= Solution
Let $V\subseteq U$ be open subsets, and let $s\in\mathcal F_3(V)$. The assumed <flasque-kernel section-lifting lemma> lifts $s$ to $\widetilde s\in\mathcal F_2(V)$. Since $\mathcal F_2$ is a <flabby sheaf>, extend $\widetilde s$ to a section on $U$, then project to $\mathcal F_3(U)$. Its restriction is $s$. Thus every restriction map of $\mathcal F_3$ is <surjective>, proving that \b[the quotient of two <flabby sheaves> in a short exact sequence is flabby].
For the construction of <sheaf cohomology>, embed a <sheaf of abelian groups> $\mathcal F$ into the <sheaf>
$$
C(\mathcal F)(U)=\prod_{P\in U}\mathcal F_P
$$
by sending a section to all its <germs>. This is <injective> because a section with every germ zero is zero. Arbitrary families of stalk elements glue point by point, and restriction maps are projections; hence $C(\mathcal F)$ is a <flabby sheaf>. Set $\mathcal Q^0=\mathcal F$, $\mathcal I^j=C(\mathcal Q^j)$ and $\mathcal Q^{j+1}=\mathcal I^j/\mathcal Q^j$. Compose the quotient map with the next embedding to obtain the <flasque resolution>
$$
0\longrightarrow\mathcal F\longrightarrow\mathcal I^0
\longrightarrow\mathcal I^1\longrightarrow\cdots.
$$
The <sheaf cohomology> groups are the <cohomology groups> of the resulting <cochain complex> of <global sections>:
$$
H^j(X,\mathcal F)=
\frac{\ker\bigl(\Gamma(X,\mathcal I^j)\to\Gamma(X,\mathcal I^{j+1})\bigr)}
{\operatorname{im}\bigl(\Gamma(X,\mathcal I^{j-1})\to\Gamma(X,\mathcal I^j)\bigr)}
\quad(j>0),
\qquad H^0(X,\mathcal F)=\Gamma(X,\mathcal F).
$$
The usual comparison of resolutions identifies this construction with the right derived functors of <global sections>; the construction above is the <Godement resolution>.
If $\mathcal F$ is itself a <flabby sheaf>, the quotient argument shows inductively that every $\mathcal Q^j$ is flabby. The assumed lifting result therefore makes
$$
0\to\Gamma(X,\mathcal Q^j)\to\Gamma(X,\mathcal I^j)
\to\Gamma(X,\mathcal Q^{j+1})\to0
$$
an <exact sequence> for every $j$. The global-section complex is consequently exact in every positive degree. In particular,
$$
\boxed{H^j(X,\mathcal F)=0\quad(j>0)\qquad\text{if }\mathcal F\text{ is flabby}.}
$$
This establishes the vanishing from the section-lifting property rather than assuming it in advance.
For the <smooth algebraic curve> $X$, a <divisor on an algebraic curve> is a finite formal sum $D=\sum_P n_PP$ of closed points with integer coefficients. At each closed point, the <local ring of a smooth algebraic curve> is a <discrete valuation ring>; write its normalized <discrete valuation> as $v_P$. For $f\in k(X)^*$ the <principal divisor> is $\operatorname{div}(f)=\sum_Pv_P(f)P$. Only finitely many coefficients are nonzero: zeros and poles are proper closed subsets in the <Noetherian Zariski topology> of a curve. The <divisor class group> is the <abelian group>
$$
\boxed{\operatorname{Cl}(X)=
\operatorname{Div}(X)/\{\operatorname{div}(f):f\in k(X)^*\}.}
$$
Equivalently, its elements are <divisors on an algebraic curve> modulo <linear equivalence of divisors>.
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