= Lower Darboux integral
{title2=$\underline{\int_a^b}f=\sup_PL(f,P)$}
The lower Darboux integral of a bounded real function on a closed interval is the <supremum> of its <lower Darboux sums> over all <partitions of an interval>. Refinement increases a <lower Darboux sum>. A common refinement of two arbitrary <partitions of an interval> proves that every <lower Darboux sum> is at most every <upper Darboux sum>, hence the lower Darboux integral never exceeds the <upper Darboux integral>.
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