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Ehrenpreis' proof is based on the existence of smooth [[bump function]]s, unique continuation of holomorphic functions, and the [[Poincaré lemma]] — the last in the form that for any smooth and compactly supported differential (0,1)-form {{mvar|ω}} on {{math|'''C'''<sup>''n''</sup>}} with {{math|{{overline|∂}}''ω'' {{=}} 0}}, there exists a smooth and compactly supported function {{mvar|η}} on {{math|'''C'''<sup>''n''</sup>}} with {{math|{{overline|∂}}''η'' {{=}} ''ω''}}. The crucial assumption {{math|''n'' ≥ 2}} is required for the validity of this Poincaré lemma; if {{math|''n'' {{=}} 1}} then it is generally impossible for {{mvar|η}} to be compactly supported.{{sfnm|1a1=Hörmander|1y=1990|1p=30}}
The [[ansatz]] for {{mvar|F}} is {{math|''φ f'' − ''v''}} for smooth functions {{mvar|φ}} and {{mvar|v}} on {{mvar|G}}; such an expression is meaningful provided that {{mvar|φ}} is identically equal to zero where {{mvar|f}} is undefined (namely on {{mvar|K}}). Furthermore, given any holomorphic function on {{mvar|G}} which is equal to {{mvar|f}} on ''some'' [[open set]], unique continuation (based on connectedness of {{math|''G'' \ ''K''}}) shows that it is equal to {{mvar|f}} on ''all'' of {{math|''G'' \ ''K''}}.
The holomorphicity of this function is identical to the condition {{math|{{overline|∂}}''v'' {{=}} ''f'' {{overline|∂}}''φ''}}. For any smooth function {{mvar|φ}}, the differential (0,1)-form {{math|''f'' {{overline|∂}}''φ''}} is {{math|{{overline|∂}}}}-closed. Choosing {{mvar|φ}} to be a [[Smoothness|smooth function]] which is identically equal to zero on {{mvar|K}} and identically equal to one on the complement of some compact subset {{mvar|L}} of {{mvar|G}}, this (0,1)-form additionally has compact support, so that the Poincaré lemma identifies an appropriate {{mvar|v}} of compact support. This defines {{mvar|F}} as a holomorphic function on {{mvar|G}}; it only remains to show (following the above comments) that it coincides with {{mvar|f}} on some open set.
On the set {{math|'''C'''<sup>''n''</sup> \ ''L''}}, {{mvar|v}} is holomorphic since {{mvar|φ}} is identically constant. Since it is zero near infinity, unique continuation applies to show that it is identically zero on some open subset of {{math|''G'' \ ''L''}}.<ref>Any connected component of {{math|'''C'''<sup>''n''</sup> \ ''L''}} must intersect {{math|''G'' \ ''L''}} in a nonempty open set. To see the nonemptiness, connect an arbitrary point {{mvar|p}} of {{math|'''C'''<sup>''n''</sup> \ ''L''}} to some point of {{mvar|L}} via a line. The intersection of the line with {{math|'''C'''<sup>''n''</sup> \ ''L''}} may have many connected components, but the component containing {{mvar|p}} gives a continuous path from {{mvar|p}} into {{math|''G'' \ ''L''}}.</ref> Thus, on this open subset, {{mvar|F}} equals {{mvar|f}} and the existence part of Hartog's theorem is proved. Uniqueness is automatic from unique continuation, based on connectedness of {{mvar|G}}.
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