= Flasque-kernel section-lifting lemma
For a <short exact sequence of sheaves> with <flasque sheaf> kernel, every section of the quotient on any open set lifts to the middle sheaf on that set. Use the <Zorn lemma> to choose a maximal partial lift, with chains glued by the <sheaf gluing axiom>. A new local lift differs from the old one on their overlap by a kernel section. Extend that difference by flasqueness, adjust the new lift, and glue, contradicting maximality unless its domain is the entire open set. Consequently a quotient of two flasque sheaves is flasque. With an <injective resolution of sheaves>, the <long exact sequence in sheaf cohomology> and repeated dimension shifts prove that flasque sheaves have zero positive-degree cohomology.
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