2 }$
However, this doesn't seem to be a correct resolution, since I don't seem to be using the information given about $S$, at least explicitly...
Any help would be appreciated.
 A: I'll work straight from the definition.
$T_1$ is sufficient for $\theta_1$ if $\theta_2$ is known.  I take that to mean that the conditional distribution of the data given $T_1$ does not change when $\theta_1$ changes but $\theta_2$ remains fixed.
Similarly the conditional distribution of the data given $T_2$ does not change when $\theta_2$ changes but $\theta_1$ remains fixed.
If $T_1$ is sufficient for $\theta_1$ when $\theta_2$ is known, then the pair $(T_1,T_2)$ is sufficient for $\theta_1$ when $\theta_2$ is known.  Similarly the pair is sufficient for $\theta_2$ when $\theta_1$ is known.
Thus the conditional distribution of the data given $(T_1,T_2)$ does not change if $\theta_1$ changes while $\theta_2$ stays fixed, and also does not change if $\theta_2$ changes while $\theta_1$ stays fixed.
Next observe that the pair $(\theta_1,\theta_2)$ can be changed by first changing $\theta_1$ while leaving $\theta_2$ fixed, and then changing $\theta_2$ while leaving $\theta_1$ fixed.
postscript: Your Fisher factorizations should say

«Suppose $T_1$ is sufficient for $\theta_1$ when $\theta_2$ is known.» Then $$f(\mathbf{x};\theta_1,\theta_2)=g_1(T_1(\mathbf{x}),\theta_1,\theta_2)h_1(x,\theta_2)$$ and similarly since $T_2$ is sufficient for $\theta_2$ when $\theta_1$ is known, $$f(\mathbf{x};\theta_1,\theta_2)=g_2(T_2(\mathbf{x}),\theta_1,\theta_2)h_2(x,\theta_1).$$

