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<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a>  &amp; <font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">f</font> holds <br/><font color="Olive" title="b1">f</font> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c1">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a>  &amp; <font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> implies <font color="Maroon" title="c2">f</font> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c1">A</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:20"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a> 
 <span class="kw">and </span><br/><a NAME="E2:20"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
<font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c1">A</font></span><br/>

<a NAME="E3:20"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
 for <font color="Olive" title="b1">D</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font><br/>  for <font color="Olive" title="b2">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">D</font> holds <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Olive" title="b1">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Olive" title="b1">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">k</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Olive" title="b1">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Olive" title="b1">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Olive" title="b1">D</font>,<font color="Olive" title="b2">k</font>)</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Olive" title="b1">D</font>,<font color="Olive" title="b2">k</font>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Olive" title="b1">D</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">D</font> be    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">D</font> holds <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Olive" title="b1">k</font>)</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Olive" title="b1">k</font>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c3">D</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">D</font> implies (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c3">D</font></span>)</span> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:20_1"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c3">D</font></span>)</span> )</span><br/>

<a NAME="E2:20_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A5</font></span>: 
( <span class="p1">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)</span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/>
<a NAME="E3:20_1"/>
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E4:20_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>)</span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/>
<a NAME="E5:20_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A6</font></span>: 
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>)</span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E6:20_1"/>
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>)</span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="integra1.html#T20" title="INTEGRA1:th.20">INTEGRA1:20</a></span>;<br/>
<a NAME="E7:20_1"/><span class="kw">then </span>
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>)</span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E8:20_1"/><span class="kw">then </span>
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>)</span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D8" title="XXREAL_2:def.8">XXREAL_2:def 8</a></span>;<br/>
<a NAME="E9:20_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A7</font></span>: 
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c3">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra3.html#D1" title="INTEGRA3:def.1">INTEGRA3:def 1</a></span>;<br/>
<a NAME="E10:20_1"/><span class="lab"><font color="Green" title="E25">A8</font></span>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="integra1.html#T8" title="INTEGRA1:th.8">INTEGRA1:8</a></span>;<br/>
<a NAME="E11:20_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A9</font></span>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E12:20_1"/>
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)</span>)</span> is  <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E1:20_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="integra1.html#T8" title="INTEGRA1:th.8">INTEGRA1:8</a>, <a class="ref" href="rfunct_2.html#T30" title="RFUNCT_2:th.30">RFUNCT_2:30</a></span>;<br/>
<a NAME="E13:20_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A10</font></span>: 
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)</span>)</span></span>)</span> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)</span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:20_1"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="ref" href="integra5.html#T15" target="_self" title="INTEGRA5:th.15">Th15</a></span>;<br/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>
<br/>.= 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E10:20_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
;<br/>
<a NAME="E15:20_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A11</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)</span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T42" title="RELAT_1:th.42">RELAT_1:42</a></span>;<br/>
<a NAME="E16:20_1"/><span class="lab"><font color="Green" title="E29">A12</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:20_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="ref" href="funct_1.html#T51" title="FUNCT_1:th.51">FUNCT_1:51</a></span>;<br/>
<a NAME="E17:20_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font></span>)</span> is  <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="integra5.html#T14" target="_self" title="INTEGRA5:th.14">Th14</a></span>;<br/>
<a NAME="E18:20_1"/><span class="kw">then </span>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)</span>)</span></span>)</span> is  <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:20_1"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="ref" href="relat_1.html#T70" title="RELAT_1:th.70">RELAT_1:70</a>, <a class="ref" href="xxreal_2.html#T45" title="XXREAL_2:th.45">XXREAL_2:45</a></span>;<br/>
<a NAME="E19:20_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A13</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:20_1"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="txt" href="#E15:20_1"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="ref" href="seq_4.html#T11" title="SEQ_4:th.11">SEQ_4:11</a>, <a class="ref" href="xreal_1.html#T48" title="XREAL_1:th.48">XREAL_1:48</a></span>;<br/>
<a NAME="E20:20_1"/>
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a></span>;<br/>
<a NAME="E21:20_1"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:20_1"><span class="lab"><font color="Green" title="E30">A13</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E22:20_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:20_1"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="txt" href="#E2:20_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E16:20_1"><span class="lab"><font color="Green" title="E29">A12</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c3">D</font></span>)</span></span><br/>

<a NAME="E23:20_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:20_1"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="txt" href="#E2:20_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E16:20_1"><span class="lab"><font color="Green" title="E29">A12</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E24:20_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">k</font>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c3">D</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:20_1"><span class="lab"><font color="Green" title="E30">A13</font></span></a>, <a class="txt" href="#E9:20_1"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:20"/><span class="lab"><font color="Green" title="E21">A14</font></span>: 
 for <font color="Olive" title="b1">D</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font> holds <br/> ( <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Olive" title="b1">D</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Olive" title="b1">D</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Olive" title="b1">D</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Olive" title="b1">D</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Olive" title="b1">D</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">D</font> be    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  = <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,$1)</span>)</span></span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">F1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E2:20_2"/><span class="lab"><font color="Green" title="E22">A15</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D</font> &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">F1</font> holds <br/><font color="Maroon" title="c4">F1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></span>
<a NAME="E3:20_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:20_2_1"/>
 ex <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:20_2_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">k1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E5:20_2"/><span class="lab"><font color="Green" title="E23">A16</font></span>: 
<font color="Maroon" title="c5">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 ;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  = <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,$1)</span>)</span></span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">F2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E7:20_2"/><span class="lab"><font color="Green" title="E24">A17</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D</font> &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">F2</font> holds <br/><font color="Maroon" title="c6">F2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">i</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></span>
<span class="kw">deffunc </span><font color="Maroon">H<sub>3</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  = <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,$1)</span>)</span></span>)</span>;<br/>
<a NAME="E8:20_2"/><span class="lab"><font color="Green" title="E25">A18</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20_2"><span class="lab"><font color="Green" title="E24">A17</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E10:20_2"/><span class="lab"><font color="Green" title="E26">A19</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k1</font> &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F</font> holds <br/><font color="Maroon" title="c7">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">i</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></span>
<a NAME="E11:20_2"/><span class="lab"><font color="Green" title="E27">A20</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">k1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:20_2"><span class="lab"><font color="Green" title="E26">A19</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E12:20_2"/><span class="lab"><font color="Green" title="E28">A21</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F2</font> holds <br/><font color="Maroon" title="c6">F2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F2</font> implies <font color="Maroon" title="c6">F2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:20_2_2"/><span class="lab"><font color="Green" title="E29">A22</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font>
 <span class="kw">and </span><br/><a NAME="E2:20_2_2"/><span class="lab"><font color="Green" title="E30">A23</font></span>: 
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">F2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_2_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:20_2_2_1"/>
( <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c8">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:20_2_2_1_1"/><span class="lab"><font color="Green" title="E31">A24</font></span>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">F2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span><br/><div class="add"><a NAME="E2:20_2_2_1_1"/><span class="lab"><font color="Green" title="E32">A25</font></span>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20_2"><span class="lab"><font color="Green" title="E24">A17</font></span></a>, <a class="txt" href="#E1:20_2_2"><span class="lab"><font color="Green" title="E29">A22</font></span></a>, <a class="txt" href="#E2:20_2_2"><span class="lab"><font color="Green" title="E30">A23</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E3:20_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A26</font></span>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">D</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><font color="Maroon" title="c6">F2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> = 
<font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">i</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20_2"><span class="lab"><font color="Green" title="E24">A17</font></span></a>, <a class="txt" href="#E8:20_2"><span class="lab"><font color="Green" title="E25">A18</font></span></a>, <a class="txt" href="#E2:20_2_2_1_1"><span class="lab"><font color="Green" title="E32">A25</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_2_2_1_1"><span class="lab"><font color="Green" title="E31">A24</font></span></a>, <a class="txt" href="#E3:20_2_2_1_1"><span class="lab"><font color="Green" title="E33">A26</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:20_2_2_1_1"/>
<font color="Maroon" title="c6">F2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_2_2_1_1"><span class="lab"><font color="Green" title="E31">A24</font></span></a>, <a class="ref" href="finseq_1.html#T41" title="FINSEQ_1:th.41">FINSEQ_1:41</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:20_2_2_1_2"/><span class="lab"><font color="Green" title="E31">A27</font></span>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">F2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span><br/><div class="add"><a NAME="E2:20_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A28</font></span>: 
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_2_2"><span class="lab"><font color="Green" title="E29">A22</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:20_2_2_1_2"/><span class="kw">then </span>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:20_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A29</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a></span>;<br/><a NAME="E5:20_2_2_1_2"/><span class="lab"><font color="Green" title="E34">A30</font></span>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20_2"><span class="lab"><font color="Green" title="E24">A17</font></span></a>, <a class="txt" href="#E1:20_2_2"><span class="lab"><font color="Green" title="E29">A22</font></span></a>, <a class="txt" href="#E2:20_2_2"><span class="lab"><font color="Green" title="E30">A23</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E6:20_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A31</font></span>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">D</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">k2</font> = <font color="Maroon" title="c8">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_2_2_1_2"><span class="lab"><font color="Green" title="E31">A27</font></span></a>, <a class="ref" href="integra1.html#T7" title="INTEGRA1:th.7">INTEGRA1:7</a></span>;<br/><a NAME="E8:20_2_2_1_2"/>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_2_2_1_2"><span class="lab"><font color="Green" title="E32">A28</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E9:20_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A32</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/><a NAME="E10:20_2_2_1_2"/>
<font color="Maroon" title="c9">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_2_2"><span class="lab"><font color="Green" title="E30">A23</font></span></a></span>;<br/><a NAME="E11:20_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A33</font></span>: 
<font color="Maroon" title="c9">k2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20_2"><span class="lab"><font color="Green" title="E24">A17</font></span></a>, <a class="ref" href="xreal_1.html#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/><a NAME="E12:20_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A34</font></span>: 
<font color="Maroon" title="c9">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20_2"><span class="lab"><font color="Green" title="E23">A16</font></span></a>, <a class="txt" href="#E9:20_2_2_1_2"><span class="lab"><font color="Green" title="E36">A32</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E13:20_2_2_1_2"/><span class="lab"><font color="Green" title="E39">A35</font></span>: 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c9">k2</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_2_1_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:20_2_2_1_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E2:20_2_2_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T20" title="XREAL_1:th.20">XREAL_1:20</a></span>;<br/>
<a NAME="E3:20_2_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">k2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:20_2_2_1_2"><span class="lab"><font color="Green" title="E37">A33</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E4:20_2_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:20_2_2_1_2"><span class="lab"><font color="Green" title="E36">A32</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E5:20_2_2_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A36</font></span>: 
<font color="Maroon" title="c9">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_2_2_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:20_2_2_1_2_1_1"/>
( <font color="Maroon" title="c9">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c9">k2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_2_2_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:20_2_2_1_2_1_1_1"/>
<font color="Maroon" title="c9">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c9">k2</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:20_2_2_1_2_1_1_1"/>
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c9">k2</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20_2_2_1_2_1"><span class="lab"><font color="Green" title="E40">A36</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_2_2_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:20_2_2_1_2_1_1_2"/>
<font color="Maroon" title="c9">k2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c9">k2</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:20_2_2_1_2_1_1_2"/>
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c9">k2</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20_2_2_1_2_1"><span class="lab"><font color="Green" title="E40">A36</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F2</font> = 
1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20_2"><span class="lab"><font color="Green" title="E24">A17</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20_2"><span class="lab"><font color="Green" title="E23">A16</font></span></a>, <a class="txt" href="#E10:20_2"><span class="lab"><font color="Green" title="E26">A19</font></span></a>, <a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a></span>
;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A37</font></span>: <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> = 
<font color="Maroon" title="c7">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p5">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_2_2"><span class="lab"><font color="Green" title="E30">A23</font></span></a>, <a class="txt" href="#E4:20_2_2_1_2"><span class="lab"><font color="Green" title="E33">A29</font></span></a>, <a class="ref" href="finseq_1.html#T23" title="FINSEQ_1:th.23">FINSEQ_1:23</a></span>
<br/>.= 
<font color="Maroon" title="c7">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a></span>
;<br/><font color="Maroon" title="c6">F2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> = 
<font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">i</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20_2"><span class="lab"><font color="Green" title="E24">A17</font></span></a>, <a class="txt" href="#E8:20_2"><span class="lab"><font color="Green" title="E25">A18</font></span></a>, <a class="txt" href="#E5:20_2_2_1_2"><span class="lab"><font color="Green" title="E34">A30</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_2_2_1_2"><span class="lab"><font color="Green" title="E31">A27</font></span></a>, <a class="txt" href="#E6:20_2_2_1_2"><span class="lab"><font color="Green" title="E35">A31</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>
;<br/><span class="kw">hence </span><a NAME="E17:20_2_2_1_2"/>
<font color="Maroon" title="c6">F2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:20_2"><span class="lab"><font color="Green" title="E26">A19</font></span></a>, <a class="txt" href="#E11:20_2"><span class="lab"><font color="Green" title="E27">A20</font></span></a>, <a class="txt" href="#E15:20_2_2_1_2"><span class="lab"><font color="Green" title="E40">A37</font></span></a>, <a class="txt" href="#E12:20_2_2_1_2"><span class="lab"><font color="Green" title="E38">A34</font></span></a>, <a class="txt" href="#E13:20_2_2_1_2"><span class="lab"><font color="Green" title="E39">A35</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">F</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>
<br/>.= 
1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20_2"><span class="lab"><font color="Green" title="E23">A16</font></span></a>, <a class="txt" href="#E10:20_2"><span class="lab"><font color="Green" title="E26">A19</font></span></a>, <a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D</font>

;<br/>
<a NAME="E14:20_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A38</font></span>: 
<font color="Maroon" title="c6">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20_2"><span class="lab"><font color="Green" title="E24">A17</font></span></a>, <a class="txt" href="#E12:20_2"><span class="lab"><font color="Green" title="E28">A21</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</a></span>;<br/>
<a NAME="E15:20_2"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E30">A39</font></span>:  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra5.html#T2" target="_self" title="INTEGRA5:th.2">Th2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D9" title="INTEGRA1:def.9">INTEGRA1:def 9</a></span>
;<br/>
<a NAME="E17:20_2"/><span class="lab"><font color="Green" title="E31">A40</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_2"><span class="lab"><font color="Green" title="E22">A15</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E18:20_2"/><span class="lab"><font color="Green" title="E32">A41</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F1</font> holds <br/><font color="Maroon" title="c4">F1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F1</font> implies <font color="Maroon" title="c4">F1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:20_2_5"/><span class="lab"><font color="Green" title="E33">A42</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font>
 <span class="kw">and </span><br/><a NAME="E2:20_2_5"/><span class="lab"><font color="Green" title="E34">A43</font></span>: 
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">F1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_5_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_2_5_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:20_2_5_1_1"/>
( <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font> or <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_2_5_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:20_2_5_1_1_1"/>
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">F1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span><br/><div class="add"><a NAME="E2:20_2_5_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_2_5"><span class="lab"><font color="Green" title="E33">A42</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E3:20_2_5_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A44</font></span>: 
( <font color="Maroon" title="c7">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">i</font>)</span>)</span></span>)</span> &amp; <font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:20_2"><span class="lab"><font color="Green" title="E26">A19</font></span></a>, <a class="txt" href="#E11:20_2"><span class="lab"><font color="Green" title="E27">A20</font></span></a></span>;<br/><a NAME="E4:20_2_5_1_1_1"/>
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_2_5"><span class="lab"><font color="Green" title="E33">A42</font></span></a>, <a class="txt" href="#E2:20_2_5"><span class="lab"><font color="Green" title="E34">A43</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E5:20_2_5_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">F1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">i</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_2"><span class="lab"><font color="Green" title="E22">A15</font></span></a>, <a class="txt" href="#E17:20_2"><span class="lab"><font color="Green" title="E31">A40</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:20_2_5_1_1_1"/>
<font color="Maroon" title="c4">F1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_2_5_1_1_1"><span class="lab"><font color="Green" title="E35">A44</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_2_5_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:20_2_5_1_1_2"/>
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">F1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span><br/><div class="add"><a NAME="E2:20_2_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A45</font></span>: 
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:20_2_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A46</font></span>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_2"><span class="lab"><font color="Green" title="E22">A15</font></span></a>, <a class="txt" href="#E5:20_2"><span class="lab"><font color="Green" title="E23">A16</font></span></a>, <a class="txt" href="#E10:20_2"><span class="lab"><font color="Green" title="E26">A19</font></span></a>, <a class="txt" href="#E2:20_2_5"><span class="lab"><font color="Green" title="E34">A43</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:20_2_5_1_1_2"/><span class="lab"><font color="Green" title="E37">A47</font></span>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_2_5"><span class="lab"><font color="Green" title="E33">A42</font></span></a>, <a class="txt" href="#E2:20_2_5"><span class="lab"><font color="Green" title="E34">A43</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E5:20_2_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A48</font></span>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_2"><span class="lab"><font color="Green" title="E22">A15</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E6:20_2_5_1_1_2"/><span class="lab"><font color="Green" title="E39">A49</font></span>: 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_5_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_5_1_1_2_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_2_5_1_1_2_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:20_2_5_1_1_2_1_1_1"/>
( <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c8">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_2_5_1_1_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:20_2_5_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:20_2_5_1_1_2_1_1_1_1"/>
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20_2_5_1_1_2"><span class="lab"><font color="Green" title="E38">A48</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_2_5_1_1_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:20_2_5_1_1_2_1_1_1_2"/>
<font color="Maroon" title="c8">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:20_2_5_1_1_2_1_1_1_2"/>
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20_2_5_1_1_2"><span class="lab"><font color="Green" title="E38">A48</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:20_2_5_1_1_2_1"/>
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E7:20_2_5_1_1_2"/>
<font color="Maroon" title="c4">F1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">i</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_2"><span class="lab"><font color="Green" title="E22">A15</font></span></a>, <a class="txt" href="#E17:20_2"><span class="lab"><font color="Green" title="E31">A40</font></span></a>, <a class="txt" href="#E4:20_2_5_1_1_2"><span class="lab"><font color="Green" title="E37">A47</font></span></a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c4">F1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> = 
<font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_2"><span class="lab"><font color="Green" title="E22">A15</font></span></a>, <a class="txt" href="#E5:20_2"><span class="lab"><font color="Green" title="E23">A16</font></span></a>, <a class="txt" href="#E10:20_2"><span class="lab"><font color="Green" title="E26">A19</font></span></a>, <a class="txt" href="#E2:20_2_5"><span class="lab"><font color="Green" title="E34">A43</font></span></a>, <a class="txt" href="#E2:20_2_5_1_1_2"><span class="lab"><font color="Green" title="E35">A45</font></span></a>, <a class="txt" href="#E6:20_2_5_1_1_2"><span class="lab"><font color="Green" title="E39">A49</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D2" title="INTEGRA1:def.2">INTEGRA1:def 2</a></span>
;<br/><span class="kw">hence </span><a NAME="E9:20_2_5_1_1_2"/>
<font color="Maroon" title="c4">F1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_2_5_1_1_2"><span class="lab"><font color="Green" title="E36">A46</font></span></a>, <a class="ref" href="finseq_1.html#T42" title="FINSEQ_1:th.42">FINSEQ_1:42</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:20_2_5"/>
<font color="Maroon" title="c4">F1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E19:20_2"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/>
<a NAME="E20:20_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A50</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra5.html#T2" target="_self" title="INTEGRA5:th.2">Th2</a></span>;<br/>
<a NAME="E21:20_2"/><span class="lab"><font color="Green" title="E34">A51</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c6">F2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_2"><span class="lab"><font color="Green" title="E22">A15</font></span></a>, <a class="txt" href="#E7:20_2"><span class="lab"><font color="Green" title="E24">A17</font></span></a>, <a class="ref" href="integra5.html#T2" target="_self" title="INTEGRA5:th.2">Th2</a></span>;<br/>
<a NAME="E22:20_2"/><span class="lab"><font color="Green" title="E35">A52</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c6">F2</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c6">F2</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">k</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:20_2_6"/><span class="lab"><font color="Green" title="E36">A53</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c6">F2</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">k</font></span><br/>

<a NAME="E2:20_2_6"/><span class="kw">then </span>
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c6">F2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:20_2"><span class="lab"><font color="Green" title="E33">A50</font></span></a>, <a class="txt" href="#E21:20_2"><span class="lab"><font color="Green" title="E34">A51</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E3:20_2_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A54</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c6">F2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E4:20_2_6"/><span class="lab"><font color="Green" title="E38">A55</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:20_2"><span class="lab"><font color="Green" title="E33">A50</font></span></a>, <a class="txt" href="#E1:20_2_6"><span class="lab"><font color="Green" title="E36">A53</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E5:20_2_6"/><span class="kw">then </span>
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E6:20_2_6"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">k</font>)</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">k</font>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c3">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E20">A3</font></span></a></span>;<br/>
<a NAME="E7:20_2_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A56</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">k</font>)</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">k</font>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c3">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_2_6"><span class="lab"><font color="Green" title="E36">A53</font></span></a>, <a class="ref" href="valued_1.html#T13" title="VALUED_1:th.13">VALUED_1:13</a></span>;<br/>
<a NAME="E8:20_2_6"/>
( <font color="Maroon" title="c4">F1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">k</font>)</span>)</span></span>)</span> &amp; <font color="Maroon" title="c6">F2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">k</font>)</span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_2"><span class="lab"><font color="Green" title="E22">A15</font></span></a>, <a class="txt" href="#E17:20_2"><span class="lab"><font color="Green" title="E31">A40</font></span></a>, <a class="txt" href="#E7:20_2"><span class="lab"><font color="Green" title="E24">A17</font></span></a>, <a class="txt" href="#E8:20_2"><span class="lab"><font color="Green" title="E25">A18</font></span></a>, <a class="txt" href="#E4:20_2_6"><span class="lab"><font color="Green" title="E38">A55</font></span></a></span>;<br/>
<a NAME="E9:20_2_6"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c6">F2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20_2_6"><span class="lab"><font color="Green" title="E39">A56</font></span></a>, <a class="txt" href="#E3:20_2_6"><span class="lab"><font color="Green" title="E37">A54</font></span></a>, <a class="ref" href="valued_1.html#T13" title="VALUED_1:th.13">VALUED_1:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:20_2_6"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c6">F2</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T44" title="RVSUM_1:th.44">RVSUM_1:44</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E23:20_2"/>
<span class="p1">(<span class="default"><a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c6">F2</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="rvsum_1.html#K3" title="RVSUM_1:func.3">multreal</a></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c6">F2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#D7" title="RVSUM_1:def.7">RVSUM_1:def 7</a></span>;<br/>
<a NAME="E24:20_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c6">F2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c6">F2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T33" title="FINSEQ_2:th.33">FINSEQ_2:33</a></span>;<br/>
<a NAME="E25:20_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c6">F2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_2"><span class="lab"><font color="Green" title="E22">A15</font></span></a>, <a class="txt" href="#E7:20_2"><span class="lab"><font color="Green" title="E24">A17</font></span></a>, <a class="txt" href="#E20:20_2"><span class="lab"><font color="Green" title="E33">A50</font></span></a>, <a class="ref" href="integra5.html#T2" target="_self" title="INTEGRA5:th.2">Th2</a></span>;<br/>
<a NAME="E26:20_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A57</font></span>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c6">F2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:20_2"><span class="lab"><font color="Green" title="E35">A52</font></span></a>, <a class="ref" href="integra5.html#T3" target="_self" title="INTEGRA5:th.3">Th3</a></span>;<br/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20_2"><span class="lab"><font color="Green" title="E23">A16</font></span></a>, <a class="txt" href="#E10:20_2"><span class="lab"><font color="Green" title="E26">A19</font></span></a>, <a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D</font>

;<br/>
<a NAME="E28:20_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_2"><span class="lab"><font color="Green" title="E22">A15</font></span></a>, <a class="txt" href="#E18:20_2"><span class="lab"><font color="Green" title="E32">A41</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</a></span>;<br/>
<a NAME="E29:20_2"/><span class="kw">then </span>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c6">F2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:20_2"><span class="lab"><font color="Green" title="E29">A38</font></span></a>, <a class="ref" href="integra5.html#T1" target="_self" title="INTEGRA5:th.1">Th1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E30:20_2"/>
<span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:20_2"><span class="lab"><font color="Green" title="E30">A39</font></span></a>, <a class="txt" href="#E26:20_2"><span class="lab"><font color="Green" title="E36">A57</font></span></a>, <a class="ref" href="rvsum_1.html#T87" title="RVSUM_1:th.87">RVSUM_1:87</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E31:20_2"/>
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span></span>)</span> holds <br/> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span></span>)</span> implies  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">k</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c9">r</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">k</font>;<br/>
<a NAME="E1:20_2_8"/><span class="lab"><font color="Green" title="E37">A58</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:20_2_8"/><span class="lab"><font color="Green" title="E38">A59</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">k</font></span><br/>

<a NAME="E3:20_2_8"/><span class="kw">then </span>
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p3">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E4:20_2_8"/><span class="kw">then </span>
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_2_8"><span class="lab"><font color="Green" title="E37">A58</font></span></a>, <a class="ref" href="integra5.html#T2" target="_self" title="INTEGRA5:th.2">Th2</a></span>;<br/>
<a NAME="E5:20_2_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A60</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E6:20_2_8"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_2_8"><span class="lab"><font color="Green" title="E38">A59</font></span></a>, <a class="ref" href="valued_1.html#T13" title="VALUED_1:th.13">VALUED_1:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:20_2_8"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E5:20_2_8"><span class="lab"><font color="Green" title="E39">A60</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E32:20_2"/>
<span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:20_2"><span class="lab"><font color="Green" title="E30">A39</font></span></a>, <a class="ref" href="rvsum_1.html#T84" title="RVSUM_1:th.84">RVSUM_1:84</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:20"/><span class="lab"><font color="Green" title="E22">A61</font></span>: 
( <font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c3">x</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E2:20_3"/><span class="lab"><font color="Green" title="E23">A62</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T4" title="SUBSET_1:th.4">SUBSET_1:4</a></span>;<br/>
<span class="lab"><font color="Green" title="E24">A63</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c1">A</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>
<br/>.= 
<font color="Maroon" title="c1">A</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:20_3"/>
<font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> </span><br/>

<a NAME="E5:20_3"/>
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c3">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_3"><span class="lab"><font color="Green" title="E23">A62</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/>
<a NAME="E6:20_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A64</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E7:20_3"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_3_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:20_3_2"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20_3"><span class="lab"><font color="Green" title="E25">A64</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/>
<a NAME="E2:20_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A65</font></span>: 
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E3:20_3"><span class="lab"><font color="Green" title="E24">A63</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E3:20_3_2"/><span class="lab"><font color="Green" title="E27">A66</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span><br/>

<a NAME="E4:20_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 
;<br/>
<a NAME="E5:20_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A67</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_3_2"><span class="lab"><font color="Green" title="E27">A66</font></span></a></span>;<br/>
<a NAME="E7:20_3_2"/>
( <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_3"><span class="lab"><font color="Green" title="E24">A63</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:20_3_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E5:20_3_2"><span class="lab"><font color="Green" title="E28">A67</font></span></a>, <a class="txt" href="#E2:20_3_2"><span class="lab"><font color="Green" title="E26">A65</font></span></a>, <a class="ref" href="rfunct_2.html#T24" title="RFUNCT_2:th.24">RFUNCT_2:24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:20_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A68</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T71" title="RFUNCT_1:th.71">RFUNCT_1:71</a></span>;<br/>
<a NAME="E9:20_3"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_3_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:20_3_3"/>
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20_3"><span class="lab"><font color="Green" title="E25">A64</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/>
<a NAME="E2:20_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A69</font></span>: 
(  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E3:20_3"><span class="lab"><font color="Green" title="E24">A63</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E3:20_3_3"/><span class="lab"><font color="Green" title="E28">A70</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span><br/>

<a NAME="E4:20_3_3"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 
;<br/>
<a NAME="E5:20_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A71</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_3_3"><span class="lab"><font color="Green" title="E28">A70</font></span></a></span>;<br/>
<a NAME="E7:20_3_3"/>
( <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_3"><span class="lab"><font color="Green" title="E24">A63</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:20_3_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E5:20_3_3"><span class="lab"><font color="Green" title="E29">A71</font></span></a>, <a class="txt" href="#E2:20_3_3"><span class="lab"><font color="Green" title="E27">A69</font></span></a>, <a class="ref" href="rfunct_2.html#T24" title="RFUNCT_2:th.24">RFUNCT_2:24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:20_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T70" title="RFUNCT_1:th.70">RFUNCT_1:70</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:20_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:20_3"><span class="lab"><font color="Green" title="E26">A68</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:20"/>
 for <font color="Olive" title="b1">T</font> being   <a href="integra2.html#NM1" title="INTEGRA2:NM.1">DivSequence</a> of <font color="Maroon" title="c1">A</font>  st  <a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Olive" title="b1">T</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Olive" title="b1">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Olive" title="b1">T</font>)</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Olive" title="b1">T</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">T</font> be   <a href="integra2.html#NM1" title="INTEGRA2:NM.1">DivSequence</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c3">T</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:20_4"/><span class="lab"><font color="Green" title="E23">A72</font></span>: 
(  <a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c3">T</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E2:20_4"/><span class="lab"><font color="Green" title="E24">A73</font></span>: 
 for <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> holds <br/> ex <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/> for <font color="Olive" title="b3">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p4">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c4">x</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E2:20_4_1"/><span class="lab"><font color="Green" title="E25">A74</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T4" title="SUBSET_1:th.4">SUBSET_1:4</a></span>;<br/>
<a NAME="E3:20_4_1"/><span class="lab"><font color="Green" title="E26">A75</font></span>: 
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">r</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> implies  ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/> for <font color="Olive" title="b2">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p4">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> )</span><br/>

<span class="kw">assume </span><a NAME="E4:20_4_1"/><span class="lab"><font color="Green" title="E27">A76</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/> for <font color="Olive" title="b2">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p4">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font></span><br/>

<a NAME="E5:20_4_1"/>
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_4_1"><span class="lab"><font color="Green" title="E25">A74</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/>
<a NAME="E6:20_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A77</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E7:20_4_1"/><span class="kw">then </span>
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/>
<a NAME="E8:20_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A78</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E6:20_4_1"><span class="lab"><font color="Green" title="E28">A77</font></span></a>, <a class="txt" href="#E3:20_4_1"><span class="lab"><font color="Green" title="E26">A75</font></span></a>, <a class="ref" href="rfunct_2.html#T24" title="RFUNCT_2:th.24">RFUNCT_2:24</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_4_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:20_4_1_1_1"/>
( <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span> or <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:20_4_1"><span class="lab"><font color="Green" title="E29">A78</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_4_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:20_4_1_1_1_1"/><span class="lab"><font color="Green" title="E30">A79</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/> for <font color="Olive" title="b2">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p4">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font></span><br/><div class="add"><a NAME="E2:20_4_1_1_1_1"/>
 for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p4">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font> implies  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p4">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:20_4_1_1_1_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p4">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">D1</font> = <font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c6">m</font> as    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font> ;<br/>
<span class="lab"><font color="Green" title="E31">A80</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">m</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p3">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T7" title="SEQ_1:th.7">SEQ_1:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">m</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T10" title="SEQ_1:th.10">SEQ_1:10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c7">D1</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">m</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra2.html#D2" title="INTEGRA2:def.2">INTEGRA2:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c7">D1</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c7">D1</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra2.html#D3" title="INTEGRA2:def.3">INTEGRA2:def 3</a></span>
;<br/>
<a NAME="E4:20_4_1_1_1_1_1"/>
( <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c7">D1</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c7">D1</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c7">D1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c7">D1</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c7">D1</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E21">A14</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:20_4_1_1_1_1_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p4">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_4_1"><span class="lab"><font color="Green" title="E27">A76</font></span></a>, <a class="txt" href="#E1:20_4_1_1_1_1"><span class="lab"><font color="Green" title="E30">A79</font></span></a>, <a class="txt" href="#E3:20_4_1_1_1_1_1"><span class="lab"><font color="Green" title="E31">A80</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:20_4_1_1_1_1"/>
 ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/> for <font color="Olive" title="b2">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p4">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_4_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:20_4_1_1_1_2"/>
<font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/> for <font color="Olive" title="b2">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p4">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font></span><br/><div class="add"><a NAME="E2:20_4_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A81</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/><a NAME="E3:20_4_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_4_1"><span class="lab"><font color="Green" title="E27">A76</font></span></a>, <a class="ref" href="xreal_1.html#T139" title="XREAL_1:th.139">XREAL_1:139</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E5:20_4_1_1_1_2"/><span class="lab"><font color="Green" title="E31">A82</font></span>: 
 for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_4"><span class="lab"><font color="Green" title="E23">A72</font></span></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></span>;<br/><a NAME="E6:20_4_1_1_1_2"/><span class="lab"><font color="Green" title="E32">A83</font></span>: 
 for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1">(<span class="default"><a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m</font> implies <span class="p1">(<span class="default"><a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:20_4_1_1_1_2_1"/>
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span></span><br/>

<a NAME="E2:20_4_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A84</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20_4_1_1_1_2"><span class="lab"><font color="Green" title="E31">A82</font></span></a></span>;<br/>
<a NAME="E3:20_4_1_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#T4" title="ABSVALUE:th.4">ABSVALUE:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:20_4_1_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_4_1_1_1_2_1"><span class="lab"><font color="Green" title="E33">A84</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E7:20_4_1_1_1_2"/>
 for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p4">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4_1_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m</font> implies  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p4">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">D</font> = <font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c7">m</font> as    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font> ;<br/>
<span class="lab"><font color="Green" title="E33">A85</font></span>:  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c8">D</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c8">D</font>)</span>)</span></span>)</span> = 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c8">D</font>)</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra2.html#D2" title="INTEGRA2:def.2">INTEGRA2:def 2</a></span>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra2.html#D3" title="INTEGRA2:def.3">INTEGRA2:def 3</a></span>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<span class="p0">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>

<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p5">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<span class="p0">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T10" title="SEQ_1:th.10">SEQ_1:10</a></span>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p4">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T7" title="SEQ_1:th.7">SEQ_1:7</a></span>
;<br/>
<span class="kw">assume </span><a NAME="E3:20_4_1_1_1_2_2"/>
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p4">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font></span><br/>

<a NAME="E4:20_4_1_1_1_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20_4_1_1_1_2"><span class="lab"><font color="Green" title="E32">A83</font></span></a></span>;<br/>
<a NAME="E5:20_4_1_1_1_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_4_1_1_1_2"><span class="lab"><font color="Green" title="E30">A81</font></span></a>, <a class="ref" href="xreal_1.html#T79" title="XREAL_1:th.79">XREAL_1:79</a></span>;<br/>
<a NAME="E6:20_4_1_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A86</font></span>: 
<span class="p1">(<span class="default"><a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c8">D</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra3.html#D2" title="INTEGRA3:def.2">INTEGRA3:def 2</a></span>;<br/>
<a NAME="E7:20_4_1_1_1_2_2"/>
<span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c8">D</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c8">D</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c8">D</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E21">A14</font></span></a></span>;<br/>
<a NAME="E8:20_4_1_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A87</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c8">D</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c8">D</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20_4_1_1_1_2_2"><span class="lab"><font color="Green" title="E34">A86</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E9:20_4_1_1_1_2_2"/>
<span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c8">D</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c8">D</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E21">A14</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:20_4_1_1_1_2_2"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p4">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:20_4_1_1_1_2_2"><span class="lab"><font color="Green" title="E35">A87</font></span></a>, <a class="txt" href="#E2:20_4_1_1_1_2_2"><span class="lab"><font color="Green" title="E33">A85</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E8:20_4_1_1_1_2"/>
 ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/> for <font color="Olive" title="b2">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p4">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E10:20_4_1"/>
 ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/> for <font color="Olive" title="b2">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p4">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:20_4"/><span class="lab"><font color="Green" title="E25">A88</font></span>: 
(  <a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>) is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>) is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20"><span class="lab"><font color="Green" title="E22">A61</font></span></a>, <a class="txt" href="#E1:20_4"><span class="lab"><font color="Green" title="E23">A72</font></span></a>, <a class="ref" href="integra4.html#T8" title="INTEGRA4:th.8">INTEGRA4:8</a>, <a class="ref" href="integra4.html#T9" title="INTEGRA4:th.9">INTEGRA4:9</a></span>;<br/>
<a NAME="E4:20_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_2.html#T11" title="SEQ_2:th.11">SEQ_2:11</a></span>;<br/>
<a NAME="E5:20_4"/><span class="kw">then </span>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_4"><span class="lab"><font color="Green" title="E24">A73</font></span></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:20_4"/>
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">T</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_4"><span class="lab"><font color="Green" title="E25">A88</font></span></a>, <a class="ref" href="seq_2.html#T12" title="SEQ_2:th.12">SEQ_2:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:20"/><span class="kw">then </span>
<font color="Maroon" title="c2">f</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c1">A</font> is  <a href="integra1.html#V3" title="INTEGRA1:attr.3">integrable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20"><span class="lab"><font color="Green" title="E22">A61</font></span></a>, <a class="ref" href="integra4.html#T12" title="INTEGRA4:th.12">INTEGRA4:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:20"/>
<font color="Maroon" title="c2">f</font> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra5.html#D1" target="_self" title="INTEGRA5:def.1">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
