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<div class="add">

<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="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">A</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><br/>  for <font color="Olive" title="b2">T</font> being   <a href="integra2.html#NM1" title="INTEGRA2:NM.1">DivSequence</a> of <font color="Maroon" title="c1">A</font>  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="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp;  <a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Olive" title="b2">T</font> is  <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">convergent_to_0</a>  &amp;  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/>(  <a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">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="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">T</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K11" title="INTEGRA1:func.11">lower_integral</a> <font color="Olive" title="b1">f</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">A</font>,<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">T</font> being   <a href="integra2.html#NM1" title="INTEGRA2:NM.1">DivSequence</a> of <font color="Maroon" title="c1">A</font>  st <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="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp;  <a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Olive" title="b1">T</font> is  <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">convergent_to_0</a>  &amp;  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/>(  <a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<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="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">T</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K11" title="INTEGRA1:func.11">lower_integral</a> <font color="Maroon" title="c2">f</font> )</span><br/><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"> ( <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="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp;  <a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c3">T</font> is  <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">convergent_to_0</a>  &amp;  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies (  <a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<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="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K11" title="INTEGRA1:func.11">lower_integral</a> <font color="Maroon" title="c2">f</font> ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:34"/><span class="lab"><font color="Green" title="E32">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="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 <span class="kw">and </span><br/><a NAME="E2:34"/><span class="lab"><font color="Green" title="E33">A2</font></span>: 
 <a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c3">T</font> is  <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">convergent_to_0</a> 
 <span class="kw">and </span><br/><a NAME="E3:34"/><span class="lab"><font color="Green" title="E34">A3</font></span>: 
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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"> (  <a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<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="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K11" title="INTEGRA1:func.11">lower_integral</a> <font color="Maroon" title="c2">f</font> )</span><br/>

<a NAME="E4:34"/><span class="lab"><font color="Green" title="E35">A4</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="fdiff_1.html#D1" title="FDIFF_1:def.1">FDIFF_1:def 1</a></span>;<br/>
<a NAME="E5:34"/><span class="lab"><font color="Green" title="E36">A5</font></span>: 
 for <font color="Olive" title="b1">D</font>, <font color="Olive" title="b2">D1</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font>  ex <font color="Olive" title="b3">D2</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b1">D</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b3">D2</font> &amp; <font color="Olive" title="b2">D1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b3">D2</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b3">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b2">D1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b1">D</font></span>)</span> &amp;  <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"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b3">D2</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> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">D</font>)</span>)</span> &amp;  <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"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b3">D2</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> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">D1</font>)</span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">D</font>, <font color="Maroon" title="c5">D1</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">  ex <font color="Olive" title="b1">D2</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Maroon" title="c4">D</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b1">D2</font> &amp; <font color="Maroon" title="c5">D1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b1">D2</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b1">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font></span>)</span> &amp;  <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"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">D2</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> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D</font>)</span>)</span> &amp;  <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"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">D2</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> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span> )</span><br/>



<span class="kw">consider </span><font color="Maroon" title="c6">D2</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E2:34_1"/><span class="lab"><font color="Green" title="E37">A6</font></span>: 
<font color="Maroon" title="c4">D</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Maroon" title="c6">D2</font>
 <span class="kw">and </span><br/><a NAME="E3:34_1"/><span class="lab"><font color="Green" title="E38">A7</font></span>: 
<font color="Maroon" title="c5">D1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Maroon" title="c6">D2</font>
 <span class="kw">and </span><br/><a NAME="E4:34_1"/><span class="lab"><font color="Green" title="E39">A8</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c6">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra3.html#T4" target="_self" title="INTEGRA3:th.4">Th4</a></span>;<br/>
<a NAME="E5:34_1"/><span class="lab"><font color="Green" title="E40">A9</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</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> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">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="#E1:34"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E3:34_1"><span class="lab"><font color="Green" title="E38">A7</font></span></a>, <a class="ref" href="integra1.html#T46" title="INTEGRA1:th.46">INTEGRA1:46</a>, <a class="ref" href="xreal_1.html#T48" title="XREAL_1:th.48">XREAL_1:48</a></span>;<br/>
<a NAME="E6:34_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</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> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">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="#E1:34"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E2:34_1"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="ref" href="integra1.html#T46" title="INTEGRA1:th.46">INTEGRA1:46</a>, <a class="ref" href="xreal_1.html#T48" title="XREAL_1:th.48">XREAL_1:48</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:34_1"/>
 ex <font color="Olive" title="b1">D2</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Maroon" title="c4">D</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b1">D2</font> &amp; <font color="Maroon" title="c5">D1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b1">D2</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b1">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font></span>)</span> &amp;  <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"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">D2</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> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D</font>)</span>)</span> &amp;  <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"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">D2</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> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_1"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E3:34_1"><span class="lab"><font color="Green" title="E38">A7</font></span></a>, <a class="txt" href="#E4:34_1"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E5:34_1"><span class="lab"><font color="Green" title="E40">A9</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="E6:34"/><span class="lab"><font color="Green" title="E37">A10</font></span>: 
 for <font color="Olive" title="b1">D</font>, <font color="Olive" title="b2">D1</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font>  st  <a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Olive" title="b2">D1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">min</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="Olive" title="b1">D</font>)</span>)</span></span>)</span> holds <br/> ex <font color="Olive" title="b3">D2</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b1">D</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b3">D2</font> &amp; <font color="Olive" title="b2">D1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b3">D2</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b3">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b2">D1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b1">D</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b3">D2</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> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">D1</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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">D</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</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="b2">D1</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">D</font>, <font color="Maroon" title="c5">D1</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"> (  <a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c5">D1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">min</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="c4">D</font>)</span>)</span></span>)</span> implies  ex <font color="Olive" title="b1">D2</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Maroon" title="c4">D</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b1">D2</font> &amp; <font color="Maroon" title="c5">D1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b1">D2</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b1">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">D2</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> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</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="c5">D1</font></span>)</span> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:34_2"/><span class="lab"><font color="Green" title="E38">A11</font></span>: 
 <a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c5">D1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">min</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="c4">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">  ex <font color="Olive" title="b1">D2</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Maroon" title="c4">D</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b1">D2</font> &amp; <font color="Maroon" title="c5">D1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b1">D2</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b1">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">D2</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> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</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="c5">D1</font></span>)</span> )</span><br/>

<a NAME="E2:34_2"/>
 ex <font color="Olive" title="b1">D2</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Maroon" title="c4">D</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b1">D2</font> &amp; <font color="Maroon" title="c5">D1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b1">D2</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b1">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">D2</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> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</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="c5">D1</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c6">D2</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E2:34_2_1"/><span class="lab"><font color="Green" title="E39">A12</font></span>: 
<font color="Maroon" title="c4">D</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Maroon" title="c6">D2</font>
 <span class="kw">and </span><br/><a NAME="E3:34_2_1"/><span class="lab"><font color="Green" title="E40">A13</font></span>: 
<font color="Maroon" title="c5">D1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Maroon" title="c6">D2</font>
 <span class="kw">and </span><br/><a NAME="E4:34_2_1"/><span class="lab"><font color="Green" title="E41">A14</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c6">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:34_2_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"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</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> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:34_2_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"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</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> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34"><span class="lab"><font color="Green" title="E36">A5</font></span></a></span>;<br/>
<a NAME="E7:34_2_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</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> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</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="c5">D1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font>) <span class="kw">-&gt; </span>   <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>  =  <a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,$1);<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</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>  = <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">PartSums</a> <span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,$1)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $2;<br/>
<a NAME="E1:34_2_1_1"/><span class="lab"><font color="Green" title="E42">A15</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</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">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></span>;<br/>
<a NAME="E2:34_2_1_1"/><span class="lab"><font color="Green" title="E43">A16</font></span>: 
 for <font color="Olive" title="b1">i</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">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">D</font> holds <br/> ex <font color="Olive" title="b2">j</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/>( <font color="Olive" title="b2">j</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="c5">D1</font> &amp; <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Olive" title="b2">j</font>) &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Olive" title="b2">j</font>)) <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Olive" title="b2">j</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</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="c5">D1</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( $1 <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">D</font> implies  ex <font color="Olive" title="b1">j</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/>( <font color="Olive" title="b1">j</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="c5">D1</font> &amp; <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Olive" title="b1">j</font>) &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Olive" title="b1">j</font>)) <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Olive" title="b1">j</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">$1 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</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="c5">D1</font></span>)</span> ) );<br/>
<span class="kw">let </span><font color="Maroon" title="c7">i</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="c7">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">D</font> implies  ex <font color="Olive" title="b1">j</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/>( <font color="Olive" title="b1">j</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="c5">D1</font> &amp; <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Olive" title="b1">j</font>) &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Olive" title="b1">j</font>)) <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Olive" title="b1">j</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</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="c5">D1</font></span>)</span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:34_2_1_1_1"/><span class="lab"><font color="Green" title="E44">A17</font></span>: 
<font color="Maroon" title="c7">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">D</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">j</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/>( <font color="Olive" title="b1">j</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="c5">D1</font> &amp; <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Olive" title="b1">j</font>) &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Olive" title="b1">j</font>)) <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Olive" title="b1">j</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</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="c5">D1</font></span>)</span> )</span><br/>

<a NAME="E2:34_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A18</font></span>: 
<font color="Maroon" title="c7">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">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="E3:34_2_1_1_1"/><span class="lab"><font color="Green" title="E46">A19</font></span>: 
 for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</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">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">D</font> &amp; <font color="Olive" title="b2">j</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="c5">D1</font> &amp; <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Olive" title="b2">j</font>) holds <br/><font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">i</font>, <font color="Maroon" title="c9">j</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">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">D</font> &amp; <font color="Maroon" title="c9">j</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="c5">D1</font> &amp; <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font>) implies <font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 )</span><br/>



<span class="kw">assume </span><a NAME="E1:34_2_1_1_1_1"/><span class="lab"><font color="Green" title="E47">A20</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="c4">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">j</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="c5">D1</font> or  not <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font>) or <font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:34_2_1_1_1_1"/><span class="lab"><font color="Green" title="E48">A21</font></span>: 
<font color="Maroon" title="c9">j</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="c5">D1</font>
 <span class="kw">and </span><br/><a NAME="E3:34_2_1_1_1_1"/><span class="lab"><font color="Green" title="E49">A22</font></span>: 
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</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="c9">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2</span><br/>

<span class="kw">assume </span><a NAME="E4:34_2_1_1_1_1"/>
<font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E5:34_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
;<br/>
<a NAME="E6:34_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A23</font></span>: 
<font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="E7:34_2_1_1_1_1"/>
<font color="Maroon" title="c9">j</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="c5">D1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_2_1_1_1_1"><span class="lab"><font color="Green" title="E48">A21</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="E8:34_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E9:34_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:34_2_1_1_1_1"><span class="lab"><font color="Green" title="E50">A23</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E10:34_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A24</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_2_1_1_1_1"><span class="lab"><font color="Green" title="E48">A21</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/>
<a NAME="E11:34_2_1_1_1_1"/><span class="lab"><font color="Green" title="E52">A25</font></span>: 
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</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> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1_1_1_1"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/>
<a NAME="E12:34_2_1_1_1_1"/>
 <a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c5">D1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">min</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="c4">D</font>)</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_1_1_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="E1:34_2_1_1_1_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="34_2_1_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E53">A26</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">  <a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c5">D1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">min</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="c4">D</font>)</span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:34_2_1_1_1_1_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</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">D</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E3:34_2_1_1_1_1_1_1_1"/><span class="kw">then </span>
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="c4">D</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E4:34_2_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A27</font></span>: 
1 <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">D</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E5:34_2_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A28</font></span>: 
1 <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">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="E6:34_2_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A29</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c4">D</font>,1)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E7:34_2_1_1_1_1_1_1_1"/>
1 <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="c4">D</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E54">A27</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/><a NAME="E8:34_2_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A30</font></span>: 
1 <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="c4">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="E9:34_2_1_1_1_1_1_1_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="c4">D</font>,1)</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="c4">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E55">A28</font></span></a>, <a class="ref" href="integra1.html#T20" title="INTEGRA1:th.20">INTEGRA1:20</a></span>;<br/><a NAME="E10:34_2_1_1_1_1_1_1_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="c4">D</font>,1)</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="c4">D</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:34_2_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E57">A30</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="E11:34_2_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A31</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="c4">D</font>,1)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">min</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="c4">D</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D7" title="XXREAL_2:def.7">XXREAL_2:def 7</a></span>;<br/><a NAME="E12:34_2_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E59">A32</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="c4">D</font>,1)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E55">A28</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E13:34_2_1_1_1_1_1_1_1"/>
<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="c5">D1</font>,<font color="Maroon" title="c9">j</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_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#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="#E11:34_2_1_1_1_1"><span class="lab"><font color="Green" title="E52">A25</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E14:34_2_1_1_1_1_1_1_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="c5">D1</font>,<font color="Maroon" title="c9">j</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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="c4">D</font>,1)</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_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="c4">D</font>,1)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:34_2_1_1_1_1"><span class="lab"><font color="Green" title="E51">A24</font></span></a>, <a class="txt" href="#E6:34_2_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E56">A29</font></span></a>, <a class="txt" href="#E12:34_2_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E59">A32</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><a NAME="E15:34_2_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A33</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="c5">D1</font>,<font color="Maroon" title="c9">j</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <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="c4">D</font>,1)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><a NAME="E16:34_2_1_1_1_1_1_1_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="c5">D1</font>,<font color="Maroon" title="c9">j</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="c5">D1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_2_1_1_1_1"><span class="lab"><font color="Green" title="E48">A21</font></span></a>, <a class="txt" href="integra3.html#E16"><span class="lab"><font color="Green" title="E14">Lm5</font></span></a></span>;<br/><a NAME="E17:34_2_1_1_1_1_1_1_1"/><span class="kw">then </span>
 <a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c5">D1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <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="c4">D</font>,1)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:34_2_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E60">A33</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E18:34_2_1_1_1_1_1_1_1"/>
 <a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c5">D1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">min</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="c4">D</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:34_2_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E58">A31</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E53">A34</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">  <a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c5">D1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">min</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="c4">D</font>)</span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:34_2_1_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">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#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <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="#E1:34_2_1_1_1_1"><span class="lab"><font color="Green" title="E47">A20</font></span></a>, <a class="ref" href="integra1.html#T7" title="INTEGRA1:th.7">INTEGRA1:7</a></span>;<br/><a NAME="E3:34_2_1_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A35</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> <font color="Maroon" title="c4">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#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 <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="E4:34_2_1_1_1_1_1_1_2"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c4">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="c4">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#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2_1_1_1_1"><span class="lab"><font color="Green" title="E47">A20</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E53">A34</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E5:34_2_1_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A36</font></span>: 
<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="c4">D</font>,<font color="Maroon" title="c8">i</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_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#NR2" title="XXREAL_0:NR.2">&gt;=</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="c4">D</font>,<font color="Maroon" title="c8">i</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_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="c4">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="#E3:34_2_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E54">A35</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E6:34_2_1_1_1_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="c4">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="c4">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="#E1:34_2_1_1_1_1"><span class="lab"><font color="Green" title="E47">A20</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E53">A34</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E7:34_2_1_1_1_1_1_1_2"/><span class="kw">then </span>
<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="c5">D1</font>,<font color="Maroon" title="c9">j</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_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="c5">D1</font>,<font color="Maroon" title="c9">j</font>)</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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="c4">D</font>,<font color="Maroon" title="c8">i</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_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="#E11:34_2_1_1_1_1"><span class="lab"><font color="Green" title="E52">A25</font></span></a>, <a class="txt" href="#E10:34_2_1_1_1_1"><span class="lab"><font color="Green" title="E51">A24</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E8:34_2_1_1_1_1_1_1_2"/><span class="kw">then </span>
<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="c5">D1</font>,<font color="Maroon" title="c9">j</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_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="c5">D1</font>,<font color="Maroon" title="c9">j</font>)</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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="c4">D</font>,<font color="Maroon" title="c8">i</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_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="c4">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="#E5:34_2_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E55">A36</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E9:34_2_1_1_1_1_1_1_2"/><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="c5">D1</font>,<font color="Maroon" title="c9">j</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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="c4">D</font>,<font color="Maroon" title="c8">i</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_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="c4">D</font>,<font color="Maroon" title="c8">i</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><a NAME="E10:34_2_1_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A37</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="c5">D1</font>,<font color="Maroon" title="c9">j</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <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="c4">D</font>,<font color="Maroon" title="c8">i</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><a NAME="E11:34_2_1_1_1_1_1_1_2"/>
<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">D</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2_1_1_1_1"><span class="lab"><font color="Green" title="E47">A20</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="E12:34_2_1_1_1_1_1_1_2"/><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> <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="c4">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="E13:34_2_1_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A38</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> <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="c4">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="E14:34_2_1_1_1_1_1_1_2"/>
 <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="c4">D</font>,<font color="Maroon" title="c8">i</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="c4">D</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="#E1:34_2_1_1_1_1"><span class="lab"><font color="Green" title="E47">A20</font></span></a>, <a class="ref" href="integra1.html#T20" title="INTEGRA1:th.20">INTEGRA1:20</a></span>;<br/><a NAME="E15:34_2_1_1_1_1_1_1_2"/><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="c4">D</font>,<font color="Maroon" title="c8">i</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="c4">D</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:34_2_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E57">A38</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="E16:34_2_1_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A39</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="c4">D</font>,<font color="Maroon" title="c8">i</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">min</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="c4">D</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D7" title="XXREAL_2:def.7">XXREAL_2:def 7</a></span>;<br/><a NAME="E17:34_2_1_1_1_1_1_1_2"/>
 <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="c5">D1</font>,<font color="Maroon" title="c9">j</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="c5">D1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_2_1_1_1_1"><span class="lab"><font color="Green" title="E48">A21</font></span></a>, <a class="txt" href="integra3.html#E16"><span class="lab"><font color="Green" title="E14">Lm5</font></span></a></span>;<br/><a NAME="E18:34_2_1_1_1_1_1_1_2"/><span class="kw">then </span>
 <a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c5">D1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <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="c4">D</font>,<font color="Maroon" title="c8">i</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:34_2_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E56">A37</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E19:34_2_1_1_1_1_1_1_2"/>
 <a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c5">D1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">min</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="c4">D</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:34_2_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E58">A39</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E13:34_2_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2"><span class="lab"><font color="Green" title="E38">A11</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="E4:34_2_1_1_1"/><span class="lab"><font color="Green" title="E47">A40</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:34_2_1_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</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">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E2:34_2_1_1_1_2"/><span class="kw">then </span>
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="c4">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E3:34_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A41</font></span>: 
1 <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">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">j</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="E5:34_2_1_1_1_2"/><span class="lab"><font color="Green" title="E49">A42</font></span>: 
<font color="Maroon" title="c8">j</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="c5">D1</font>
 <span class="kw">and </span><br/><a NAME="E6:34_2_1_1_1_2"/><span class="lab"><font color="Green" title="E50">A43</font></span>: 
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra3.html#T3" target="_self" title="INTEGRA3:th.3">Th3</a>, <a class="ref" href="integra1.html#T6" title="INTEGRA1:th.6">INTEGRA1:6</a></span>;<br/>
<a NAME="E7:34_2_1_1_1_2"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)) <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</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="c5">D1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:34_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E51">A44</font></span>: 
<font color="Maroon" title="c8">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1_1_1"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E48">A41</font></span></a>, <a class="txt" href="#E5:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="txt" href="#E6:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E50">A43</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">j1</font> = <font color="Maroon" title="c8">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="#E5:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="ref" href="integra1.html#T7" title="INTEGRA1:th.7">INTEGRA1:7</a></span>;<br/>
<a NAME="E3:34_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E52">A45</font></span>: 
<font color="Maroon" title="c9">j1</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="c5">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E51">A44</font></span></a>, <a class="ref" href="integra1.html#T7" title="INTEGRA1:th.7">INTEGRA1:7</a></span>;<br/>
<a NAME="E4:34_2_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">j1</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="c5">D1</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="E5:34_2_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">j1</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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/>
<a NAME="E6:34_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A46</font></span>: 
<font color="Maroon" title="c9">j1</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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</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="E7:34_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E54">A47</font></span>: 
<font color="Maroon" title="c8">j</font> <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="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E51">A44</font></span></a>, <a class="ref" href="integra1.html#T7" title="INTEGRA1:th.7">INTEGRA1:7</a></span>;<br/>
<a NAME="E8:34_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A48</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</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">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E9:34_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A49</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</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="c6">D2</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="E10:34_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A50</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E11:34_2_1_1_1_2_1"/><span class="kw">then </span>
 <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c6">D2</font>,1,<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span>) is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E55">A48</font></span></a>, <a class="ref" href="integra1.html#T35" title="INTEGRA1:th.35">INTEGRA1:35</a></span>;<br/>
<a NAME="E12:34_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A51</font></span>: 
<font color="Maroon" title="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E57">A50</font></span></a>, <a class="ref" href="finseq_6.html#T116" title="FINSEQ_6:th.116">FINSEQ_6:116</a></span>;<br/>
<a NAME="E13:34_2_1_1_1_2_1"/>
<font color="Maroon" title="c8">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">j</font> <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="E14:34_2_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">j</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="E15:34_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A52</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E5:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E52">A45</font></span></a>, <a class="ref" href="integra3.html#T8" target="_self" title="INTEGRA3:th.8">Th8</a></span>;<br/>
<a NAME="E16:34_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A53</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</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>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</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="E17:34_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E61">A54</font></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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p3">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</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"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p3">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</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"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></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="c5">D1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:34_2_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E62">A55</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c10">ID1</font> = <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">ID2</font> = <font color="Maroon" title="c10">ID1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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>  ;<br/>
<span class="kw">assume </span><a NAME="E3:34_2_1_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:34_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A56</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</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="E5:34_2_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E64">A57</font></span>: 
<font color="Maroon" title="c10">ID1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">ID2</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="E6:34_2_1_1_1_2_1_1_1"/><span class="kw">then </span>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">ID2</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="E7:34_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A58</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">ID2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E57">A50</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E8:34_2_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E66">A59</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</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">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E5:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E9:34_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A60</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</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">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E10:34_2_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">ID2</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">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E63">A56</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E11:34_2_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">ID2</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="c6">D2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E65">A58</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E12:34_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A61</font></span>: 
<font color="Maroon" title="c11">ID2</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">D2</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="E13:34_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A62</font></span>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">ID2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E63">A56</font></span></a>, <a class="txt" href="#E8:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E66">A59</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/>
<a NAME="E14:34_2_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E70">A63</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">ID1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E57">A50</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E15:34_2_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E71">A64</font></span>: 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E5:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E16:34_2_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c10">ID1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E63">A56</font></span></a>, <a class="txt" href="#E5:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E64">A57</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E17:34_2_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">ID1</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">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E67">A60</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E18:34_2_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">ID1</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="c6">D2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E70">A63</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E19:34_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A65</font></span>: 
<font color="Maroon" title="c10">ID1</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">D2</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="E20:34_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A66</font></span>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">ID1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">ID2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E64">A57</font></span></a>, <a class="txt" href="#E12:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E68">A61</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/>
<a NAME="E21:34_2_1_1_1_2_1_1_1"/>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">ID1</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="E22:34_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A67</font></span>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">ID1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E55">A48</font></span></a>, <a class="txt" href="#E19:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A65</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/>
<a NAME="E23:34_2_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E75">A68</font></span>: 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E52">A45</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E24:34_2_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E76">A69</font></span>: 
(  not <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">ID1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font> &amp;  not <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">ID2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_2_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:34_2_1_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E77">A70</font></span>: 
( <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">ID1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font> or <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">ID2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_1_1_2_1_1_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="E1:34_2_1_1_1_2_1_1_1_1_1"/>
( <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">ID1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font> or <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">ID2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E77">A70</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_1_1_2_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">ID1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">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="E3:34_2_1_1_1_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E78">A71</font></span>: 
<font color="Maroon" title="c12">n</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="c5">D1</font>
 <span class="kw">and </span><br/><a NAME="E4:34_2_1_1_1_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E79">A72</font></span>: 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">ID1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T3" title="PARTFUN1:th.3">PARTFUN1:3</a></span>;<br/><a NAME="E5:34_2_1_1_1_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E52">A45</font></span></a>, <a class="txt" href="#E22:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A67</font></span></a>, <a class="txt" href="#E23:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E75">A68</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E78">A71</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E79">A72</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/><a NAME="E6:34_2_1_1_1_2_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A73</font></span>: 
<font color="Maroon" title="c8">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">n</font> <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="xreal_1.html#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/><a NAME="E7:34_2_1_1_1_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">ID1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E73">A66</font></span></a>, <a class="txt" href="#E13:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E69">A62</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E8:34_2_1_1_1_2_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A64</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E78">A71</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E79">A72</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/><span class="kw">hence </span><a NAME="E9:34_2_1_1_1_2_1_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:34_2_1_1_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E80">A73</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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="34_2_1_1_1_2_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_2_1_1_1_1_1_2"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">ID2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">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="E3:34_2_1_1_1_2_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E78">A74</font></span>: 
<font color="Maroon" title="c12">n</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="c5">D1</font>
 <span class="kw">and </span><br/><a NAME="E4:34_2_1_1_1_2_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E79">A75</font></span>: 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">ID2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T3" title="PARTFUN1:th.3">PARTFUN1:3</a></span>;<br/><a NAME="E5:34_2_1_1_1_2_1_1_1_1_1_2"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">ID2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A67</font></span></a>, <a class="txt" href="#E20:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E73">A66</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E6:34_2_1_1_1_2_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E52">A45</font></span></a>, <a class="txt" href="#E23:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E75">A68</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E78">A74</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E79">A75</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/><a NAME="E7:34_2_1_1_1_2_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A76</font></span>: 
<font color="Maroon" title="c8">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">n</font> <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="xreal_1.html#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/><a NAME="E8:34_2_1_1_1_2_1_1_1_1_1_2"/>
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="txt" href="#E13:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E69">A62</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A64</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E78">A74</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E79">A75</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/><span class="kw">hence </span><a NAME="E9:34_2_1_1_1_2_1_1_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:34_2_1_1_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E80">A76</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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 NAME="E25:34_2_1_1_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="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E51">A44</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/>
<a NAME="E26:34_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A77</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="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E5:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E27:34_2_1_1_1_2_1_1_1"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E51">A44</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/>
<a NAME="E28:34_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A78</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E52">A45</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E29:34_2_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">ID2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1"><span class="lab"><font color="Green" title="E41">A14</font></span></a>, <a class="txt" href="#E12:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E68">A61</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="E30:34_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A79</font></span>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">ID2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E76">A69</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E31:34_2_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">ID1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1"><span class="lab"><font color="Green" title="E41">A14</font></span></a>, <a class="txt" href="#E19:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A65</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="E32:34_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A80</font></span>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">ID1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E76">A69</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E33:34_2_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">ID2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A67</font></span></a>, <a class="txt" href="#E20:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E73">A66</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E34:34_2_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">ID2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E69">A62</font></span></a>, <a class="txt" href="#E28:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E78">A78</font></span></a>, <a class="txt" href="#E26:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E77">A77</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/>
<a NAME="E35:34_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A81</font></span>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">ID2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</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="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E79">A79</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E36:34_2_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">ID1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E73">A66</font></span></a>, <a class="txt" href="#E13:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E69">A62</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E37:34_2_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">ID1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A67</font></span></a>, <a class="txt" href="#E28:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E78">A78</font></span></a>, <a class="txt" href="#E26:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E77">A77</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/>
<a NAME="E38:34_2_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">ID1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</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="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E80">A80</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E39:34_2_1_1_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="txt" href="#E5:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="txt" href="#E5:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E64">A57</font></span></a>, <a class="txt" href="#E19:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A65</font></span></a>, <a class="txt" href="#E12:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E68">A61</font></span></a>, <a class="txt" href="#E35:34_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E81">A81</font></span></a>, <a class="ref" href="integra3.html#T5" target="_self" title="INTEGRA3:th.5">Th5</a>, <a class="ref" href="seq_4.html#T138" title="SEQ_4:th.138">SEQ_4:138</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="E2:34_2_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E63">A82</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</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="#E10:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E57">A50</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E3:34_2_1_1_1_2_1_1"/>
<font color="Maroon" title="c8">j</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="c5">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E4:34_2_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A83</font></span>: 
<font color="Maroon" title="c8">j</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> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/>
<a NAME="E5:34_2_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E65">A84</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E6:34_2_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A85</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E64">A83</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>;<br/>
<a NAME="E7:34_2_1_1_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c8">j</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="integra3.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/>
<a NAME="E8:34_2_1_1_1_2_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E65">A84</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E64">A83</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>;<br/>
<a NAME="E9:34_2_1_1_1_2_1_1"/><span class="kw">then </span>
 <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</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"><span class="p3">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">j</font></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="#E6:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E66">A85</font></span></a>, <a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>;<br/>
<a NAME="E10:34_2_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A86</font></span>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finsop_1.html#T11" title="FINSOP_1:th.11">FINSOP_1:11</a></span>;<br/>
<a NAME="E11:34_2_1_1_1_2_1_1"/>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</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">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E5:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E12:34_2_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A87</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</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="c6">D2</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="E13:34_2_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A88</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E14:34_2_1_1_1_2_1_1"/>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E68">A87</font></span></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/>
<a NAME="E15:34_2_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A89</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</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> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E16:34_2_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A90</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</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>  <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> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E60">A53</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E17:34_2_1_1_1_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E63">A82</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E18:34_2_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A91</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="c6">D2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/>
<a NAME="E19:34_2_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A92</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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">D2</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="E20:34_2_1_1_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E60">A53</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/>
<a NAME="E21:34_2_1_1_1_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</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> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E62">A55</font></span></a></span>;<br/>
<a NAME="E22:34_2_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A93</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E60">A53</font></span></a>, <a class="txt" href="#E13:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E69">A88</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E70">A89</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E63">A82</font></span></a>, <a class="txt" href="#E16:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E71">A90</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>;<br/>
<a NAME="E23:34_2_1_1_1_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <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="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E24:34_2_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A94</font></span>: 
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> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E60">A53</font></span></a>, <a class="txt" href="#E13:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E69">A88</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E70">A89</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E63">A82</font></span></a>, <a class="txt" href="#E16:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E71">A90</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_2_1_1_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_1_1_2_1_1_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="E1:34_2_1_1_1_2_1_1_2_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E75">A94</font></span></a>, <a class="txt" href="#E22:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E74">A93</font></span></a>, <a class="txt" href="integra3.html#E2"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_1_1_2_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_2_1_1_2_1_1"/><span class="lab"><font color="Green" title="E76">A95</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span> <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"> <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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p3">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</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"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p3">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</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"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></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="c5">D1</font></span>)</span></span><br/><div class="add"><a NAME="E2:34_2_1_1_1_2_1_1_2_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="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E51">A44</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E3:34_2_1_1_1_2_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A96</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="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E5:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/><a NAME="E4:34_2_1_1_1_2_1_1_2_1_1"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E51">A44</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E5:34_2_1_1_1_2_1_1_2_1_1"/><span class="kw">then </span>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E52">A45</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/><a NAME="E6:34_2_1_1_1_2_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A97</font></span>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E77">A96</font></span></a>, <a class="ref" href="integra1.html#T4" title="INTEGRA1:th.4">INTEGRA1:4</a></span>;<br/><a NAME="E7:34_2_1_1_1_2_1_1_2_1_1"/><span class="lab"><font color="Green" title="E79">A98</font></span>: 
 <a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c5">D1</font> <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="integra3.html#T9" target="_self" title="INTEGRA3:th.9">Th9</a></span>;<br/><a NAME="E8:34_2_1_1_1_2_1_1_2_1_1"/><span class="lab"><font color="Green" title="E80">A99</font></span>: 
<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> <font color="Maroon" title="c2">f</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_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> <font color="Maroon" title="c2">f</font></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="#E1:34"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="integra3.html#E14"><span class="lab"><font color="Green" title="E12">Lm3</font></span></a>, <a class="ref" href="xreal_1.html#T48" title="XREAL_1:th.48">XREAL_1:48</a></span>;<br/><a NAME="E9:34_2_1_1_1_2_1_1_2_1_1"/><span class="lab"><font color="Green" title="E81">A100</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</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">D2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E5:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/><a NAME="E10:34_2_1_1_1_2_1_1_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E60">A53</font></span></a>, <a class="txt" href="#E13:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E69">A88</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E70">A89</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E63">A82</font></span></a>, <a class="txt" href="#E16:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E71">A90</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E76">A95</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>;<br/><a NAME="E11:34_2_1_1_1_2_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A101</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="#E16:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E60">A53</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/><a NAME="E12:34_2_1_1_1_2_1_1_2_1_1"/><span class="kw">then </span>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E56">A49</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E13:34_2_1_1_1_2_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A102</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="c6">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:34_2_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E81">A100</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E14:34_2_1_1_1_2_1_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:34_2_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E82">A101</font></span></a></span>;<br/><a NAME="E15:34_2_1_1_1_2_1_1_2_1_1"/><span class="kw">then </span>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E57">A50</font></span></a>, <a class="txt" href="#E11:34_2_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E82">A101</font></span></a>, <a class="txt" href="#E9:34_2_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E81">A100</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E16:34_2_1_1_1_2_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A103</font></span>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:34_2_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E78">A97</font></span></a>, <a class="txt" href="#E13:34_2_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E83">A102</font></span></a>, <a class="ref" href="integra1.html#T4" title="INTEGRA1:th.4">INTEGRA1:4</a></span>;<br/><a NAME="E17:34_2_1_1_1_2_1_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E69">A88</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E70">A89</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E63">A82</font></span></a>, <a class="txt" href="#E16:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E71">A90</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>;<br/><a NAME="E18:34_2_1_1_1_2_1_1_2_1_1"/><span class="kw">then </span>
 <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>) <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"><span class="p3">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="#E1:34_2_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E76">A95</font></span></a>, <a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>;<br/><span class="kw">then </span> <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span> = 
<span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finsop_1.html#T11" title="FINSOP_1:th.11">FINSOP_1:11</a></span>
<br/>.= 
<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"><font color="Maroon" title="c2">f</font> <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="c6">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</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="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E73">A92</font></span></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="txt" href="#E10:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E67">A86</font></span></a>, <a class="txt" href="#E11:34_2_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E82">A101</font></span></a>, <a class="txt" href="#E16:34_2_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E84">A103</font></span></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>
;<br/><span class="kw">hence </span><a NAME="E20:34_2_1_1_1_2_1_1_2_1_1"/>
<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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p3">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</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"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p3">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</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"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></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="c5">D1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:34_2_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E79">A98</font></span></a>, <a class="txt" href="#E8:34_2_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E80">A99</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_1_1_2_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_2_1_1_2_1_2"/><span class="lab"><font color="Green" title="E76">A104</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <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="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p3">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</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"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p3">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</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"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></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="c5">D1</font></span>)</span></span><br/><div class="add"><a NAME="E2:34_2_1_1_1_2_1_1_2_1_2"/><span class="lab"><font color="Green" title="E77">A105</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E69">A88</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E70">A89</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E63">A82</font></span></a>, <a class="txt" href="#E16:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E71">A90</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>;<br/><a NAME="E3:34_2_1_1_1_2_1_1_2_1_2"/><span class="lab"><font color="Green" title="E78">A106</font></span>: 
2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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> <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="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/><span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2 = 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E60">A53</font></span></a>, <a class="txt" href="#E13:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E69">A88</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E70">A89</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E63">A82</font></span></a>, <a class="txt" href="#E16:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E71">A90</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E76">A104</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="#E3:34_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E78">A106</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>

;<br/><a NAME="E5:34_2_1_1_1_2_1_1_2_1_2"/><span class="kw">then </span>
 <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E76">A104</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E77">A105</font></span></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/><a NAME="E6:34_2_1_1_1_2_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A107</font></span>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span> <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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T77" title="RVSUM_1:th.77">RVSUM_1:77</a></span>;<br/><a NAME="E7:34_2_1_1_1_2_1_1_2_1_2"/><span class="lab"><font color="Green" title="E80">A108</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="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a></span>;<br/><a NAME="E8:34_2_1_1_1_2_1_1_2_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="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E51">A44</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E9:34_2_1_1_1_2_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A109</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="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E5:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/><a NAME="E10:34_2_1_1_1_2_1_1_2_1_2"/><span class="lab"><font color="Green" title="E82">A110</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="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</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="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a></span>;<br/><a NAME="E11:34_2_1_1_1_2_1_1_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E60">A53</font></span></a>, <a class="txt" href="#E13:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E69">A88</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E70">A89</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E63">A82</font></span></a>, <a class="txt" href="#E16:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E71">A90</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E76">A104</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>;<br/><a NAME="E12:34_2_1_1_1_2_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A111</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E60">A53</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/><a NAME="E13:34_2_1_1_1_2_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A112</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <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">D2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E5:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/><a NAME="E14:34_2_1_1_1_2_1_1_2_1_2"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E51">A44</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E15:34_2_1_1_1_2_1_1_2_1_2"/><span class="kw">then </span>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E52">A45</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/><a NAME="E16:34_2_1_1_1_2_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A113</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="c5">D1</font>,<font color="Maroon" title="c8">j</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"><span class="p3">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:34_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E81">A109</font></span></a>, <a class="txt" href="#E12:34_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E83">A111</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><a NAME="E17:34_2_1_1_1_2_1_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E63">A82</font></span></a>, <a class="txt" href="#E16:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E71">A90</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E18:34_2_1_1_1_2_1_1_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="c6">D2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/><a NAME="E19:34_2_1_1_1_2_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A114</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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">D2</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="E20:34_2_1_1_1_2_1_1_2_1_2"/><span class="lab"><font color="Green" title="E87">A115</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E57">A50</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E21:34_2_1_1_1_2_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A116</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="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:34_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E86">A114</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E22:34_2_1_1_1_2_1_1_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 ;<br/><a NAME="E23:34_2_1_1_1_2_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A117</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:34_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E86">A114</font></span></a>, <a class="txt" href="#E20:34_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E87">A115</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E24:34_2_1_1_1_2_1_1_2_1_2"/><span class="lab"><font color="Green" title="E90">A118</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="#E2:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E63">A82</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E25:34_2_1_1_1_2_1_1_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><a NAME="E26:34_2_1_1_1_2_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A119</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:34_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E84">A112</font></span></a>, <a class="txt" href="#E24:34_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E90">A118</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E27:34_2_1_1_1_2_1_1_2_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="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:34_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E84">A112</font></span></a>, <a class="txt" href="#E24:34_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E90">A118</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><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="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:34_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E91">A119</font></span></a>, <a class="txt" href="#E16:34_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E85">A113</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>
<br/>.= 
<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="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>)</span>)</span></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="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="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:34_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E89">A117</font></span></a>, <a class="txt" href="#E21:34_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E88">A116</font></span></a></span>
;<br/><a NAME="E29:34_2_1_1_1_2_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A120</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="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><a NAME="E30:34_2_1_1_1_2_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A121</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">j</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"><font color="Maroon" title="c2">f</font> <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="c5">D1</font>,<font color="Maroon" title="c8">j</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"><span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>)</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/><a NAME="E31:34_2_1_1_1_2_1_1_2_1_2"/><span class="lab"><font color="Green" title="E94">A122</font></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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</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"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</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"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></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"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_2_1_1_2_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c10">ID2</font> = <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">ID1</font> = <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">B</font> =  <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="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">C</font> =  <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="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>)</span>)</span>;<br/>
<a NAME="E1:34_2_1_1_1_2_1_1_2_1_2_3"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</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="#E5:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="ref" href="integra1.html#T8" title="INTEGRA1:th.8">INTEGRA1:8</a></span>;<br/>
<a NAME="E2:34_2_1_1_1_2_1_1_2_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A123</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="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span></span>)</span></span>)</span> <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> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="integra3.html#E15"><span class="lab"><font color="Green" title="E13">Lm4</font></span></a></span>;<br/>
<a NAME="E3:34_2_1_1_1_2_1_1_2_1_2_3"/>
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E72">A91</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="E4:34_2_1_1_1_2_1_1_2_1_2_3"/><span class="kw">then </span>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <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="ref" href="integra1.html#T8" title="INTEGRA1:th.8">INTEGRA1:8</a></span>;<br/>
<a NAME="E5:34_2_1_1_1_2_1_1_2_1_2_3"/><span class="kw">then </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="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span></span>)</span> <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> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="integra3.html#E15"><span class="lab"><font color="Green" title="E13">Lm4</font></span></a></span>;<br/>
<a NAME="E6:34_2_1_1_1_2_1_1_2_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A124</font></span>: 
<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"><font color="Maroon" title="c2">f</font> <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="c6">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</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="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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> <font color="Maroon" title="c2">f</font></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="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:34_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E80">A108</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/>
<a NAME="E7:34_2_1_1_1_2_1_1_2_1_2_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E60">A53</font></span></a>, <a class="txt" href="#E13:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E69">A88</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E70">A89</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E63">A82</font></span></a>, <a class="txt" href="#E16:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E71">A90</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E76">A104</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>;<br/>
<a NAME="E8:34_2_1_1_1_2_1_1_2_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A125</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E60">A53</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/>
<a NAME="E9:34_2_1_1_1_2_1_1_2_1_2_3"/><span class="lab"><font color="Green" title="E98">A126</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</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">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E5:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E10:34_2_1_1_1_2_1_1_2_1_2_3"/><span class="kw">then </span>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>) <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="#E8:34_2_1_1_1_2_1_1_2_1_2_3"><span class="lab"><font color="Green" title="E97">A125</font></span></a>, <a class="ref" href="integra1.html#T8" title="INTEGRA1:th.8">INTEGRA1:8</a></span>;<br/>
<a NAME="E11:34_2_1_1_1_2_1_1_2_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A127</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="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>)</span>)</span></span>)</span></span>)</span> <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> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="integra3.html#E15"><span class="lab"><font color="Green" title="E13">Lm4</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">A</font> =  <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="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span></span>)</span></span>)</span> as   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  ;<br/>
<a NAME="E13:34_2_1_1_1_2_1_1_2_1_2_3"/><span class="lab"><font color="Green" title="E100">A128</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">A</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">A</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</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="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:34_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E93">A121</font></span></a></span>;<br/>
<a NAME="E14:34_2_1_1_1_2_1_1_2_1_2_3"/>
<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"><font color="Maroon" title="c2">f</font> <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="c5">D1</font>,<font color="Maroon" title="c8">j</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="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>)</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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> <font color="Maroon" title="c2">f</font></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="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:34_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E82">A110</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_2_1_1_2_1_2_3"><span class="lab"><font color="Green" title="E95">A123</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/>
<a NAME="E15:34_2_1_1_1_2_1_1_2_1_2_3"/><span class="kw">then </span>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span></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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>)</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E67">A86</font></span></a>, <a class="txt" href="#E13:34_2_1_1_1_2_1_1_2_1_2_3"><span class="lab"><font color="Green" title="E100">A128</font></span></a>, <a class="ref" href="xreal_1.html#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/>
<a NAME="E16:34_2_1_1_1_2_1_1_2_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A129</font></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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</font>)</span>)</span></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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>)</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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"><font color="Maroon" title="c2">f</font> <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="c5">D1</font>,<font color="Maroon" title="c8">j</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="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span>
 
<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="E17:34_2_1_1_1_2_1_1_2_1_2_3"/>
<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"><font color="Maroon" title="c2">f</font> <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="c5">D1</font>,<font color="Maroon" title="c8">j</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="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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> <font color="Maroon" title="c2">f</font></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="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:34_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E80">A108</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_2_1_1_2_1_2_3"><span class="lab"><font color="Green" title="E95">A123</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/>
<a NAME="E18:34_2_1_1_1_2_1_1_2_1_2_3"/><span class="kw">then </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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</font>)</span>)</span></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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>)</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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> <font color="Maroon" title="c2">f</font></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="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:34_2_1_1_1_2_1_1_2_1_2_3"><span class="lab"><font color="Green" title="E101">A129</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E19:34_2_1_1_1_2_1_1_2_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A130</font></span>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>)</span>)</span></span>)</span></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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span></span>)</span>
 
<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 href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span> = 
<span class="p1">(<span class="default"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>)</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:34_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E79">A107</font></span></a>, <a class="txt" href="#E9:34_2_1_1_1_2_1_1_2_1_2_3"><span class="lab"><font color="Green" title="E98">A126</font></span></a>, <a class="txt" href="#E8:34_2_1_1_1_2_1_1_2_1_2_3"><span class="lab"><font color="Green" title="E97">A125</font></span></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>)</span>)</span></span>)</span></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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:34_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E73">A92</font></span></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>
;<br/>
<a NAME="E21:34_2_1_1_1_2_1_1_2_1_2_3"/><span class="kw">then </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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</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"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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> <font color="Maroon" title="c2">f</font></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="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:34_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E82">A110</font></span></a>, <a class="txt" href="#E11:34_2_1_1_1_2_1_1_2_1_2_3"><span class="lab"><font color="Green" title="E99">A127</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/>
<a NAME="E22:34_2_1_1_1_2_1_1_2_1_2_3"/><span class="kw">then </span>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</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"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>)</span>)</span></span>)</span></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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span></span>)</span>
 
<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="E23:34_2_1_1_1_2_1_1_2_1_2_3"/><span class="kw">then </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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</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"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>)</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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"><font color="Maroon" title="c2">f</font> <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="c6">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</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="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span>
 
<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="E24:34_2_1_1_1_2_1_1_2_1_2_3"/><span class="kw">then </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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</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"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>)</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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> <font color="Maroon" title="c2">f</font></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="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:34_2_1_1_1_2_1_1_2_1_2_3"><span class="lab"><font color="Green" title="E96">A124</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E25:34_2_1_1_1_2_1_1_2_1_2_3"/><span class="kw">then </span>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</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"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>)</span>)</span></span>)</span></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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span></span>)</span>
 
<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="E26:34_2_1_1_1_2_1_1_2_1_2_3"/><span class="kw">then </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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</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"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</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"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>)</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span></span>)</span></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"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>)</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:34_2_1_1_1_2_1_1_2_1_2_3"><span class="lab"><font color="Green" title="E102">A130</font></span></a>, <a class="ref" href="xreal_1.html#T13" title="XREAL_1:th.13">XREAL_1:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E27:34_2_1_1_1_2_1_1_2_1_2_3"/>
<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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</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"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</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"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></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"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></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"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E32:34_2_1_1_1_2_1_1_2_1_2"/>
<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> <font color="Maroon" title="c2">f</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_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> <font color="Maroon" title="c2">f</font></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="#E1:34"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="integra3.html#E14"><span class="lab"><font color="Green" title="E12">Lm3</font></span></a>, <a class="ref" href="xreal_1.html#T48" title="XREAL_1:th.48">XREAL_1:48</a></span>;<br/><a NAME="E33:34_2_1_1_1_2_1_1_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></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="c5">D1</font>,<font color="Maroon" title="c8">j</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"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></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="c5">D1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="txt" href="integra3.html#E16"><span class="lab"><font color="Green" title="E14">Lm5</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/><span class="kw">hence </span><a NAME="E34:34_2_1_1_1_2_1_1_2_1_2"/>
<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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p3">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</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"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p3">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</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"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></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="c5">D1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:34_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E92">A120</font></span></a>, <a class="txt" href="#E31:34_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E94">A122</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E26:34_2_1_1_1_2_1_1"/>
<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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p3">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</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"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p3">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</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"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></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="c5">D1</font></span>)</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="E18:34_2_1_1_1_2_1"/>
<font color="Maroon" title="c8">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">j</font> <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="E19:34_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A131</font></span>: 
<font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">j</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="E20:34_2_1_1_1_2_1"/>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</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">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E5:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E21:34_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A132</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</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="c6">D2</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="E22:34_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A133</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E23:34_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E65">A134</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</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">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E56">A49</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E24:34_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A135</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="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>;<br/>
<a NAME="E25:34_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E67">A136</font></span>: 
<font color="Maroon" title="c9">j1</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="c5">D1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E54">A47</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="E26:34_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A137</font></span>: 
<font color="Maroon" title="c9">j1</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="c5">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E27:34_2_1_1_1_2_1"/>
 for <font color="Olive" title="b1">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</font> <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"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> holds <br/><font color="Olive" title="b1">x1</font> <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"><font color="Maroon" title="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_2_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">x1</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="c10">x1</font> <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"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> implies <font color="Maroon" title="c10">x1</font> <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"><font color="Maroon" title="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:34_2_1_1_1_2_1_2"/>
<font color="Maroon" title="c10">x1</font> <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"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span>
 ; <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="c10">x1</font> <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"><font color="Maroon" title="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">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> <span class="kw"> such that </span><br/><a NAME="E3:34_2_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E69">A138</font></span>: 
<font color="Maroon" title="c11">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="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:34_2_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E70">A139</font></span>: 
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T3" title="PARTFUN1:th.3">PARTFUN1:3</a></span>;<br/>
<a NAME="E5:34_2_1_1_1_2_1_2"/>
<font color="Maroon" title="c11">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="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E69">A138</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="E6:34_2_1_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A140</font></span>: 
<font color="Maroon" title="c11">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> <font color="Maroon" title="c9">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E68">A137</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>;<br/>
<a NAME="E7:34_2_1_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A141</font></span>: 
<font color="Maroon" title="c11">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="c5">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E52">A45</font></span></a>, <a class="ref" href="rfinseq.html#T6" title="RFINSEQ:th.6">RFINSEQ:6</a></span>;<br/>
<a NAME="E8:34_2_1_1_1_2_1_2"/>
<font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:34_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E71">A140</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E9:34_2_1_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E54">A47</font></span></a>, <a class="txt" href="#E7:34_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E72">A141</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/>
<a NAME="E10:34_2_1_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E7:34_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E72">A141</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E11:34_2_1_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A142</font></span>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E7:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E54">A47</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E12:34_2_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E74">A143</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E52">A45</font></span></a>, <a class="txt" href="#E6:34_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E71">A140</font></span></a>, <a class="ref" href="rfinseq.html#T6" title="RFINSEQ:th.6">RFINSEQ:6</a></span>;<br/>
<a NAME="E13:34_2_1_1_1_2_1_2"/>
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:34_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E72">A141</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="E14:34_2_1_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c6">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1"><span class="lab"><font color="Green" title="E41">A14</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E70">A139</font></span></a>, <a class="txt" href="#E12:34_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E74">A143</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">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="E16:34_2_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E75">A144</font></span>: 
<font color="Maroon" title="c12">n</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">D2</font>
 <span class="kw">and </span><br/><a NAME="E17:34_2_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E76">A145</font></span>: 
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T3" title="PARTFUN1:th.3">PARTFUN1:3</a></span>;<br/>
<a NAME="E18:34_2_1_1_1_2_1_2"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E70">A139</font></span></a>, <a class="txt" href="#E12:34_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E74">A143</font></span></a>, <a class="txt" href="#E7:34_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E72">A141</font></span></a>, <a class="txt" href="#E17:34_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E76">A145</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E19:34_2_1_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A146</font></span>: 
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E55">A48</font></span></a>, <a class="txt" href="#E16:34_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E75">A144</font></span></a>, <a class="txt" href="#E11:34_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E73">A142</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/>
<a NAME="E20:34_2_1_1_1_2_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:34_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E75">A144</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E21:34_2_1_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A147</font></span>: 
<font color="Maroon" title="c12">n</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="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:34_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E77">A146</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E22:34_2_1_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">n</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="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E65">A134</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>;<br/>
<a NAME="E23:34_2_1_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A148</font></span>: 
<font color="Maroon" title="c12">n</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="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</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="E24:34_2_1_1_1_2_1_2"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E55">A48</font></span></a>, <a class="txt" href="#E21:34_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E78">A147</font></span></a>, <a class="ref" href="rfinseq.html#T6" title="RFINSEQ:th.6">RFINSEQ:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E25:34_2_1_1_1_2_1_2"/>
<font color="Maroon" title="c10">x1</font> <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"><font color="Maroon" title="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:34_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E76">A145</font></span></a>, <a class="txt" href="#E23:34_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E79">A148</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</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="E28:34_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A149</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="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E29:34_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E70">A150</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E67">A136</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E30:34_2_1_1_1_2_1"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E50">A43</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/>
<a NAME="E31:34_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A151</font></span>: 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E51">A44</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/>
<a NAME="E32:34_2_1_1_1_2_1"/>
 for <font color="Olive" title="b1">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</font> <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"><font color="Maroon" title="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span></span>)</span> holds <br/><font color="Olive" title="b1">x1</font> <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"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_2_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">x1</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="c10">x1</font> <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"><font color="Maroon" title="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span></span>)</span> implies <font color="Maroon" title="c10">x1</font> <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"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:34_2_1_1_1_2_1_3"/>
<font color="Maroon" title="c10">x1</font> <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"><font color="Maroon" title="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</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"> <font color="Maroon" title="c10">x1</font> <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"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">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> <span class="kw"> such that </span><br/><a NAME="E3:34_2_1_1_1_2_1_3"/><span class="lab"><font color="Green" title="E72">A152</font></span>: 
<font color="Maroon" title="c11">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="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:34_2_1_1_1_2_1_3"/><span class="lab"><font color="Green" title="E73">A153</font></span>: 
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T3" title="PARTFUN1:th.3">PARTFUN1:3</a></span>;<br/>
<a NAME="E5:34_2_1_1_1_2_1_3"/>
<font color="Maroon" title="c11">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="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1_1_1_2_1_3"><span class="lab"><font color="Green" title="E72">A152</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="E6:34_2_1_1_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A154</font></span>: 
<font color="Maroon" title="c11">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="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E65">A134</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>;<br/>
<a NAME="E7:34_2_1_1_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A155</font></span>: 
<font color="Maroon" title="c11">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="c6">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E55">A48</font></span></a>, <a class="ref" href="rfinseq.html#T6" title="RFINSEQ:th.6">RFINSEQ:6</a></span>;<br/>
<a NAME="E8:34_2_1_1_1_2_1_3"/><span class="lab"><font color="Green" title="E76">A156</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="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E68">A137</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>;<br/>
<a NAME="E9:34_2_1_1_1_2_1_3"/>
<font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:34_2_1_1_1_2_1_3"><span class="lab"><font color="Green" title="E74">A154</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E10:34_2_1_1_1_2_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E55">A48</font></span></a>, <a class="txt" href="#E7:34_2_1_1_1_2_1_3"><span class="lab"><font color="Green" title="E75">A155</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/>
<a NAME="E11:34_2_1_1_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A157</font></span>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E7:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E54">A47</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E12:34_2_1_1_1_2_1_3"/><span class="lab"><font color="Green" title="E78">A158</font></span>: 
( <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font> implies <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <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"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_2_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:34_2_1_1_1_2_1_3_1"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</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">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <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"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">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> <span class="kw"> such that </span><br/><a NAME="E3:34_2_1_1_1_2_1_3_1"/><span class="lab"><font color="Green" title="E79">A159</font></span>: 
<font color="Maroon" title="c12">m</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="c5">D1</font>
 <span class="kw">and </span><br/><a NAME="E4:34_2_1_1_1_2_1_3_1"/><span class="lab"><font color="Green" title="E80">A160</font></span>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T3" title="PARTFUN1:th.3">PARTFUN1:3</a></span>;<br/>
<a NAME="E5:34_2_1_1_1_2_1_3_1"/>
<font color="Maroon" title="c12">m</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="c5">D1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E79">A159</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="E6:34_2_1_1_1_2_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A161</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E7:34_2_1_1_1_2_1_3_1"/><span class="lab"><font color="Green" title="E82">A162</font></span>: 
<font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E52">A45</font></span></a>, <a class="txt" href="#E11:34_2_1_1_1_2_1_3"><span class="lab"><font color="Green" title="E77">A157</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E79">A159</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E80">A160</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/>
<a NAME="E8:34_2_1_1_1_2_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">m</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="c9">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:34_2_1_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E81">A161</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E9:34_2_1_1_1_2_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A163</font></span>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E52">A45</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E80">A160</font></span></a>, <a class="ref" href="rfinseq.html#T6" title="RFINSEQ:th.6">RFINSEQ:6</a></span>;<br/>
<a NAME="E10:34_2_1_1_1_2_1_3_1"/>
<font color="Maroon" title="c12">m</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="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:34_2_1_1_1_2_1_3"><span class="lab"><font color="Green" title="E76">A156</font></span></a>, <a class="txt" href="#E6:34_2_1_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E81">A161</font></span></a>, <a class="txt" href="#E7:34_2_1_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E82">A162</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:34_2_1_1_1_2_1_3_1"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <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"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:34_2_1_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E83">A163</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</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="E13:34_2_1_1_1_2_1_3"/><span class="lab"><font color="Green" title="E79">A164</font></span>: 
( <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font> implies <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">j1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_2_1_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:34_2_1_1_1_2_1_3_2"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</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">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">j1</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">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="E3:34_2_1_1_1_2_1_3_2"/><span class="lab"><font color="Green" title="E80">A165</font></span>: 
<font color="Maroon" title="c12">n</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">D</font>
 <span class="kw">and </span><br/><a NAME="E4:34_2_1_1_1_2_1_3_2"/><span class="lab"><font color="Green" title="E81">A166</font></span>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T3" title="PARTFUN1:th.3">PARTFUN1:3</a></span>;<br/>
<a NAME="E5:34_2_1_1_1_2_1_3_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1_1_1_2_1_3_2"><span class="lab"><font color="Green" title="E80">A165</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E6:34_2_1_1_1_2_1_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E48">A41</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_2_1_3_2"><span class="lab"><font color="Green" title="E80">A165</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_2_1_3_2"><span class="lab"><font color="Green" title="E81">A166</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/>
<a NAME="E7:34_2_1_1_1_2_1_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E71">A151</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:34_2_1_1_1_2_1_3_2"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:34_2_1_1_1_2_1_3"><span class="lab"><font color="Green" title="E77">A157</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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>
<a NAME="E14:34_2_1_1_1_2_1_3"/><span class="lab"><font color="Green" title="E80">A167</font></span>: 
( <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font> implies <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <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"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_2_1_3_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:34_2_1_1_1_2_1_3_3"/>
<font color="Maroon" title="c9">j1</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="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E70">A150</font></span></a>, <a class="txt" href="#E8:34_2_1_1_1_2_1_3"><span class="lab"><font color="Green" title="E76">A156</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E2:34_2_1_1_1_2_1_3_3"/><span class="kw">then </span>
<font color="Maroon" title="c9">j1</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="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</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="E3:34_2_1_1_1_2_1_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A168</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">j1</font> <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"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:34_2_1_1_1_2_1_3_3"/><span class="lab"><font color="Green" title="E82">A169</font></span>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</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">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <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"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span></span><br/>

<a NAME="E5:34_2_1_1_1_2_1_3_3"/>
<font color="Maroon" title="c9">j1</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="c9">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E70">A150</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:34_2_1_1_1_2_1_3_3"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <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"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E52">A45</font></span></a>, <a class="txt" href="#E13:34_2_1_1_1_2_1_3"><span class="lab"><font color="Green" title="E79">A164</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_2_1_3_3"><span class="lab"><font color="Green" title="E82">A169</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_2_1_3_3"><span class="lab"><font color="Green" title="E81">A168</font></span></a>, <a class="ref" href="rfinseq.html#T6" title="RFINSEQ:th.6">RFINSEQ:6</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="E15:34_2_1_1_1_2_1_3"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c6">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:34_2_1_1_1_2_1_3"><span class="lab"><font color="Green" title="E75">A155</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:34_2_1_1_1_2_1_3"/>
<font color="Maroon" title="c10">x1</font> <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"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1"><span class="lab"><font color="Green" title="E41">A14</font></span></a>, <a class="txt" href="#E8:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E55">A48</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_2_1_3"><span class="lab"><font color="Green" title="E73">A153</font></span></a>, <a class="txt" href="#E6:34_2_1_1_1_2_1_3"><span class="lab"><font color="Green" title="E74">A154</font></span></a>, <a class="txt" href="#E14:34_2_1_1_1_2_1_3"><span class="lab"><font color="Green" title="E80">A167</font></span></a>, <a class="txt" href="#E12:34_2_1_1_1_2_1_3"><span class="lab"><font color="Green" title="E78">A158</font></span></a>, <a class="ref" href="rfinseq.html#T6" title="RFINSEQ:th.6">RFINSEQ:6</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</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="E33:34_2_1_1_1_2_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="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E34:34_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A170</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="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</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"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E69">A149</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E35:34_2_1_1_1_2_1"/>
 <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c5">D1</font>,1,<font color="Maroon" title="c9">j1</font>) is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E51">A44</font></span></a>, <a class="txt" href="#E29:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E70">A150</font></span></a>, <a class="ref" href="integra1.html#T7" title="INTEGRA1:th.7">INTEGRA1:7</a>, <a class="ref" href="integra1.html#T35" title="INTEGRA1:th.35">INTEGRA1:35</a></span>;<br/>
<a NAME="E36:34_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A171</font></span>: 
<font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E70">A150</font></span></a>, <a class="ref" href="finseq_6.html#T116" title="FINSEQ_6:th.116">FINSEQ_6:116</a></span>;<br/>
<a NAME="E37:34_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A172</font></span>: 
<font color="Maroon" title="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E58">A51</font></span></a>, <a class="txt" href="#E34:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E72">A170</font></span></a>, <a class="ref" href="integra3.html#T6" target="_self" title="INTEGRA3:th.6">Th6</a></span>;<br/>
<a NAME="E38:34_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E75">A173</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 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j1</font> holds <br/><font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Olive" title="b1">k</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_2_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">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"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j1</font> implies <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font>) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:34_2_1_1_1_2_1_4"/><span class="lab"><font color="Green" title="E76">A174</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">and </span><br/><a NAME="E2:34_2_1_1_1_2_1_4"/><span class="lab"><font color="Green" title="E77">A175</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j1</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="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font>)</span><br/>

<span class="kw">assume </span><a NAME="E3:34_2_1_1_1_2_1_4"/><span class="lab"><font color="Green" title="E78">A176</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_2_1_4_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_1_1_2_1_4_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="E1:34_2_1_1_1_2_1_4_1_1"/>
( <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font>) or <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font>) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1_1_1_2_1_4"><span class="lab"><font color="Green" title="E78">A176</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="34_2_1_1_1_2_1_4_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_2_1_4_1_1_1"/><span class="lab"><font color="Green" title="E79">A177</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:34_2_1_1_1_2_1_4_1_1_1"/>
<font color="Maroon" title="c10">k</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="c5">D1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E68">A137</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_2_1_4"><span class="lab"><font color="Green" title="E77">A175</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E3:34_2_1_1_1_2_1_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A178</font></span>: 
<font color="Maroon" title="c10">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="c5">D1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2_1_1_1_2_1_4"><span class="lab"><font color="Green" title="E76">A174</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E4:34_2_1_1_1_2_1_4_1_1_1"/><span class="kw">then </span>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">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="c6">D2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/><a NAME="E5:34_2_1_1_1_2_1_4_1_1_1"/><span class="kw">then </span>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">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="c6">D2</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="E6:34_2_1_1_1_2_1_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A179</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E7:34_2_1_1_1_2_1_4_1_1_1"/><span class="lab"><font color="Green" title="E82">A180</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_2_1_1_1_2_1_4"><span class="lab"><font color="Green" title="E77">A175</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_2_1_4_1_1_1"><span class="lab"><font color="Green" title="E79">A177</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E8:34_2_1_1_1_2_1_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A181</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">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> <font color="Maroon" title="c9">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:34_2_1_1_1_2_1_4_1_1_1"><span class="lab"><font color="Green" title="E81">A179</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E9:34_2_1_1_1_2_1_4_1_1_1"/>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E52">A45</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_2_1_4"><span class="lab"><font color="Green" title="E77">A175</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_2_1_4_1_1_1"><span class="lab"><font color="Green" title="E80">A178</font></span></a>, <a class="ref" href="integra3.html#T7" target="_self" title="INTEGRA3:th.7">Th7</a></span>;<br/><a NAME="E10:34_2_1_1_1_2_1_4_1_1_1"/><span class="kw">then </span>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">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="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:34_2_1_1_1_2_1_4_1_1_1"><span class="lab"><font color="Green" title="E81">A179</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E11:34_2_1_1_1_2_1_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A182</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E55">A48</font></span></a>, <a class="ref" href="rfinseq.html#T6" title="RFINSEQ:th.6">RFINSEQ:6</a></span>;<br/><a NAME="E12:34_2_1_1_1_2_1_4_1_1_1"/>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</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="c5">D1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E68">A137</font></span></a>, <a class="txt" href="#E7:34_2_1_1_1_2_1_4_1_1_1"><span class="lab"><font color="Green" title="E82">A180</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E13:34_2_1_1_1_2_1_4_1_1_1"/><span class="kw">then </span>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">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="c5">D1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:34_2_1_1_1_2_1_4_1_1_1"><span class="lab"><font color="Green" title="E81">A179</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E14:34_2_1_1_1_2_1_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A183</font></span>: 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2_1_1_1_2_1_4_1_1_1"><span class="lab"><font color="Green" title="E79">A177</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_2_1_4_1_1_1"><span class="lab"><font color="Green" title="E80">A178</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/><a NAME="E15:34_2_1_1_1_2_1_4_1_1_1"/>
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_2_1_4_1_1_1"><span class="lab"><font color="Green" title="E80">A178</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/><span class="kw">hence </span><a NAME="E16:34_2_1_1_1_2_1_4_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E52">A45</font></span></a>, <a class="txt" href="#E37:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E74">A172</font></span></a>, <a class="txt" href="#E11:34_2_1_1_1_2_1_4_1_1_1"><span class="lab"><font color="Green" title="E84">A182</font></span></a>, <a class="txt" href="#E14:34_2_1_1_1_2_1_4_1_1_1"><span class="lab"><font color="Green" title="E85">A183</font></span></a>, <a class="txt" href="#E8:34_2_1_1_1_2_1_4_1_1_1"><span class="lab"><font color="Green" title="E83">A181</font></span></a>, <a class="ref" href="rfinseq.html#T6" title="RFINSEQ:th.6">RFINSEQ:6</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="34_2_1_1_1_2_1_4_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_2_1_4_1_1_2"/><span class="lab"><font color="Green" title="E79">A184</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:34_2_1_1_1_2_1_4_1_1_2"/>
<font color="Maroon" title="c10">k</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="c5">D1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E68">A137</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_2_1_4"><span class="lab"><font color="Green" title="E77">A175</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E3:34_2_1_1_1_2_1_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A185</font></span>: 
<font color="Maroon" title="c10">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="c5">D1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2_1_1_1_2_1_4"><span class="lab"><font color="Green" title="E76">A174</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E4:34_2_1_1_1_2_1_4_1_1_2"/><span class="kw">then </span>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E52">A45</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_2_1_4"><span class="lab"><font color="Green" title="E77">A175</font></span></a>, <a class="ref" href="integra3.html#T7" target="_self" title="INTEGRA3:th.7">Th7</a></span>;<br/><a NAME="E5:34_2_1_1_1_2_1_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A186</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2_1_1_1_2_1_4_1_1_2"><span class="lab"><font color="Green" title="E79">A184</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E6:34_2_1_1_1_2_1_4_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">k</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">D2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E65">A134</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E7:34_2_1_1_1_2_1_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A187</font></span>: 
<font color="Maroon" title="c10">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="c6">D2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2_1_1_1_2_1_4"><span class="lab"><font color="Green" title="E76">A174</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E8:34_2_1_1_1_2_1_4_1_1_2"/>
<font color="Maroon" title="c10">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> <font color="Maroon" title="c9">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2_1_1_1_2_1_4"><span class="lab"><font color="Green" title="E76">A174</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_2_1_4"><span class="lab"><font color="Green" title="E77">A175</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E9:34_2_1_1_1_2_1_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A188</font></span>: 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E54">A47</font></span></a>, <a class="ref" href="rfinseq.html#T6" title="RFINSEQ:th.6">RFINSEQ:6</a></span>;<br/><a NAME="E10:34_2_1_1_1_2_1_4_1_1_2"/>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">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="c6">D2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_2_1_4_1_1_2"><span class="lab"><font color="Green" title="E80">A185</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/><a NAME="E11:34_2_1_1_1_2_1_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A189</font></span>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2_1_1_1_2_1_4_1_1_2"><span class="lab"><font color="Green" title="E79">A184</font></span></a>, <a class="txt" href="#E7:34_2_1_1_1_2_1_4_1_1_2"><span class="lab"><font color="Green" title="E82">A187</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/><a NAME="E12:34_2_1_1_1_2_1_4_1_1_2"/><span class="lab"><font color="Green" title="E85">A190</font></span>: 
<font color="Maroon" title="c10">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="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2_1_1_1_2_1_4"><span class="lab"><font color="Green" title="E76">A174</font></span></a>, <a class="txt" href="#E5:34_2_1_1_1_2_1_4_1_1_2"><span class="lab"><font color="Green" title="E81">A186</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E13:34_2_1_1_1_2_1_4_1_1_2"/>
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_2_1_4_1_1_2"><span class="lab"><font color="Green" title="E80">A185</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/><span class="kw">hence </span><a NAME="E14:34_2_1_1_1_2_1_4_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E55">A48</font></span></a>, <a class="txt" href="#E37:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E74">A172</font></span></a>, <a class="txt" href="#E9:34_2_1_1_1_2_1_4_1_1_2"><span class="lab"><font color="Green" title="E83">A188</font></span></a>, <a class="txt" href="#E11:34_2_1_1_1_2_1_4_1_1_2"><span class="lab"><font color="Green" title="E84">A189</font></span></a>, <a class="txt" href="#E12:34_2_1_1_1_2_1_4_1_1_2"><span class="lab"><font color="Green" title="E85">A190</font></span></a>, <a class="ref" href="rfinseq.html#T6" title="RFINSEQ:th.6">RFINSEQ:6</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="E5:34_2_1_1_1_2_1_4"/>
contradiction
 ; <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="E39:34_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E76">A191</font></span>: 
 for <font color="Olive" title="b1">k</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">k</font> &amp; <font color="Olive" title="b1">k</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</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="34_2_1_1_1_2_1_5"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:34_2_1_1_1_2_1_5"/>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</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="c6">D2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E55">A48</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="E2:34_2_1_1_1_2_1_5"/><span class="kw">then </span>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/>
<a NAME="E3:34_2_1_1_1_2_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A192</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</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/>
<span class="kw">let </span><font color="Maroon" title="c10">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"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c10">k</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">k</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E4:34_2_1_1_1_2_1_5"/><span class="lab"><font color="Green" title="E78">A193</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">and </span><br/><a NAME="E5:34_2_1_1_1_2_1_5"/><span class="lab"><font color="Green" title="E79">A194</font></span>: 
<font color="Maroon" title="c10">k</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c11">k</font> = <font color="Maroon" title="c10">k</font> 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="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E7:34_2_1_1_1_2_1_5"/><span class="lab"><font color="Green" title="E80">A195</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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</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="c5">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/>
<a NAME="E8:34_2_1_1_1_2_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A196</font></span>: 
<font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E68">A137</font></span></a>, <a class="txt" href="#E5:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E79">A194</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>;<br/>
<a NAME="E9:34_2_1_1_1_2_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c11">k</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="c5">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E68">A137</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E10:34_2_1_1_1_2_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A197</font></span>: 
<font color="Maroon" title="c11">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="c5">D1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E78">A193</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E11:34_2_1_1_1_2_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A198</font></span>: 
<font color="Maroon" title="c11">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="c5">D1</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="E12:34_2_1_1_1_2_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A199</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">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="c6">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E13:34_2_1_1_1_2_1_5"/><span class="lab"><font color="Green" title="E85">A200</font></span>: 
<font color="Maroon" title="c11">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> <font color="Maroon" title="c9">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E78">A193</font></span></a>, <a class="txt" href="#E8:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E81">A196</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E14:34_2_1_1_1_2_1_5"/><span class="kw">then </span>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">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> <font color="Maroon" title="c9">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E75">A173</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E78">A193</font></span></a>, <a class="txt" href="#E8:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E81">A196</font></span></a></span>;<br/>
<a NAME="E15:34_2_1_1_1_2_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A201</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">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="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E70">A150</font></span></a>, <a class="txt" href="#E38:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E75">A173</font></span></a></span>;<br/>
<a NAME="E16:34_2_1_1_1_2_1_5"/><span class="kw">then </span>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E17:34_2_1_1_1_2_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A202</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</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">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E65">A134</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E18:34_2_1_1_1_2_1_5"/><span class="lab"><font color="Green" title="E88">A203</font></span>: 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E11:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E83">A198</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E19:34_2_1_1_1_2_1_5"/><span class="lab"><font color="Green" title="E89">A204</font></span>: 
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">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="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c6">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>)</span>)</span>)</span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_2_1_5_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_1_1_2_1_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="E1:34_2_1_1_1_2_1_5_1_1"/>
( <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c11">k</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="34_2_1_1_1_2_1_5_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_2_1_5_1_1_1"/><span class="lab"><font color="Green" title="E90">A205</font></span>: 
<font color="Maroon" title="c11">k</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#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">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="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c6">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>)</span>)</span>)</span>)</span> )</span><br/><div class="add"><a NAME="E2:34_2_1_1_1_2_1_5_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A206</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="c5">D1</font>,<font color="Maroon" title="c11">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E83">A198</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E3:34_2_1_1_1_2_1_5_1_1_1"/><span class="lab"><font color="Green" title="E92">A207</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E83">A198</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_2_1_5_1_1_1"><span class="lab"><font color="Green" title="E90">A205</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E4:34_2_1_1_1_2_1_5_1_1_1"/>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E70">A150</font></span></a>, <a class="txt" href="#E38:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E75">A173</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_2_1_5_1_1_1"><span class="lab"><font color="Green" title="E90">A205</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:34_2_1_1_1_2_1_5_1_1_1"/>
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">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="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c6">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>)</span>)</span>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E84">A199</font></span></a>, <a class="txt" href="#E18:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E88">A203</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_2_1_5_1_1_1"><span class="lab"><font color="Green" title="E92">A207</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_2_1_5_1_1_1"><span class="lab"><font color="Green" title="E91">A206</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="34_2_1_1_1_2_1_5_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_2_1_5_1_1_2"/><span class="lab"><font color="Green" title="E90">A208</font></span>: 
<font color="Maroon" title="c11">k</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#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">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="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c6">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>)</span>)</span>)</span>)</span> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">k1</font> = <font color="Maroon" title="c11">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="#E11:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E83">A198</font></span></a>, <a class="ref" href="integra1.html#T7" title="INTEGRA1:th.7">INTEGRA1:7</a></span>;<br/><a NAME="E3:34_2_1_1_1_2_1_5_1_1_2"/>
<font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font> <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#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E4:34_2_1_1_1_2_1_5_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</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="E5:34_2_1_1_1_2_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A209</font></span>: 
<font color="Maroon" title="c12">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E81">A196</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E6:34_2_1_1_1_2_1_5_1_1_2"/><span class="lab"><font color="Green" title="E92">A210</font></span>: 
<font color="Maroon" title="c11">k</font> <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="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E83">A198</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_2_1_5_1_1_2"><span class="lab"><font color="Green" title="E90">A208</font></span></a>, <a class="ref" href="integra1.html#T7" title="INTEGRA1:th.7">INTEGRA1:7</a></span>;<br/><a NAME="E7:34_2_1_1_1_2_1_5_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">k1</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="c5">D1</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="E8:34_2_1_1_1_2_1_5_1_1_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E9:34_2_1_1_1_2_1_5_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c12">k1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E75">A173</font></span></a>, <a class="txt" href="#E5:34_2_1_1_1_2_1_5_1_1_2"><span class="lab"><font color="Green" title="E91">A209</font></span></a></span>;<br/><a NAME="E10:34_2_1_1_1_2_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A211</font></span>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c12">k1</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E75">A173</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E78">A193</font></span></a>, <a class="txt" href="#E8:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E81">A196</font></span></a></span>;<br/><a NAME="E11:34_2_1_1_1_2_1_5_1_1_2"/><span class="lab"><font color="Green" title="E94">A212</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E75">A173</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E78">A193</font></span></a>, <a class="txt" href="#E8:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E81">A196</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_2_1_5_1_1_2"><span class="lab"><font color="Green" title="E90">A208</font></span></a></span>;<br/><a NAME="E12:34_2_1_1_1_2_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A213</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>)</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</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="#E12:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E84">A199</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E13:34_2_1_1_1_2_1_5_1_1_2"/><span class="lab"><font color="Green" title="E96">A214</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="c6">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>)</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E84">A199</font></span></a>, <a class="txt" href="#E11:34_2_1_1_1_2_1_5_1_1_2"><span class="lab"><font color="Green" title="E94">A212</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E14:34_2_1_1_1_2_1_5_1_1_2"/><span class="lab"><font color="Green" title="E97">A215</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="c5">D1</font>,<font color="Maroon" title="c11">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E83">A198</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_2_1_5_1_1_2"><span class="lab"><font color="Green" title="E90">A208</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E15:34_2_1_1_1_2_1_5_1_1_2"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <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="#E11:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E83">A198</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_2_1_5_1_1_2"><span class="lab"><font color="Green" title="E90">A208</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="E16:34_2_1_1_1_2_1_5_1_1_2"/>
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">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="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c6">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>)</span>)</span>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E11:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E83">A198</font></span></a>, <a class="txt" href="#E14:34_2_1_1_1_2_1_5_1_1_2"><span class="lab"><font color="Green" title="E97">A215</font></span></a>, <a class="txt" href="#E6:34_2_1_1_1_2_1_5_1_1_2"><span class="lab"><font color="Green" title="E92">A210</font></span></a>, <a class="txt" href="#E12:34_2_1_1_1_2_1_5_1_1_2"><span class="lab"><font color="Green" title="E95">A213</font></span></a>, <a class="txt" href="#E13:34_2_1_1_1_2_1_5_1_1_2"><span class="lab"><font color="Green" title="E96">A214</font></span></a>, <a class="txt" href="#E10:34_2_1_1_1_2_1_5_1_1_2"><span class="lab"><font color="Green" title="E93">A211</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</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 NAME="E20:34_2_1_1_1_2_1_5"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><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="c6">D2</font>,<span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>)</span>)</span>)</span>)</span></span>)</span>,<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="c6">D2</font>,<span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>)</span>)</span>)</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T4" title="INTEGRA1:th.4">INTEGRA1:4</a></span>;<br/>
<a NAME="E21:34_2_1_1_1_2_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A216</font></span>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E89">A204</font></span></a>, <a class="ref" href="integra1.html#T4" title="INTEGRA1:th.4">INTEGRA1:4</a></span>;<br/>
<a NAME="E22:34_2_1_1_1_2_1_5"/><span class="lab"><font color="Green" title="E91">A217</font></span>: 
<font color="Maroon" title="c11">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="c5">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E82">A197</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="E23:34_2_1_1_1_2_1_5"/>
<font color="Maroon" title="c9">j1</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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E52">A45</font></span></a>, <a class="txt" href="#E7:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E80">A195</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="E24:34_2_1_1_1_2_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c9">j1</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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</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/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E92">A218</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> = 
<span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E85">A200</font></span></a>, <a class="ref" href="rfinseq.html#T6" title="RFINSEQ:th.6">RFINSEQ:6</a></span>
<br/>.= 
<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"><font color="Maroon" title="c2">f</font> <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="c6">D2</font>,<span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>)</span>)</span>)</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="c6">D2</font>,<span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>)</span>)</span>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E91">A217</font></span></a>, <a class="txt" href="#E21:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E90">A216</font></span></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>
;<br/>
<a NAME="E26:34_2_1_1_1_2_1_5"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E75">A173</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E78">A193</font></span></a>, <a class="txt" href="#E8:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E81">A196</font></span></a></span>;<br/>
<a NAME="E27:34_2_1_1_1_2_1_5"/><span class="kw">then </span>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">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="c6">D2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E87">A202</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E28:34_2_1_1_1_2_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A219</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">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="c6">D2</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="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E75">A173</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E78">A193</font></span></a>, <a class="txt" href="#E8:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E81">A196</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E86">A201</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E77">A192</font></span></a>, <a class="ref" href="rfinseq.html#T6" title="RFINSEQ:th.6">RFINSEQ:6</a></span>
<br/>.= 
<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"><font color="Maroon" title="c2">f</font> <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="c6">D2</font>,<span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>)</span>)</span>)</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="c6">D2</font>,<span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font>)</span>)</span>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E93">A219</font></span></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E30:34_2_1_1_1_2_1_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:34_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E92">A218</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="E40:34_2_1_1_1_2_1"/>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</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">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E7:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E54">A47</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E41:34_2_1_1_1_2_1"/><span class="kw">then </span>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</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">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E42:34_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A220</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</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> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/>
<a NAME="E43:34_2_1_1_1_2_1"/>
<font color="Maroon" title="c9">j1</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="c5">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E54">A47</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E44:34_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A221</font></span>: 
<font color="Maroon" title="c9">j1</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> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/>
<a NAME="E45:34_2_1_1_1_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</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="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E58">A51</font></span></a>, <a class="txt" href="#E36:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E73">A171</font></span></a>, <a class="txt" href="#E34:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E72">A170</font></span></a>, <a class="ref" href="integra3.html#T6" target="_self" title="INTEGRA3:th.6">Th6</a></span>;<br/>
<a NAME="E46:34_2_1_1_1_2_1"/><span class="kw">then </span>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E68">A137</font></span></a>, <a class="txt" href="#E24:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E66">A135</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>;<br/>
<a NAME="E47:34_2_1_1_1_2_1"/><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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E78">A221</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>;<br/>
<a NAME="E48:34_2_1_1_1_2_1"/><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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E77">A220</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>;<br/>
<a NAME="E49:34_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A222</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E76">A191</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</a></span>;<br/>
<a NAME="E50:34_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E80">A223</font></span>: 
<font color="Maroon" title="c8">j</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="c5">D1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E49">A42</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="E51:34_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A224</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E52:34_2_1_1_1_2_1"/>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</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">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E63">A132</font></span></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/>
<a NAME="E53:34_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A225</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</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">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</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="E54:34_2_1_1_1_2_1"/>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</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">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E63">A132</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E55:34_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A226</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</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">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/>
<a NAME="E56:34_2_1_1_1_2_1"/>
<font color="Maroon" title="c8">j</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">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E80">A223</font></span></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/>
<a NAME="E57:34_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A227</font></span>: 
<font color="Maroon" title="c8">j</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">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</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="E58:34_2_1_1_1_2_1"/>
<font color="Maroon" title="c8">j</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="c5">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E80">A223</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E59:34_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A228</font></span>: 
<font color="Maroon" title="c8">j</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">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/>
<a NAME="E60:34_2_1_1_1_2_1"/>
<font color="Maroon" title="c9">j1</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="c5">D1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E52">A45</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="E61:34_2_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">j1</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">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/>
<a NAME="E62:34_2_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">j1</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">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</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="E63:34_2_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>) <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D20" title="INTEGRA1:def.20">INTEGRA1:def 20</a></span>;<br/>
<span class="kw">then </span><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>) <a href="real_1.html#K7" title="REAL_1:func.7">+</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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</font>)</span>)</span></span>)</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"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>) <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T75" title="RVSUM_1:th.75">RVSUM_1:75</a></span>
<br/>.= 
 <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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),1,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p3">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c8">j</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E70">A150</font></span></a>, <a class="ref" href="finseq_6.html#T116" title="FINSEQ_6:th.116">FINSEQ_6:116</a></span>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),1,<font color="Maroon" title="c8">j</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E70">A150</font></span></a>, <a class="txt" href="#E59:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E85">A228</font></span></a>, <a class="txt" href="#E19:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E62">A131</font></span></a>, <a class="ref" href="integra2.html#T4" title="INTEGRA2:th.4">INTEGRA2:4</a></span>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>) <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E81">A224</font></span></a>, <a class="ref" href="finseq_6.html#T116" title="FINSEQ_6:th.116">FINSEQ_6:116</a></span>
;<br/>
<a NAME="E65:34_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A229</font></span>: 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>) <a href="real_1.html#K7" title="REAL_1:func.7">+</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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p3">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E84">A227</font></span></a>, <a class="ref" href="integra1.html#D20" title="INTEGRA1:def.20">INTEGRA1:def 20</a></span>;<br/>
<a NAME="E66:34_2_1_1_1_2_1"/>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</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="c6">D2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E55">A48</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="E67:34_2_1_1_1_2_1"/><span class="kw">then </span>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</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">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/>
<a NAME="E68:34_2_1_1_1_2_1"/><span class="kw">then </span>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</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">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</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="E69:34_2_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D20" title="INTEGRA1:def.20">INTEGRA1:def 20</a></span>;<br/>
<span class="kw">then </span><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)) <a href="real_1.html#K7" title="REAL_1:func.7">+</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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p3">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span></span>)</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"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T75" title="RVSUM_1:th.75">RVSUM_1:75</a></span>
<br/>.= 
 <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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),1,<span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E57">A50</font></span></a>, <a class="ref" href="finseq_6.html#T116" title="FINSEQ_6:th.116">FINSEQ_6:116</a></span>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),1,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E57">A50</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E59">A52</font></span></a>, <a class="txt" href="#E55:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E83">A226</font></span></a>, <a class="ref" href="integra2.html#T4" title="INTEGRA2:th.4">INTEGRA2:4</a></span>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E64">A133</font></span></a>, <a class="ref" href="finseq_6.html#T116" title="FINSEQ_6:th.116">FINSEQ_6:116</a></span>
;<br/>
<a NAME="E71:34_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A230</font></span>: 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)) <a href="real_1.html#K7" title="REAL_1:func.7">+</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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p3">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>))
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E82">A225</font></span></a>, <a class="ref" href="integra1.html#D20" title="INTEGRA1:def.20">INTEGRA1:def 20</a></span>;<br/>
<a NAME="E72:34_2_1_1_1_2_1"/>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</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="c6">D2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E55">A48</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="E73:34_2_1_1_1_2_1"/><span class="kw">then </span>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/>
<a NAME="E74:34_2_1_1_1_2_1"/><span class="kw">then </span>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</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/>
<span class="kw">then </span><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)) = 
 <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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D20" title="INTEGRA1:def.20">INTEGRA1:def 20</a></span>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E79">A222</font></span></a>, <a class="txt" href="#E6:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E53">A46</font></span></a>, <a class="ref" href="integra1.html#D20" title="INTEGRA1:def.20">INTEGRA1:def 20</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E76:34_2_1_1_1_2_1"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)) <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</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="c5">D1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E61">A54</font></span></a>, <a class="txt" href="#E71:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E87">A230</font></span></a>, <a class="txt" href="#E65:34_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E86">A229</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="E8:34_2_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="txt" href="#E6:34_2_1_1_1_2"><span class="lab"><font color="Green" title="E50">A43</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">reconsider </span><font color="Maroon" title="c8">i</font> = <font color="Maroon" title="c7">i</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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="#E2:34_2_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E6:34_2_1_1_1"/><span class="lab"><font color="Green" title="E48">A231</font></span>: 
 for <font color="Olive" title="b1">i</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">i</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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">S<sub>1</sub></font>[<font color="Maroon" title="c9">i</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<a NAME="E1:34_2_1_1_1_3"/><span class="lab"><font color="Green" title="E49">A232</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:34_2_1_1_1_3"/><span class="lab"><font color="Green" title="E50">A233</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">i</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">S<sub>1</sub></font>[<font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E3:34_2_1_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_3_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:34_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E51">A234</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:34_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E52">A235</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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">D</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">j</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/>( <font color="Olive" title="b1">j</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="c5">D1</font> &amp; <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Olive" title="b1">j</font>) &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Olive" title="b1">j</font>)) <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Olive" title="b1">j</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="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</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="c5">D1</font></span>)</span> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">j</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="E4:34_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E53">A236</font></span>: 
<font color="Maroon" title="c10">j</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="c5">D1</font>
 <span class="kw">and </span><br/><a NAME="E5:34_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E54">A237</font></span>: 
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra3.html#T3" target="_self" title="INTEGRA3:th.3">Th3</a>, <a class="ref" href="integra1.html#T6" title="INTEGRA1:th.6">INTEGRA1:6</a></span>;<br/>
<a NAME="E6:34_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E55">A238</font></span>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E7:34_2_1_1_1_3_1"/>
<font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c4">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E52">A235</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E8:34_2_1_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">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">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E51">A234</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E9:34_2_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A239</font></span>: 
<font color="Maroon" title="c9">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">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2_1_1_1_3"><span class="lab"><font color="Green" title="E49">A232</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E10:34_2_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A240</font></span>: 
<font color="Maroon" title="c9">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">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">consider </span><font color="Maroon" title="c11">n1</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="E12:34_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E58">A241</font></span>: 
<font color="Maroon" title="c11">n1</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="c5">D1</font>
 <span class="kw">and </span><br/><a NAME="E13:34_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E59">A242</font></span>: 
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)
 <span class="kw">and </span><br/><a NAME="E14:34_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E60">A243</font></span>: 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)) <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</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="c5">D1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_2_1_1_1_3"><span class="lab"><font color="Green" title="E50">A233</font></span></a>, <a class="txt" href="#E9:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E56">A239</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="E15:34_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E61">A244</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n1</font> <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#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E16:34_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E62">A245</font></span>: 
<font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:34_2_1_1_1_3_1_1"/><span class="lab"><font color="Green" title="E63">A246</font></span>: 
<font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_3_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_1_1_3_1_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="E1:34_2_1_1_1_3_1_1_1_1"/>
( <font color="Maroon" title="c11">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j</font> or <font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">j</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2_1_1_1_3_1_1"><span class="lab"><font color="Green" title="E63">A246</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="34_2_1_1_1_3_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_3_1_1_1_1_1"/><span class="lab"><font color="Green" title="E64">A247</font></span>: 
<font color="Maroon" title="c11">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:34_2_1_1_1_3_1_1_1_1_1"/>
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E57">A240</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="E3:34_2_1_1_1_3_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A248</font></span>: 
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</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="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E59">A242</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E64">A247</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E4:34_2_1_1_1_3_1_1_1_1_1"/>
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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> <font color="Maroon" title="c4">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E52">A235</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="E5:34_2_1_1_1_3_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A249</font></span>: 
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</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="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E54">A237</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E6:34_2_1_1_1_3_1_1_1_1_1"/>
<font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" title="XREAL_1:th.29">XREAL_1:29</a></span>;<br/><span class="kw">hence </span><a NAME="E7:34_2_1_1_1_3_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E52">A235</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</font></span></a>, <a class="txt" href="#E10:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E57">A240</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E65">A248</font></span></a>, <a class="txt" href="#E5:34_2_1_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E66">A249</font></span></a>, <a class="ref" href="integra3.html#T5" target="_self" title="INTEGRA3:th.5">Th5</a>, <a class="ref" href="seq_4.html#T138" title="SEQ_4:th.138">SEQ_4:138</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="34_2_1_1_1_3_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_3_1_1_1_1_2"/>
<font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:34_2_1_1_1_3_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A250</font></span>: 
<font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">j</font> <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="E3:34_2_1_1_1_3_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A251</font></span>: 
<font color="Maroon" title="c11">n1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">j</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="E4:34_2_1_1_1_3_1_1_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E5:34_2_1_1_1_3_1_1_1_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="c10">j</font> <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="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E6:34_2_1_1_1_3_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A252</font></span>: 
<font color="Maroon" title="c11">n1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_2_1_1_1_3_1_1_1_1_2"><span class="lab"><font color="Green" title="E64">A250</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E7:34_2_1_1_1_3_1_1_1_1_2"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E59">A242</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/><a NAME="E8:34_2_1_1_1_3_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A253</font></span>: 
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <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="#E12:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E58">A241</font></span></a>, <a class="txt" href="#E6:34_2_1_1_1_3_1_1_1_1_2"><span class="lab"><font color="Green" title="E66">A252</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E9:34_2_1_1_1_3_1_1_1_1_2"/>
<font color="Maroon" title="c11">n1</font> <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="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E58">A241</font></span></a>, <a class="txt" href="#E6:34_2_1_1_1_3_1_1_1_1_2"><span class="lab"><font color="Green" title="E66">A252</font></span></a>, <a class="ref" href="integra1.html#T7" title="INTEGRA1:th.7">INTEGRA1:7</a></span>;<br/><a NAME="E10:34_2_1_1_1_3_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <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="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_3_1_1_1_1_2"><span class="lab"><font color="Green" title="E65">A251</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/><a NAME="E11:34_2_1_1_1_3_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A254</font></span>: 
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:34_2_1_1_1_3_1_1_1_1_2"><span class="lab"><font color="Green" title="E67">A253</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E12:34_2_1_1_1_3_1_1_1_1_2"/><span class="lab"><font color="Green" title="E69">A255</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" title="XREAL_1:th.29">XREAL_1:29</a></span>;<br/><a NAME="E13:34_2_1_1_1_3_1_1_1_1_2"/><span class="lab"><font color="Green" title="E70">A256</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="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_3_1_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_1_1_3_1_1_1_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="E1:34_2_1_1_1_3_1_1_1_1_2_1_1"/>
( <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">j</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="34_2_1_1_1_3_1_1_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_3_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c10">j</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="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">j</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:34_2_1_1_1_3_1_1_1_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="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</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="34_2_1_1_1_3_1_1_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_3_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c10">j</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="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">j</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:34_2_1_1_1_3_1_1_1_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="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</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 NAME="E14:34_2_1_1_1_3_1_1_1_1_2"/>
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E54">A237</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/><a NAME="E15:34_2_1_1_1_3_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:34_2_1_1_1_3_1_1_1_1_2"><span class="lab"><font color="Green" title="E70">A256</font></span></a>, <a class="txt" href="#E11:34_2_1_1_1_3_1_1_1_1_2"><span class="lab"><font color="Green" title="E68">A254</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E16:34_2_1_1_1_3_1_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E52">A235</font></span></a>, <a class="txt" href="#E10:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E57">A240</font></span></a>, <a class="txt" href="#E12:34_2_1_1_1_3_1_1_1_1_2"><span class="lab"><font color="Green" title="E69">A255</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3: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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:34_2_1_1_1_3_1_1"/>
contradiction
 ; <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="E17:34_2_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A257</font></span>: 
<font color="Maroon" title="c11">n1</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> <font color="Maroon" title="c10">j</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="E18:34_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E64">A258</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E58">A241</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E19:34_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E65">A259</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</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">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E12:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E58">A241</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E20:34_2_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A260</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E21:34_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E67">A261</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</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">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E22:34_2_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A262</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E23:34_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E69">A263</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</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">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E67">A261</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E24:34_2_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A264</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</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">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/>
<a NAME="E25:34_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E71">A265</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E26:34_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E72">A266</font></span>: 
<font color="Maroon" title="c10">j</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="c5">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E27:34_2_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A267</font></span>: 
<font color="Maroon" title="c11">n1</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="c5">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A257</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E28:34_2_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A268</font></span>: 
<font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="c5">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E61">A244</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E29:34_2_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A269</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <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">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E30:34_2_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A270</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E31:34_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E77">A271</font></span>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E28:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E74">A268</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E32:34_2_1_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</font></span></a>, <a class="txt" href="#E17:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A257</font></span></a>, <a class="txt" href="#E28:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E74">A268</font></span></a>, <a class="txt" href="#E6:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E55">A238</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/>
<a NAME="E33:34_2_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A272</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E75">A269</font></span></a>, <a class="txt" href="#E21:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E67">A261</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/>
<a NAME="E34:34_2_1_1_1_3_1"/><span class="kw">then </span>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</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="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E35:34_2_1_1_1_3_1"/><span class="kw">then </span>
1 <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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span>
 
<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/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E79">A273</font></span>: <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 = 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <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> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E78">A272</font></span></a>, <a class="txt" href="#E30:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E76">A270</font></span></a>, <a class="txt" href="#E23:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E69">A263</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>
<br/>.= 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E28:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E74">A268</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>
;<br/>
<a NAME="E37:34_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E80">A274</font></span>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E12:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E58">A241</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E38:34_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E81">A275</font></span>: 
<font color="Maroon" title="c10">j</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">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/>
<a NAME="E39:34_2_1_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">j</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">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E71">A265</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E40:34_2_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A276</font></span>: 
<font color="Maroon" title="c10">j</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">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</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="E41:34_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E83">A277</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <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">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E75">A269</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E42:34_2_1_1_1_3_1"/>
<font color="Maroon" title="c11">n1</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="c5">D1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E58">A241</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="E43:34_2_1_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">n1</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">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/>
<a NAME="E44:34_2_1_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">n1</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">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</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><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>) = 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>) <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c11">n1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D20" title="INTEGRA1:def.20">INTEGRA1:def 20</a></span>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),1,<font color="Maroon" title="c11">n1</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E64">A258</font></span></a>, <a class="ref" href="finseq_6.html#T116" title="FINSEQ_6:th.116">FINSEQ_6:116</a></span>
;<br/>
<span class="kw">then </span><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>) <a href="real_1.html#K7" title="REAL_1:func.7">+</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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c10">j</font>)</span>)</span></span>)</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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),1,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c10">j</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T75" title="RVSUM_1:th.75">RVSUM_1:75</a></span>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),1,<font color="Maroon" title="c10">j</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A245</font></span></a>, <a class="txt" href="#E18:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E64">A258</font></span></a>, <a class="txt" href="#E38:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E81">A275</font></span></a>, <a class="ref" href="integra2.html#T4" title="INTEGRA2:th.4">INTEGRA2:4</a></span>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>) <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E71">A265</font></span></a>, <a class="ref" href="finseq_6.html#T116" title="FINSEQ_6:th.116">FINSEQ_6:116</a></span>
;<br/>
<a NAME="E47:34_2_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A278</font></span>: 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>) <a href="real_1.html#K7" title="REAL_1:func.7">+</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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c10">j</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E82">A276</font></span></a>, <a class="ref" href="integra1.html#D20" title="INTEGRA1:def.20">INTEGRA1:def 20</a></span>;<br/>
<a NAME="E48:34_2_1_1_1_3_1"/>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</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="c6">D2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E67">A261</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="E49:34_2_1_1_1_3_1"/><span class="kw">then </span>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</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">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/>
<a NAME="E50:34_2_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A279</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</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">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</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="E51:34_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E86">A280</font></span>: 
<font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E58">A241</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E52:34_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E87">A281</font></span>: 
<font color="Maroon" title="c10">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A257</font></span></a>, <a class="ref" href="xreal_1.html#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/>
<a NAME="E53:34_2_1_1_1_3_1"/>
<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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</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"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c10">j</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"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></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="c5">D1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_3_1_5"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_1_1_3_1_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="E1:34_2_1_1_1_3_1_5_1_1"/>
( <font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j</font> or <font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">j</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A257</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="34_2_1_1_1_3_1_5_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_3_1_5_1_1_1"/><span class="lab"><font color="Green" title="E88">A282</font></span>: 
<font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j</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="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</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"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c10">j</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"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></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="c5">D1</font></span>)</span></span><br/><div class="add"><a NAME="E2:34_2_1_1_1_3_1_5_1_1_1"/><span class="lab"><font color="Green" title="E89">A283</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:34_2_1_1_1_3_1_5_1_1_1_1"/><span class="lab"><font color="Green" title="E90">A284</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="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</font></span></a>, <a class="txt" href="#E16:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A245</font></span></a>, <a class="txt" href="#E51:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E86">A280</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/>
<a NAME="E2:34_2_1_1_1_3_1_5_1_1_1_1"/><span class="lab"><font color="Green" title="E91">A285</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <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="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</font></span></a>, <a class="txt" href="#E16:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A245</font></span></a>, <a class="txt" href="#E51:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E86">A280</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/>
<a NAME="E3:34_2_1_1_1_3_1_5_1_1_1_1"/><span class="lab"><font color="Green" title="E92">A286</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</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="#E20:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E66">A260</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:34_2_1_1_1_3_1_5_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E5:34_2_1_1_1_3_1_5_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A287</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</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="E6:34_2_1_1_1_3_1_5_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A288</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <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">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E69">A263</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E7:34_2_1_1_1_3_1_5_1_1_1_1"/><span class="lab"><font color="Green" title="E95">A289</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E8:34_2_1_1_1_3_1_5_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A290</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<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="E9:34_2_1_1_1_3_1_5_1_1_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E66">A260</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E10:34_2_1_1_1_3_1_5_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A291</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <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">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:34_2_1_1_1_3_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E94">A288</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E11:34_2_1_1_1_3_1_5_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A292</font></span>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E67">A261</font></span></a>, <a class="txt" href="#E5:34_2_1_1_1_3_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E93">A287</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/>
<a NAME="E12:34_2_1_1_1_3_1_5_1_1_1_1"/><span class="lab"><font color="Green" title="E99">A293</font></span>: 
 not <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</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> <font color="Maroon" title="c5">D1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:34_2_1_1_1_3_1_5_1_1_1_1_1"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</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> <font color="Maroon" title="c5">D1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">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="E3:34_2_1_1_1_3_1_5_1_1_1_1_1"/><span class="lab"><font color="Green" title="E100">A294</font></span>: 
<font color="Maroon" title="c12">k1</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="c5">D1</font>
 <span class="kw">and </span><br/><a NAME="E4:34_2_1_1_1_3_1_5_1_1_1_1_1"/><span class="lab"><font color="Green" title="E101">A295</font></span>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T3" title="PARTFUN1:th.3">PARTFUN1:3</a></span>;<br/>
<a NAME="E5:34_2_1_1_1_3_1_5_1_1_1_1_1"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E67">A261</font></span></a>, <a class="txt" href="#E5:34_2_1_1_1_3_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E93">A287</font></span></a>, <a class="txt" href="#E10:34_2_1_1_1_3_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E97">A291</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/>
<a NAME="E6:34_2_1_1_1_3_1_5_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A296</font></span>: 
<font color="Maroon" title="c12">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</font></span></a>, <a class="txt" href="#E6:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E55">A238</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_1_1_1"><span class="lab"><font color="Green" title="E100">A294</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_3_1_5_1_1_1_1_1"><span class="lab"><font color="Green" title="E101">A295</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/>
<a NAME="E7:34_2_1_1_1_3_1_5_1_1_1_1_1"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E65">A259</font></span></a>, <a class="txt" href="#E8:34_2_1_1_1_3_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E96">A290</font></span></a>, <a class="txt" href="#E10:34_2_1_1_1_3_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E97">A291</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/>
<a NAME="E8:34_2_1_1_1_3_1_5_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">k1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E58">A241</font></span></a>, <a class="txt" href="#E37:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E80">A274</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_1_1_1"><span class="lab"><font color="Green" title="E100">A294</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_3_1_5_1_1_1_1_1"><span class="lab"><font color="Green" title="E101">A295</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:34_2_1_1_1_3_1_5_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E88">A282</font></span></a>, <a class="txt" href="#E6:34_2_1_1_1_3_1_5_1_1_1_1_1"><span class="lab"><font color="Green" title="E102">A296</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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="E13:34_2_1_1_1_3_1_5_1_1_1_1"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</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> <font color="Maroon" title="c6">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:34_2_1_1_1_3_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E97">A291</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="E14:34_2_1_1_1_3_1_5_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A297</font></span>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</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> <font color="Maroon" title="c4">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1"><span class="lab"><font color="Green" title="E41">A14</font></span></a>, <a class="txt" href="#E12:34_2_1_1_1_3_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E99">A293</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E15:34_2_1_1_1_3_1_5_1_1_1_1"/><span class="lab"><font color="Green" title="E101">A298</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <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="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</font></span></a>, <a class="txt" href="#E16:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A245</font></span></a>, <a class="txt" href="#E51:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E86">A280</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/>
<a NAME="E16:34_2_1_1_1_3_1_5_1_1_1_1"/><span class="lab"><font color="Green" title="E102">A299</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="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</font></span></a>, <a class="txt" href="#E16:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A245</font></span></a>, <a class="txt" href="#E51:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E86">A280</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/>
<a NAME="E17:34_2_1_1_1_3_1_5_1_1_1_1"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E65">A259</font></span></a>, <a class="txt" href="#E8:34_2_1_1_1_3_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E96">A290</font></span></a>, <a class="txt" href="#E10:34_2_1_1_1_3_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E97">A291</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/>
<a NAME="E18:34_2_1_1_1_3_1_5_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E80">A274</font></span></a>, <a class="txt" href="#E6:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E55">A238</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E88">A282</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_3_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E101">A298</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E90">A284</font></span></a>, <a class="txt" href="#E11:34_2_1_1_1_3_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E98">A292</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/>
<a NAME="E19:34_2_1_1_1_3_1_5_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A300</font></span>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</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="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:34_2_1_1_1_3_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E100">A297</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E20:34_2_1_1_1_3_1_5_1_1_1_1"/><span class="lab"><font color="Green" title="E104">A301</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_3_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E93">A287</font></span></a>, <a class="txt" href="#E7:34_2_1_1_1_3_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E95">A289</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E21:34_2_1_1_1_3_1_5_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E69">A263</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E22:34_2_1_1_1_3_1_5_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A302</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E92">A286</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E23:34_2_1_1_1_3_1_5_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A303</font></span>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E67">A261</font></span></a>, <a class="txt" href="#E20:34_2_1_1_1_3_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E104">A301</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/>
<a NAME="E24:34_2_1_1_1_3_1_5_1_1_1_1"/><span class="lab"><font color="Green" title="E107">A304</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</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="E25:34_2_1_1_1_3_1_5_1_1_1_1"/><span class="lab"><font color="Green" title="E108">A305</font></span>: 
 not <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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> <font color="Maroon" title="c5">D1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:34_2_1_1_1_3_1_5_1_1_1_1_2"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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> <font color="Maroon" title="c5">D1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">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="E3:34_2_1_1_1_3_1_5_1_1_1_1_2"/><span class="lab"><font color="Green" title="E109">A306</font></span>: 
<font color="Maroon" title="c12">k1</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="c5">D1</font>
 <span class="kw">and </span><br/><a NAME="E4:34_2_1_1_1_3_1_5_1_1_1_1_2"/><span class="lab"><font color="Green" title="E110">A307</font></span>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T3" title="PARTFUN1:th.3">PARTFUN1:3</a></span>;<br/>
<a NAME="E5:34_2_1_1_1_3_1_5_1_1_1_1_2"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E67">A261</font></span></a>, <a class="txt" href="#E20:34_2_1_1_1_3_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E104">A301</font></span></a>, <a class="txt" href="#E22:34_2_1_1_1_3_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E105">A302</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/>
<a NAME="E6:34_2_1_1_1_3_1_5_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">A308</font></span>: 
<font color="Maroon" title="c12">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</font></span></a>, <a class="txt" href="#E6:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E55">A238</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_1_1_2"><span class="lab"><font color="Green" title="E109">A306</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_3_1_5_1_1_1_1_2"><span class="lab"><font color="Green" title="E110">A307</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/>
<a NAME="E7:34_2_1_1_1_3_1_5_1_1_1_1_2"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E65">A259</font></span></a>, <a class="txt" href="#E24:34_2_1_1_1_3_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E107">A304</font></span></a>, <a class="txt" href="#E22:34_2_1_1_1_3_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E105">A302</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/>
<a NAME="E8:34_2_1_1_1_3_1_5_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">k1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E58">A241</font></span></a>, <a class="txt" href="#E37:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E80">A274</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_1_1_2"><span class="lab"><font color="Green" title="E109">A306</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_3_1_5_1_1_1_1_2"><span class="lab"><font color="Green" title="E110">A307</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:34_2_1_1_1_3_1_5_1_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E88">A282</font></span></a>, <a class="txt" href="#E6:34_2_1_1_1_3_1_5_1_1_1_1_2"><span class="lab"><font color="Green" title="E111">A308</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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="E26:34_2_1_1_1_3_1_5_1_1_1_1"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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> <font color="Maroon" title="c6">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:34_2_1_1_1_3_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E105">A302</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="E27:34_2_1_1_1_3_1_5_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A309</font></span>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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> <font color="Maroon" title="c4">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1"><span class="lab"><font color="Green" title="E41">A14</font></span></a>, <a class="txt" href="#E25:34_2_1_1_1_3_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E108">A305</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E28:34_2_1_1_1_3_1_5_1_1_1_1"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E65">A259</font></span></a>, <a class="txt" href="#E24:34_2_1_1_1_3_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E107">A304</font></span></a>, <a class="txt" href="#E22:34_2_1_1_1_3_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E105">A302</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/>
<a NAME="E29:34_2_1_1_1_3_1_5_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E80">A274</font></span></a>, <a class="txt" href="#E6:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E55">A238</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E88">A282</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_3_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E91">A285</font></span></a>, <a class="txt" href="#E16:34_2_1_1_1_3_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E102">A299</font></span></a>, <a class="txt" href="#E23:34_2_1_1_1_3_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E106">A303</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/>
<a NAME="E30:34_2_1_1_1_3_1_5_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</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="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:34_2_1_1_1_3_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E109">A309</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E31:34_2_1_1_1_3_1_5_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</font></span></a>, <a class="txt" href="#E19:34_2_1_1_1_3_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E103">A300</font></span></a>, <a class="ref" href="integra3.html#T5" target="_self" title="INTEGRA3:th.5">Th5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E32:34_2_1_1_1_3_1_5_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:34_2_1_1_1_3_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E95">A289</font></span></a>, <a class="txt" href="#E22:34_2_1_1_1_3_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E105">A302</font></span></a>, <a class="txt" href="#E10:34_2_1_1_1_3_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E97">A291</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3: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><a NAME="E3:34_2_1_1_1_3_1_5_1_1_1"/><span class="lab"><font color="Green" title="E90">A310</font></span>: 
( <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>) implies <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:34_2_1_1_1_3_1_5_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span><br/>

<a NAME="E2:34_2_1_1_1_3_1_5_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A311</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</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="E3:34_2_1_1_1_3_1_5_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E89">A283</font></span></a>, <a class="ref" href="xreal_1.html#T20" title="XREAL_1:th.20">XREAL_1:20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:34_2_1_1_1_3_1_5_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_2_1_1_1_3_1_5_1_1_1_2"><span class="lab"><font color="Green" title="E91">A311</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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><a NAME="E4:34_2_1_1_1_3_1_5_1_1_1"/><span class="lab"><font color="Green" title="E91">A312</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</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#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E5:34_2_1_1_1_3_1_5_1_1_1"/><span class="lab"><font color="Green" title="E92">A313</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</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">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E69">A263</font></span></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/><a NAME="E6:34_2_1_1_1_3_1_5_1_1_1"/>
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</font></span></a>, <a class="txt" href="#E12:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E58">A241</font></span></a>, <a class="txt" href="#E16:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A245</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/><a NAME="E7:34_2_1_1_1_3_1_5_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A314</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E65">A259</font></span></a>, <a class="txt" href="#E37:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E80">A274</font></span></a>, <a class="txt" href="#E21:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E67">A261</font></span></a>, <a class="txt" href="#E6:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E55">A238</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/><a NAME="E8:34_2_1_1_1_3_1_5_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A315</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</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>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</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="E9:34_2_1_1_1_3_1_5_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E69">A263</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E10:34_2_1_1_1_3_1_5_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>)
 <span class="kw">by</span> <span class="lab"><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="E95">A316</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>)</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="#E22:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E68">A262</font></span></a>, <a class="txt" href="#E8:34_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E94">A315</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E91">A312</font></span></a>, <a class="txt" href="#E5:34_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E92">A313</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="#E8:34_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E94">A315</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span>

;<br/><a NAME="E12:34_2_1_1_1_3_1_5_1_1_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</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>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:34_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E93">A314</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E13:34_2_1_1_1_3_1_5_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A317</font></span>: 
( <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>) or <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E14:34_2_1_1_1_3_1_5_1_1_1"/><span class="lab"><font color="Green" title="E97">A318</font></span>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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> <font color="Maroon" title="c2">f</font></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="c5">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_1_4"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_1_4_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="E1:34_2_1_1_1_3_1_5_1_1_1_4_1"/>
( <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:34_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E96">A317</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E90">A310</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_1_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_3_1_5_1_1_1_4_1_1"/><span class="lab"><font color="Green" title="E98">A319</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <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="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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> <font color="Maroon" title="c2">f</font></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="c5">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:34_2_1_1_1_3_1_5_1_1_1_4_1_1"/><span class="lab"><font color="Green" title="E99">A320</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="#E20:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E66">A260</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:34_2_1_1_1_3_1_5_1_1_1_4_1_1"/><span class="kw">then </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="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E67">A261</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E98">A319</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E4:34_2_1_1_1_3_1_5_1_1_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A321</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="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E98">A319</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/><a NAME="E5:34_2_1_1_1_3_1_5_1_1_1_4_1_1"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="#E21:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E67">A261</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E98">A319</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_3_1_5_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E99">A320</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E6:34_2_1_1_1_3_1_5_1_1_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A322</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E12:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E58">A241</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/><a NAME="E7:34_2_1_1_1_3_1_5_1_1_1_4_1_1"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="#E16:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A245</font></span></a>, <a class="txt" href="#E51:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E86">A280</font></span></a>, <a class="txt" href="#E28:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E74">A268</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E88">A282</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E8:34_2_1_1_1_3_1_5_1_1_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A323</font></span>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A245</font></span></a>, <a class="txt" href="#E51:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E86">A280</font></span></a>, <a class="txt" href="#E28:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E74">A268</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E88">A282</font></span></a>, <a class="txt" href="#E6:34_2_1_1_1_3_1_5_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E101">A322</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_3_1_5_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E100">A321</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E9:34_2_1_1_1_3_1_5_1_1_1_4_1_1"/><span class="lab"><font color="Green" title="E103">A324</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="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a></span>;<br/><a NAME="E10:34_2_1_1_1_3_1_5_1_1_1_4_1_1"/>
1 <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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="#E1:34_2_1_1_1_3_1_5_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E98">A319</font></span></a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 = 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="#E4:34_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E91">A312</font></span></a>, <a class="txt" href="#E5:34_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E92">A313</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>

;<br/><a NAME="E12:34_2_1_1_1_3_1_5_1_1_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A325</font></span>: 
 <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>) <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">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="#E11:34_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E95">A316</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E98">A319</font></span></a>, <a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>;<br/><a NAME="E13:34_2_1_1_1_3_1_5_1_1_1_4_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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"><font color="Maroon" title="c2">f</font> <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="c6">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</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="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E67">A261</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E98">A319</font></span></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/><a NAME="E14:34_2_1_1_1_3_1_5_1_1_1_4_1_1"/><span class="kw">then </span>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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> <font color="Maroon" title="c2">f</font></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="c5">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E21:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E67">A261</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E98">A319</font></span></a>, <a class="txt" href="#E8:34_2_1_1_1_3_1_5_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E102">A323</font></span></a>, <a class="txt" href="#E9:34_2_1_1_1_3_1_5_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E103">A324</font></span></a>, <a class="ref" href="integra3.html#T18" target="_self" title="INTEGRA3:th.18">Th18</a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/><span class="kw">hence </span><a NAME="E15:34_2_1_1_1_3_1_5_1_1_1_4_1_1"/>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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> <font color="Maroon" title="c2">f</font></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="c5">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:34_2_1_1_1_3_1_5_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E104">A325</font></span></a>, <a class="ref" href="finsop_1.html#T11" title="FINSOP_1:th.11">FINSOP_1:11</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="34_2_1_1_1_3_1_5_1_1_1_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_3_1_5_1_1_1_4_1_2"/><span class="lab"><font color="Green" title="E98">A326</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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> <font color="Maroon" title="c2">f</font></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="c5">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:34_2_1_1_1_3_1_5_1_1_1_4_1_2"/>
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <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="#E20:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E66">A260</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E3:34_2_1_1_1_3_1_5_1_1_1_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A327</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><a NAME="E4:34_2_1_1_1_3_1_5_1_1_1_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A328</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="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E67">A261</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_1_4_1_2"><span class="lab"><font color="Green" title="E98">A326</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E5:34_2_1_1_1_3_1_5_1_1_1_4_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><a NAME="E6:34_2_1_1_1_3_1_5_1_1_1_4_1_2"/><span class="kw">then </span>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E67">A261</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_1_4_1_2"><span class="lab"><font color="Green" title="E98">A326</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_1_4_1_2"><span class="lab"><font color="Green" title="E99">A327</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E7:34_2_1_1_1_3_1_5_1_1_1_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A329</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="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E55">A238</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_3_1_5_1_1_1_4_1_2"><span class="lab"><font color="Green" title="E100">A328</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><a NAME="E8:34_2_1_1_1_3_1_5_1_1_1_4_1_2"/><span class="lab"><font color="Green" title="E102">A330</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="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A245</font></span></a>, <a class="txt" href="#E51:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E86">A280</font></span></a>, <a class="txt" href="#E28:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E74">A268</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E88">A282</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E9:34_2_1_1_1_3_1_5_1_1_1_4_1_2"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="#E16:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A245</font></span></a>, <a class="txt" href="#E51:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E86">A280</font></span></a>, <a class="txt" href="#E28:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E74">A268</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E88">A282</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E10:34_2_1_1_1_3_1_5_1_1_1_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A331</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="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:34_2_1_1_1_3_1_5_1_1_1_4_1_2"><span class="lab"><font color="Green" title="E102">A330</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><a NAME="E11:34_2_1_1_1_3_1_5_1_1_1_4_1_2"/><span class="lab"><font color="Green" title="E104">A332</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="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</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="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a></span>;<br/><a NAME="E12:34_2_1_1_1_3_1_5_1_1_1_4_1_2"/><span class="lab"><font color="Green" title="E105">A333</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</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">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E69">A263</font></span></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/><a NAME="E13:34_2_1_1_1_3_1_5_1_1_1_4_1_2"/><span class="lab"><font color="Green" title="E106">A334</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="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a></span>;<br/><a NAME="E14:34_2_1_1_1_3_1_5_1_1_1_4_1_2"/><span class="lab"><font color="Green" title="E107">A335</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</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#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E15:34_2_1_1_1_3_1_5_1_1_1_4_1_2"/><span class="lab"><font color="Green" title="E108">A336</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</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> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E16:34_2_1_1_1_3_1_5_1_1_1_4_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E69">A263</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_1_4_1_2"><span class="lab"><font color="Green" title="E98">A326</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E17:34_2_1_1_1_3_1_5_1_1_1_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A337</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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">D2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:34_2_1_1_1_3_1_5_1_1_1_4_1_2"><span class="lab"><font color="Green" title="E107">A335</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E18:34_2_1_1_1_3_1_5_1_1_1_4_1_2"/><span class="kw">then </span>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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"><font color="Maroon" title="c2">f</font> <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="c6">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</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="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/><a NAME="E19:34_2_1_1_1_3_1_5_1_1_1_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A338</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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> <font color="Maroon" title="c2">f</font></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="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E17:34_2_1_1_1_3_1_5_1_1_1_4_1_2"><span class="lab"><font color="Green" title="E109">A337</font></span></a>, <a class="txt" href="#E13:34_2_1_1_1_3_1_5_1_1_1_4_1_2"><span class="lab"><font color="Green" title="E106">A334</font></span></a>, <a class="ref" href="integra3.html#T18" target="_self" title="INTEGRA3:th.18">Th18</a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/><a NAME="E20:34_2_1_1_1_3_1_5_1_1_1_4_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <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="#E1:34_2_1_1_1_3_1_5_1_1_1_4_1_2"><span class="lab"><font color="Green" title="E98">A326</font></span></a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">A339</font></span>: <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2 = 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="#E14:34_2_1_1_1_3_1_5_1_1_1_4_1_2"><span class="lab"><font color="Green" title="E107">A335</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_3_1_5_1_1_1_4_1_2"><span class="lab"><font color="Green" title="E108">A336</font></span></a>, <a class="txt" href="#E12:34_2_1_1_1_3_1_5_1_1_1_4_1_2"><span class="lab"><font color="Green" title="E105">A333</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>

;<br/><a NAME="E22:34_2_1_1_1_3_1_5_1_1_1_4_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_1_4_1_2"><span class="lab"><font color="Green" title="E98">A326</font></span></a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 = 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="#E1:34_2_1_1_1_3_1_5_1_1_1_4_1_2"><span class="lab"><font color="Green" title="E98">A326</font></span></a>, <a class="txt" href="#E14:34_2_1_1_1_3_1_5_1_1_1_4_1_2"><span class="lab"><font color="Green" title="E107">A335</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_3_1_5_1_1_1_4_1_2"><span class="lab"><font color="Green" title="E108">A336</font></span></a>, <a class="txt" href="#E12:34_2_1_1_1_3_1_5_1_1_1_4_1_2"><span class="lab"><font color="Green" title="E105">A333</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>

;<br/><a NAME="E24:34_2_1_1_1_3_1_5_1_1_1_4_1_2"/><span class="kw">then </span>
 <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:34_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E95">A316</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_1_4_1_2"><span class="lab"><font color="Green" title="E98">A326</font></span></a>, <a class="txt" href="#E21:34_2_1_1_1_3_1_5_1_1_1_4_1_2"><span class="lab"><font color="Green" title="E111">A339</font></span></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/><a NAME="E25:34_2_1_1_1_3_1_5_1_1_1_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E112">A340</font></span>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T77" title="RVSUM_1:th.77">RVSUM_1:77</a></span>;<br/><a NAME="E26:34_2_1_1_1_3_1_5_1_1_1_4_1_2"/><span class="lab"><font color="Green" title="E113">A341</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="#E20:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E66">A260</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E27:34_2_1_1_1_3_1_5_1_1_1_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">A342</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="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:34_2_1_1_1_3_1_5_1_1_1_4_1_2"><span class="lab"><font color="Green" title="E109">A337</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E28:34_2_1_1_1_3_1_5_1_1_1_4_1_2"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="#E17:34_2_1_1_1_3_1_5_1_1_1_4_1_2"><span class="lab"><font color="Green" title="E109">A337</font></span></a>, <a class="txt" href="#E26:34_2_1_1_1_3_1_5_1_1_1_4_1_2"><span class="lab"><font color="Green" title="E113">A341</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E29:34_2_1_1_1_3_1_5_1_1_1_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">A343</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="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E80">A274</font></span></a>, <a class="txt" href="#E27:34_2_1_1_1_3_1_5_1_1_1_4_1_2"><span class="lab"><font color="Green" title="E114">A342</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><a NAME="E30:34_2_1_1_1_3_1_5_1_1_1_4_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <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"><font color="Maroon" title="c2">f</font> <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="c6">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>)</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="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E67">A261</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_1_4_1_2"><span class="lab"><font color="Green" title="E98">A326</font></span></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/><a NAME="E31:34_2_1_1_1_3_1_5_1_1_1_4_1_2"/><span class="kw">then </span>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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> <font color="Maroon" title="c2">f</font></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="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E21:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E67">A261</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_1_4_1_2"><span class="lab"><font color="Green" title="E98">A326</font></span></a>, <a class="txt" href="#E11:34_2_1_1_1_3_1_5_1_1_1_4_1_2"><span class="lab"><font color="Green" title="E104">A332</font></span></a>, <a class="ref" href="integra3.html#T18" target="_self" title="INTEGRA3:th.18">Th18</a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/><a NAME="E32:34_2_1_1_1_3_1_5_1_1_1_4_1_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"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</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"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span></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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>)</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:34_2_1_1_1_3_1_5_1_1_1_4_1_2"><span class="lab"><font color="Green" title="E112">A340</font></span></a>, <a class="txt" href="#E19:34_2_1_1_1_3_1_5_1_1_1_4_1_2"><span class="lab"><font color="Green" title="E110">A338</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E33:34_2_1_1_1_3_1_5_1_1_1_4_1_2"/>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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> <font color="Maroon" title="c2">f</font></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="c5">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E88">A282</font></span></a>, <a class="txt" href="#E29:34_2_1_1_1_3_1_5_1_1_1_4_1_2"><span class="lab"><font color="Green" title="E115">A343</font></span></a>, <a class="txt" href="#E7:34_2_1_1_1_3_1_5_1_1_1_4_1_2"><span class="lab"><font color="Green" title="E101">A329</font></span></a>, <a class="txt" href="#E10:34_2_1_1_1_3_1_5_1_1_1_4_1_2"><span class="lab"><font color="Green" title="E103">A331</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 NAME="E15:34_2_1_1_1_3_1_5_1_1_1"/><span class="lab"><font color="Green" title="E98">A344</font></span>: 
<font color="Maroon" title="c11">n1</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">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E73">A267</font></span></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="E99">A345</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c10">j</font>)</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="#E15:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E61">A244</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E88">A282</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">j</font></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="#E1:34_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E88">A282</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>
<br/>.= 
1

;<br/><a NAME="E17:34_2_1_1_1_3_1_5_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="c10">j</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="#E17:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A257</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E18:34_2_1_1_1_3_1_5_1_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <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/><span class="kw">then </span><span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 = 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>) <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <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> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E61">A244</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E88">A282</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E98">A344</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>
<br/>.= 
<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"><font color="Maroon" title="c2">f</font> <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="c5">D1</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</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="c5">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E74">A268</font></span></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>
;<br/><a NAME="E20:34_2_1_1_1_3_1_5_1_1_1"/><span class="kw">then </span>
 <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c10">j</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"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p0">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></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="#E16:34_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E99">A345</font></span></a>, <a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>;<br/><a NAME="E21:34_2_1_1_1_3_1_5_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A346</font></span>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c10">j</font>)</span>)</span> <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"><font color="Maroon" title="c2">f</font> <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="c5">D1</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</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="c5">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finsop_1.html#T11" title="FINSOP_1:th.11">FINSOP_1:11</a></span>;<br/><a NAME="E22:34_2_1_1_1_3_1_5_1_1_1"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <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="#E28:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E74">A268</font></span></a>, <a class="ref" href="integra1.html#T8" title="INTEGRA1:th.8">INTEGRA1:8</a></span>;<br/><a NAME="E23:34_2_1_1_1_3_1_5_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A347</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="c5">D1</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span></span>)</span> <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> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="integra3.html#E15"><span class="lab"><font color="Green" title="E13">Lm4</font></span></a></span>;<br/><a NAME="E24:34_2_1_1_1_3_1_5_1_1_1"/>
<font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="c5">D1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E74">A268</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="E25:34_2_1_1_1_3_1_5_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="c5">D1</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="E26:34_2_1_1_1_3_1_5_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A348</font></span>: 
<font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="c5">D1</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="E27:34_2_1_1_1_3_1_5_1_1_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="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</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="c5">D1</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E74">A268</font></span></a>, <a class="ref" href="integra1.html#T20" title="INTEGRA1:th.20">INTEGRA1:20</a></span>;<br/><a NAME="E28:34_2_1_1_1_3_1_5_1_1_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="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</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="c5">D1</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:34_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E102">A348</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="E29:34_2_1_1_1_3_1_5_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A349</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="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</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="c5">D1</font>
 <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="E30:34_2_1_1_1_3_1_5_1_1_1"/>
<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> <font color="Maroon" title="c2">f</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_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> <font color="Maroon" title="c2">f</font></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="#E1:34"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="integra3.html#E14"><span class="lab"><font color="Green" title="E12">Lm3</font></span></a>, <a class="ref" href="xreal_1.html#T48" title="XREAL_1:th.48">XREAL_1:48</a></span>;<br/><a NAME="E31:34_2_1_1_1_3_1_5_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A350</font></span>: 
<span class="p1">(<span class="default"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></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="c5">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</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"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></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="c5">D1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:34_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E103">A349</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/><a NAME="E32:34_2_1_1_1_3_1_5_1_1_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="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a></span>;<br/><a NAME="E33:34_2_1_1_1_3_1_5_1_1_1"/><span class="kw">then </span>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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> <font color="Maroon" title="c2">f</font></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="c5">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:34_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E100">A346</font></span></a>, <a class="txt" href="#E23:34_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E101">A347</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/><a NAME="E34:34_2_1_1_1_3_1_5_1_1_1"/><span class="kw">then </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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</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"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c10">j</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"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span></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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:34_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E97">A318</font></span></a>, <a class="ref" href="xreal_1.html#T13" title="XREAL_1:th.13">XREAL_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E35:34_2_1_1_1_3_1_5_1_1_1"/>
<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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</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"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c10">j</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"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></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="c5">D1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:34_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E104">A350</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_3_1_5_1_1_2"/><span class="lab"><font color="Green" title="E88">A351</font></span>: 
<font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">j</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="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</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"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c10">j</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"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></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="c5">D1</font></span>)</span></span><br/><div class="add"><a NAME="E2:34_2_1_1_1_3_1_5_1_1_2"/><span class="lab"><font color="Green" title="E89">A352</font></span>: 
<font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">n1</font> <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="E3:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A353</font></span>: 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E58">A241</font></span></a>, <a class="txt" href="#E28:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E74">A268</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">B</font> being   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>, <font color="Maroon" title="c13">MD1</font>, <font color="Maroon" title="c14">MD2</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c12">B</font><span class="kw"> such that </span><br/><a NAME="E5:34_2_1_1_1_3_1_5_1_1_2"/><span class="lab"><font color="Green" title="E91">A354</font></span>: 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c12">B</font>
 <span class="kw">and </span><br/><a NAME="E6:34_2_1_1_1_3_1_5_1_1_2"/>
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c12">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">MD2</font> <a href="seq_1.html#K1" title="SEQ_1:func.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="c14">MD2</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:34_2_1_1_1_3_1_5_1_1_2"/><span class="lab"><font color="Green" title="E92">A355</font></span>: 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c12">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">MD1</font> <a href="seq_1.html#K1" title="SEQ_1:func.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="c13">MD1</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:34_2_1_1_1_3_1_5_1_1_2"/><span class="lab"><font color="Green" title="E93">A356</font></span>: 
<font color="Maroon" title="c13">MD1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Maroon" title="c14">MD2</font>
 <span class="kw">and </span><br/><a NAME="E9:34_2_1_1_1_3_1_5_1_1_2"/><span class="lab"><font color="Green" title="E94">A357</font></span>: 
<font color="Maroon" title="c13">MD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c10">j</font>)
 <span class="kw">and </span><br/><a NAME="E10:34_2_1_1_1_3_1_5_1_1_2"/><span class="lab"><font color="Green" title="E95">A358</font></span>: 
<font color="Maroon" title="c14">MD2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c6">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</font></span></a>, <a class="txt" href="#E17:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A257</font></span></a>, <a class="txt" href="#E28:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E74">A268</font></span></a>, <a class="txt" href="#E36:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E79">A273</font></span></a>, <a class="ref" href="integra3.html#T15" target="_self" title="INTEGRA3:th.15">Th15</a></span>;<br/><a NAME="E11:34_2_1_1_1_3_1_5_1_1_2"/><span class="lab"><font color="Green" title="E96">A359</font></span>: 
 <a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c13">MD1</font> <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="integra3.html#T9" target="_self" title="INTEGRA3:th.9">Th9</a></span>;<br/><a NAME="E12:34_2_1_1_1_3_1_5_1_1_2"/><span class="lab"><font color="Green" title="E97">A360</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="#E17:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A257</font></span></a>, <a class="txt" href="#E25:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E71">A265</font></span></a>, <a class="txt" href="#E26:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E61">A244</font></span></a>, <a class="txt" href="#E27:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E73">A267</font></span></a>, <a class="txt" href="#E9:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E94">A357</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A361</font></span>: <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="c13">MD1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="#E17:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A257</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>
<br/>.= 
<font color="Maroon" title="c10">j</font>

;<br/><a NAME="E14:34_2_1_1_1_3_1_5_1_1_2"/>
<font color="Maroon" title="c10">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A257</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/><a NAME="E15:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A362</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">n1</font>
 ;<br/><a NAME="E16:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A363</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A257</font></span></a>, <a class="txt" href="#E25:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E71">A265</font></span></a>, <a class="txt" href="#E26:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E61">A244</font></span></a>, <a class="txt" href="#E27:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E73">A267</font></span></a>, <a class="txt" href="#E9:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E94">A357</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>;<br/><a NAME="E17:34_2_1_1_1_3_1_5_1_1_2"/><span class="lab"><font color="Green" title="E101">A364</font></span>: 
<font color="Maroon" title="c12">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">x1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c15">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">B</font> or <font color="Maroon" title="c15">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )</span><br/>

<a NAME="E1:34_2_1_1_1_3_1_5_1_1_2_2"/><span class="lab"><font color="Green" title="E102">A365</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</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="ref" href="integra1.html#D2" title="INTEGRA1:def.2">INTEGRA1:def 2</a></span>;<br/>
<a NAME="E2:34_2_1_1_1_3_1_5_1_1_2_2"/>
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E58">A241</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="E3:34_2_1_1_1_3_1_5_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A366</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> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_2"><span class="lab"><font color="Green" title="E102">A365</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:34_2_1_1_1_3_1_5_1_1_2_2"/><span class="lab"><font color="Green" title="E104">A367</font></span>: 
<font color="Maroon" title="c15">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">B</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="c15">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">x1</font> = <font color="Maroon" title="c15">x1</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E6:34_2_1_1_1_3_1_5_1_1_2_2"/><span class="lab"><font color="Green" title="E105">A368</font></span>: 
<font color="Maroon" title="c16">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">MD1</font> <a href="seq_1.html#K1" title="SEQ_1:func.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="c13">MD1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E92">A355</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_3_1_5_1_1_2_2"><span class="lab"><font color="Green" title="E104">A367</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/>
<a NAME="E7:34_2_1_1_1_3_1_5_1_1_2_2"/>
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</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="E8:34_2_1_1_1_3_1_5_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A369</font></span>: 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">j</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="#E1:34_2_1_1_1_3_1_5_1_1_2_2"><span class="lab"><font color="Green" title="E102">A365</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/>
<a NAME="E9:34_2_1_1_1_3_1_5_1_1_2_2"/>
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E91">A354</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_3_1_5_1_1_2_2"><span class="lab"><font color="Green" title="E104">A367</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/>
<a NAME="E10:34_2_1_1_1_3_1_5_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A370</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> <font color="Maroon" title="c16">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_2_2"><span class="lab"><font color="Green" title="E103">A366</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<font color="Maroon" title="c13">MD1</font> <a href="seq_1.html#K1" title="SEQ_1:func.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="c13">MD1</font></span>)</span> = 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">n1</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> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A257</font></span></a>, <a class="txt" href="#E52:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E87">A281</font></span></a>, <a class="txt" href="#E26:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E61">A244</font></span></a>, <a class="txt" href="#E9:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E94">A357</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E99">A362</font></span></a>, <a class="txt" href="#E16:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E100">A363</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>
<br/>.= 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">j</font>

;<br/>
<a NAME="E12:34_2_1_1_1_3_1_5_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">x1</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="#E6:34_2_1_1_1_3_1_5_1_1_2_2"><span class="lab"><font color="Green" title="E105">A368</font></span></a>, <a class="txt" href="#E8:34_2_1_1_1_3_1_5_1_1_2_2"><span class="lab"><font color="Green" title="E106">A369</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:34_2_1_1_1_3_1_5_1_1_2_2"/>
<font color="Maroon" title="c15">x1</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="#E10:34_2_1_1_1_3_1_5_1_1_2_2"><span class="lab"><font color="Green" title="E107">A370</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">g</font> = <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c12">B</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c12">B</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T32" title="FUNCT_2:th.32">FUNCT_2:32</a></span>;<br/><span class="lab"><font color="Green" title="E102">A371</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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c13">MD1</font>)</span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="#E17:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A257</font></span></a>, <a class="txt" href="#E25:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E71">A265</font></span></a>, <a class="txt" href="#E26:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E61">A244</font></span></a>, <a class="txt" href="#E27:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E73">A267</font></span></a>, <a class="txt" href="#E9:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E94">A357</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="#E17:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A257</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>
;<br/><a NAME="E20:34_2_1_1_1_3_1_5_1_1_2"/><span class="lab"><font color="Green" title="E103">A372</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</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="c13">MD1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></span>;<br/><a NAME="E21:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A373</font></span>: 
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="c13">MD1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E22:34_2_1_1_1_3_1_5_1_1_2"/><span class="lab"><font color="Green" title="E105">A374</font></span>: 
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</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="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_2_4"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_2_4_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="E1:34_2_1_1_1_3_1_5_1_1_2_4_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</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="34_2_1_1_1_3_1_5_1_1_2_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_3_1_5_1_1_2_4_1_1"/><span class="lab"><font color="Green" title="E106">A375</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</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#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</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="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> )</span><br/><div class="add"><a NAME="E2:34_2_1_1_1_3_1_5_1_1_2_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A376</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="c13">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">MD1</font> <a href="seq_1.html#K1" title="SEQ_1:func.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="c13">MD1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E103">A372</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E3:34_2_1_1_1_3_1_5_1_1_2_4_1_1"/><span class="lab"><font color="Green" title="E108">A377</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="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</font></span></a>, <a class="txt" href="#E16:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A245</font></span></a>, <a class="txt" href="#E51:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E86">A280</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E4:34_2_1_1_1_3_1_5_1_1_2_4_1_1"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <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="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</font></span></a>, <a class="txt" href="#E16:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A245</font></span></a>, <a class="txt" href="#E51:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E86">A280</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="E5:34_2_1_1_1_3_1_5_1_1_2_4_1_1"/>
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</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="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E71">A265</font></span></a>, <a class="txt" href="#E26:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="txt" href="#E5:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E91">A354</font></span></a>, <a class="txt" href="#E9:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E94">A357</font></span></a>, <a class="txt" href="#E13:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E98">A361</font></span></a>, <a class="txt" href="#E20:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E103">A372</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_4_1_1"><span class="lab"><font color="Green" title="E106">A375</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_3_1_5_1_1_2_4_1_1"><span class="lab"><font color="Green" title="E107">A376</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_2_4_1_1"><span class="lab"><font color="Green" title="E108">A377</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</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="34_2_1_1_1_3_1_5_1_1_2_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_3_1_5_1_1_2_4_1_2"/><span class="lab"><font color="Green" title="E106">A378</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</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#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</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="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> )</span><br/><div class="add"><a NAME="E2:34_2_1_1_1_3_1_5_1_1_2_4_1_2"/><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="c13">MD1</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="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">MD1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E103">A372</font></span></a>, <a class="ref" href="integra1.html#T7" title="INTEGRA1:th.7">INTEGRA1:7</a></span>;<br/><a NAME="E3:34_2_1_1_1_3_1_5_1_1_2_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A379</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E4:34_2_1_1_1_3_1_5_1_1_2_4_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</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="c13">MD1</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#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E5:34_2_1_1_1_3_1_5_1_1_2_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A380</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</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="c13">MD1</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="E6:34_2_1_1_1_3_1_5_1_1_2_4_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="c13">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">MD1</font> <a href="seq_1.html#K1" title="SEQ_1:func.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="c13">MD1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E103">A372</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_4_1_2"><span class="lab"><font color="Green" title="E106">A378</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E7:34_2_1_1_1_3_1_5_1_1_2_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A381</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="c13">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A257</font></span></a>, <a class="txt" href="#E26:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E61">A244</font></span></a>, <a class="txt" href="#E9:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E94">A357</font></span></a>, <a class="txt" href="#E12:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E97">A360</font></span></a>, <a class="txt" href="#E13:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E98">A361</font></span></a>, <a class="txt" href="#E21:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E104">A373</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>;<br/><a NAME="E8:34_2_1_1_1_3_1_5_1_1_2_4_1_2"/><span class="lab"><font color="Green" title="E110">A382</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</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> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j</font> <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="#E16:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E100">A363</font></span></a></span>;<br/><a NAME="E9:34_2_1_1_1_3_1_5_1_1_2_4_1_2"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">MD1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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="c13">MD1</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="#E20:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E103">A372</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_4_1_2"><span class="lab"><font color="Green" title="E106">A378</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E10:34_2_1_1_1_3_1_5_1_1_2_4_1_2"/><span class="kw">then </span>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <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="#E17:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A257</font></span></a>, <a class="txt" href="#E26:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E61">A244</font></span></a>, <a class="txt" href="#E9:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E94">A357</font></span></a>, <a class="txt" href="#E12:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E97">A360</font></span></a>, <a class="txt" href="#E8:34_2_1_1_1_3_1_5_1_1_2_4_1_2"><span class="lab"><font color="Green" title="E110">A382</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_2_4_1_2"><span class="lab"><font color="Green" title="E107">A379</font></span></a>, <a class="txt" href="#E5:34_2_1_1_1_3_1_5_1_1_2_4_1_2"><span class="lab"><font color="Green" title="E108">A380</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>;<br/><span class="kw">hence </span><a NAME="E11:34_2_1_1_1_3_1_5_1_1_2_4_1_2"/>
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</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="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</font></span></a>, <a class="txt" href="#E16:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A245</font></span></a>, <a class="txt" href="#E51:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E86">A280</font></span></a>, <a class="txt" href="#E7:34_2_1_1_1_3_1_5_1_1_2_4_1_2"><span class="lab"><font color="Green" title="E109">A381</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 NAME="E23:34_2_1_1_1_3_1_5_1_1_2"/><span class="lab"><font color="Green" title="E106">A383</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</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="c13">MD1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></span>;<br/><a NAME="E24:34_2_1_1_1_3_1_5_1_1_2"/><span class="lab"><font color="Green" title="E107">A384</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="c13">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">MD1</font> <a href="seq_1.html#K1" title="SEQ_1:func.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="c13">MD1</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_2_5"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_2_5_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="E1:34_2_1_1_1_3_1_5_1_1_2_5_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</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="34_2_1_1_1_3_1_5_1_1_2_5_1_1"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_3_1_5_1_1_2_5_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</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="c13">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">MD1</font> <a href="seq_1.html#K1" title="SEQ_1:func.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="c13">MD1</font></span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:34_2_1_1_1_3_1_5_1_1_2_5_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="c13">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">MD1</font> <a href="seq_1.html#K1" title="SEQ_1:func.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="c13">MD1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E106">A383</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="34_2_1_1_1_3_1_5_1_1_2_5_1_2"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_3_1_5_1_1_2_5_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</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="c13">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">MD1</font> <a href="seq_1.html#K1" title="SEQ_1:func.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="c13">MD1</font></span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:34_2_1_1_1_3_1_5_1_1_2_5_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="c13">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">MD1</font> <a href="seq_1.html#K1" title="SEQ_1:func.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="c13">MD1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E106">A383</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 NAME="E25:34_2_1_1_1_3_1_5_1_1_2"/>
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E58">A241</font></span></a>, <a class="txt" href="#E28:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E74">A268</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E89">A352</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/><a NAME="E26:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E65">A259</font></span></a>, <a class="txt" href="#E37:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E80">A274</font></span></a>, <a class="txt" href="#E29:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E75">A269</font></span></a>, <a class="txt" href="#E31:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E77">A271</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/><a NAME="E27:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A385</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</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>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
 <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="E28:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A386</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E83">A277</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E29:34_2_1_1_1_3_1_5_1_1_2"/>
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c12">B</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> <font color="Maroon" title="c12">B</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E91">A354</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><a NAME="E30:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span>
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c12">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</font></span></a>, <a class="txt" href="#E16:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A245</font></span></a>, <a class="txt" href="#E51:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E86">A280</font></span></a>, <a class="txt" href="#E7:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E92">A355</font></span></a>, <a class="txt" href="#E22:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E105">A374</font></span></a>, <a class="txt" href="#E24:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E107">A384</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E31:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A387</font></span>: 
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c12">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</font></span></a>, <a class="txt" href="#E12:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E58">A241</font></span></a>, <a class="txt" href="#E16:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A245</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/><a NAME="E32:34_2_1_1_1_3_1_5_1_1_2"/><span class="lab"><font color="Green" title="E111">A388</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</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="#E20:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E66">A260</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E33:34_2_1_1_1_3_1_5_1_1_2"/><span class="lab"><font color="Green" title="E112">A389</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</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="E34:34_2_1_1_1_3_1_5_1_1_2"/><span class="lab"><font color="Green" title="E113">A390</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_2_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:34_2_1_1_1_3_1_5_1_1_2_6"/>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:34_2_1_1_1_3_1_5_1_1_2_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">A391</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</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="#E27:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E108">A385</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E3:34_2_1_1_1_3_1_5_1_1_2_6"/><span class="lab"><font color="Green" title="E115">A392</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E111">A388</font></span></a>, <a class="txt" href="#E28:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E109">A386</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E4:34_2_1_1_1_3_1_5_1_1_2_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A393</font></span>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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> <font color="Maroon" title="c6">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_2_6_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_2_6_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="E1:34_2_1_1_1_3_1_5_1_1_2_6_1_1"/>
( <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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> <font color="Maroon" title="c5">D1</font> or <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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> <font color="Maroon" title="c4">D</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1"><span class="lab"><font color="Green" title="E41">A14</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_3_1_5_1_1_2_6"><span class="lab"><font color="Green" title="E116">A393</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_2_6_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_3_1_5_1_1_2_6_1_1_1"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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> <font color="Maroon" title="c5">D1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">n2</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="E3:34_2_1_1_1_3_1_5_1_1_2_6_1_1_1"/><span class="lab"><font color="Green" title="E117">A394</font></span>: 
<font color="Maroon" title="c16">n2</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="c5">D1</font>
 <span class="kw">and </span><br/><a NAME="E4:34_2_1_1_1_3_1_5_1_1_2_6_1_1_1"/><span class="lab"><font color="Green" title="E118">A395</font></span>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">n2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T3" title="PARTFUN1:th.3">PARTFUN1:3</a></span>;<br/><a NAME="E5:34_2_1_1_1_3_1_5_1_1_2_6_1_1_1"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E65">A259</font></span></a>, <a class="txt" href="#E33:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E112">A389</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_2_6"><span class="lab"><font color="Green" title="E115">A392</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/><a NAME="E6:34_2_1_1_1_3_1_5_1_1_2_6_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">n2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E58">A241</font></span></a>, <a class="txt" href="#E37:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E80">A274</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_2_6_1_1_1"><span class="lab"><font color="Green" title="E117">A394</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_3_1_5_1_1_2_6_1_1_1"><span class="lab"><font color="Green" title="E118">A395</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/><a NAME="E7:34_2_1_1_1_3_1_5_1_1_2_6_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">A396</font></span>: 
<font color="Maroon" title="c11">n1</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> <font color="Maroon" title="c16">n2</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="E8:34_2_1_1_1_3_1_5_1_1_2_6_1_1_1"/>
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">n2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E75">A269</font></span></a>, <a class="txt" href="#E31:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E77">A271</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_3_1_5_1_1_2_6"><span class="lab"><font color="Green" title="E114">A391</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_2_6"><span class="lab"><font color="Green" title="E115">A392</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_3_1_5_1_1_2_6_1_1_1"><span class="lab"><font color="Green" title="E118">A395</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E9:34_2_1_1_1_3_1_5_1_1_2_6_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E74">A268</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_2_6_1_1_1"><span class="lab"><font color="Green" title="E117">A394</font></span></a>, <a class="txt" href="#E7:34_2_1_1_1_3_1_5_1_1_2_6_1_1_1"><span class="lab"><font color="Green" title="E119">A396</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</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="34_2_1_1_1_3_1_5_1_1_2_6_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_3_1_5_1_1_2_6_1_1_2"/><span class="lab"><font color="Green" title="E117">A397</font></span>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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> <font color="Maroon" title="c4">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:34_2_1_1_1_3_1_5_1_1_2_6_1_1_2"/><span class="lab"><font color="Green" title="E118">A398</font></span>: 
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</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> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E59">A242</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/><a NAME="E3:34_2_1_1_1_3_1_5_1_1_2_6_1_1_2"/><span class="lab"><font color="Green" title="E119">A399</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="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_2_6_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_2_6_1_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="E1:34_2_1_1_1_3_1_5_1_1_2_6_1_1_2_1_1"/>
( <font color="Maroon" title="c11">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c11">n1</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="34_2_1_1_1_3_1_5_1_1_2_6_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_3_1_5_1_1_2_6_1_1_2_1_1_1"/>
<font color="Maroon" title="c11">n1</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="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n1</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:34_2_1_1_1_3_1_5_1_1_2_6_1_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="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E58">A241</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="34_2_1_1_1_3_1_5_1_1_2_6_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_3_1_5_1_1_2_6_1_1_2_1_1_2"/>
<font color="Maroon" title="c11">n1</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="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n1</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:34_2_1_1_1_3_1_5_1_1_2_6_1_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="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E58">A241</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">consider </span><font color="Maroon" title="c16">n2</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="E5:34_2_1_1_1_3_1_5_1_1_2_6_1_1_2"/><span class="lab"><font color="Green" title="E120">A400</font></span>: 
<font color="Maroon" title="c16">n2</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">D</font>
 <span class="kw">and </span><br/><a NAME="E6:34_2_1_1_1_3_1_5_1_1_2_6_1_1_2"/><span class="lab"><font color="Green" title="E121">A401</font></span>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">n2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_6_1_1_2"><span class="lab"><font color="Green" title="E117">A397</font></span></a>, <a class="ref" href="partfun1.html#T3" title="PARTFUN1:th.3">PARTFUN1:3</a></span>;<br/><a NAME="E7:34_2_1_1_1_3_1_5_1_1_2_6_1_1_2"/>
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">n2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E65">A259</font></span></a>, <a class="txt" href="#E37:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E80">A274</font></span></a>, <a class="txt" href="#E33:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E112">A389</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_2_6"><span class="lab"><font color="Green" title="E115">A392</font></span></a>, <a class="txt" href="#E6:34_2_1_1_1_3_1_5_1_1_2_6_1_1_2"><span class="lab"><font color="Green" title="E121">A401</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/><a NAME="E8:34_2_1_1_1_3_1_5_1_1_2_6_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">n2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_2_1_1_1_3_1_5_1_1_2_6_1_1_2"><span class="lab"><font color="Green" title="E118">A398</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_2_6_1_1_2"><span class="lab"><font color="Green" title="E119">A399</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E9:34_2_1_1_1_3_1_5_1_1_2_6_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">n2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E57">A240</font></span></a>, <a class="txt" href="#E5:34_2_1_1_1_3_1_5_1_1_2_6_1_1_2"><span class="lab"><font color="Green" title="E120">A400</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/><a NAME="E10:34_2_1_1_1_3_1_5_1_1_2_6_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E122">A402</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">n2</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="E11:34_2_1_1_1_3_1_5_1_1_2_6_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="c10">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E88">A351</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E12:34_2_1_1_1_3_1_5_1_1_2_6_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E123">A403</font></span>: 
<font color="Maroon" title="c11">n1</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> <font color="Maroon" title="c10">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <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="E13:34_2_1_1_1_3_1_5_1_1_2_6_1_1_2"/>
<font color="Maroon" title="c10">j</font> <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="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</font></span></a>, <a class="txt" href="#E16:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A245</font></span></a>, <a class="txt" href="#E51:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E86">A280</font></span></a>, <a class="ref" href="integra1.html#T7" title="INTEGRA1:th.7">INTEGRA1:7</a></span>;<br/><a NAME="E14:34_2_1_1_1_3_1_5_1_1_2_6_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A404</font></span>: 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <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="#E28:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E74">A268</font></span></a>, <a class="txt" href="#E12:34_2_1_1_1_3_1_5_1_1_2_6_1_1_2"><span class="lab"><font color="Green" title="E123">A403</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/><a NAME="E15:34_2_1_1_1_3_1_5_1_1_2_6_1_1_2"/><span class="lab"><font color="Green" title="E125">A405</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E54">A237</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/><a NAME="E16:34_2_1_1_1_3_1_5_1_1_2_6_1_1_2"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <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="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</font></span></a>, <a class="txt" href="#E16:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A245</font></span></a>, <a class="txt" href="#E51:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E86">A280</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E17:34_2_1_1_1_3_1_5_1_1_2_6_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E126">A406</font></span>: 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:34_2_1_1_1_3_1_5_1_1_2_6_1_1_2"><span class="lab"><font color="Green" title="E124">A404</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_3_1_5_1_1_2_6_1_1_2"><span class="lab"><font color="Green" title="E125">A405</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E18:34_2_1_1_1_3_1_5_1_1_2_6_1_1_2"/>
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">n2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E75">A269</font></span></a>, <a class="txt" href="#E31:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E77">A271</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_3_1_5_1_1_2_6"><span class="lab"><font color="Green" title="E114">A391</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_2_6"><span class="lab"><font color="Green" title="E115">A392</font></span></a>, <a class="txt" href="#E6:34_2_1_1_1_3_1_5_1_1_2_6_1_1_2"><span class="lab"><font color="Green" title="E121">A401</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/><a NAME="E19:34_2_1_1_1_3_1_5_1_1_2_6_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">n2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:34_2_1_1_1_3_1_5_1_1_2_6_1_1_2"><span class="lab"><font color="Green" title="E126">A406</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E20:34_2_1_1_1_3_1_5_1_1_2_6_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E52">A235</font></span></a>, <a class="txt" href="#E5:34_2_1_1_1_3_1_5_1_1_2_6_1_1_2"><span class="lab"><font color="Green" title="E120">A400</font></span></a>, <a class="txt" href="#E10:34_2_1_1_1_3_1_5_1_1_2_6_1_1_2"><span class="lab"><font color="Green" title="E122">A402</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</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="E6:34_2_1_1_1_3_1_5_1_1_2_6"/>
contradiction
 ; <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="E35:34_2_1_1_1_3_1_5_1_1_2"/><span class="lab"><font color="Green" title="E114">A407</font></span>: 
<font color="Maroon" title="c10">j</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">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/><a NAME="E36:34_2_1_1_1_3_1_5_1_1_2"/><span class="lab"><font color="Green" title="E115">A408</font></span>: 
 for <font color="Olive" title="b1">k</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">k</font> &amp; <font color="Olive" title="b1">k</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> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c13">MD1</font>)</span>)</span> holds <br/><span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c13">MD1</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c10">j</font>)</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="34_2_1_1_1_3_1_5_1_1_2_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">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"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">k</font> &amp; <font color="Maroon" title="c16">k</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> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c13">MD1</font>)</span>)</span> implies <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c13">MD1</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">k</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:34_2_1_1_1_3_1_5_1_1_2_7"/><span class="lab"><font color="Green" title="E116">A409</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">k</font>
 <span class="kw">and </span><br/><a NAME="E2:34_2_1_1_1_3_1_5_1_1_2_7"/><span class="lab"><font color="Green" title="E117">A410</font></span>: 
<font color="Maroon" title="c16">k</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> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c13">MD1</font>)</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"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c13">MD1</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">k</font></span><br/>

<a NAME="E3:34_2_1_1_1_3_1_5_1_1_2_7"/><span class="lab"><font color="Green" title="E118">A411</font></span>: 
<font color="Maroon" title="c16">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c13">MD1</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E116">A409</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E117">A410</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E4:34_2_1_1_1_3_1_5_1_1_2_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">A412</font></span>: 
<font color="Maroon" title="c16">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="c13">MD1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/>
<a NAME="E5:34_2_1_1_1_3_1_5_1_1_2_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E120">A413</font></span>: 
<font color="Maroon" title="c16">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="c13">MD1</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="E6:34_2_1_1_1_3_1_5_1_1_2_7"/>
<font color="Maroon" title="c16">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="c13">MD1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E119">A412</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="E7:34_2_1_1_1_3_1_5_1_1_2_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E121">A414</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c13">MD1</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">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"><font color="Maroon" title="c15">g</font> <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="c13">MD1</font>,<font color="Maroon" title="c16">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="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c17">k2</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="E9:34_2_1_1_1_3_1_5_1_1_2_7"/><span class="lab"><font color="Green" title="E122">A415</font></span>: 
<font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="Maroon" title="c17">k2</font>
 ;<br/>
<a NAME="E10:34_2_1_1_1_3_1_5_1_1_2_7"/><span class="lab"><font color="Green" title="E123">A416</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E116">A409</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E11:34_2_1_1_1_3_1_5_1_1_2_7"/>
<font color="Maroon" title="c16">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="#E19:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E102">A371</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E117">A410</font></span></a></span>;<br/>
<a NAME="E12:34_2_1_1_1_3_1_5_1_1_2_7"/><span class="kw">then </span>
<font color="Maroon" title="c16">k</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</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="E13:34_2_1_1_1_3_1_5_1_1_2_7"/><span class="kw">then </span>
<font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">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="c5">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="txt" href="#E9:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E122">A415</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E14:34_2_1_1_1_3_1_5_1_1_2_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A417</font></span>: 
<font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">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="c5">D1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E123">A416</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E15:34_2_1_1_1_3_1_5_1_1_2_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E125">A418</font></span>: 
<font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">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="c5">D1</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="E16:34_2_1_1_1_3_1_5_1_1_2_7"/>
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="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E64">A258</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E116">A409</font></span></a>, <a class="txt" href="#E9:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E122">A415</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<a NAME="E17:34_2_1_1_1_3_1_5_1_1_2_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E126">A419</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</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="E18:34_2_1_1_1_3_1_5_1_1_2_7"/><span class="lab"><font color="Green" title="E127">A420</font></span>: 
<font color="Maroon" title="c17">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="#E9:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E122">A415</font></span></a></span>;<br/>
<a NAME="E19:34_2_1_1_1_3_1_5_1_1_2_7"/><span class="lab"><font color="Green" title="E128">A421</font></span>: 
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)</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="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)</span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_2_7_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_2_7_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="E1:34_2_1_1_1_3_1_5_1_1_2_7_1_1"/>
( <font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c16">k</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="34_2_1_1_1_3_1_5_1_1_2_7_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_3_1_5_1_1_2_7_1_1_1"/><span class="lab"><font color="Green" title="E129">A422</font></span>: 
<font color="Maroon" title="c16">k</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#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)</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="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)</span>)</span> )</span><br/><div class="add"><a NAME="E2:34_2_1_1_1_3_1_5_1_1_2_7_1_1_1"/><span class="kw">then </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="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">MD1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E120">A413</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E3:34_2_1_1_1_3_1_5_1_1_2_7_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E130">A423</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="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="#E17:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A257</font></span></a>, <a class="txt" href="#E26:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E61">A244</font></span></a>, <a class="txt" href="#E9:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E94">A357</font></span></a>, <a class="txt" href="#E19:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E102">A371</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E116">A409</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E117">A410</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E118">A411</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>;<br/><a NAME="E4:34_2_1_1_1_3_1_5_1_1_2_7_1_1_1"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E91">A354</font></span></a>, <a class="txt" href="#E5:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E120">A413</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_7_1_1_1"><span class="lab"><font color="Green" title="E129">A422</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="E5:34_2_1_1_1_3_1_5_1_1_2_7_1_1_1"/>
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)</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="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E127">A420</font></span></a>, <a class="txt" href="#E17:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E126">A419</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E125">A418</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_7_1_1_1"><span class="lab"><font color="Green" title="E129">A422</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_2_7_1_1_1"><span class="lab"><font color="Green" title="E130">A423</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="34_2_1_1_1_3_1_5_1_1_2_7_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_3_1_5_1_1_2_7_1_1_2"/><span class="lab"><font color="Green" title="E129">A424</font></span>: 
<font color="Maroon" title="c16">k</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#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)</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="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)</span>)</span> )</span><br/><div class="add"><a NAME="E2:34_2_1_1_1_3_1_5_1_1_2_7_1_1_2"/><span class="kw">then </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="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">MD1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E120">A413</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E3:34_2_1_1_1_3_1_5_1_1_2_7_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E130">A425</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="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="#E17:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A257</font></span></a>, <a class="txt" href="#E26:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E61">A244</font></span></a>, <a class="txt" href="#E9:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E94">A357</font></span></a>, <a class="txt" href="#E19:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E102">A371</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E116">A409</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E117">A410</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E118">A411</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>;<br/><a NAME="E4:34_2_1_1_1_3_1_5_1_1_2_7_1_1_2"/><span class="lab"><font color="Green" title="E131">A426</font></span>: 
<font color="Maroon" title="c16">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="#E19:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E102">A371</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E117">A410</font></span></a>, <a class="ref" href="xreal_1.html#T146" title="XREAL_1:th.146">XREAL_1:146</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E5:34_2_1_1_1_3_1_5_1_1_2_7_1_1_2"/><span class="lab"><font color="Green" title="E132">A427</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">MD1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">k</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="#E5:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E120">A413</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_7_1_1_2"><span class="lab"><font color="Green" title="E129">A424</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E6:34_2_1_1_1_3_1_5_1_1_2_7_1_1_2"/><span class="lab"><font color="Green" title="E133">A428</font></span>: 
<font color="Maroon" title="c16">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <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="c13">MD1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E120">A413</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_7_1_1_2"><span class="lab"><font color="Green" title="E129">A424</font></span></a>, <a class="ref" href="integra1.html#T7" title="INTEGRA1:th.7">INTEGRA1:7</a></span>;<br/><a NAME="E7:34_2_1_1_1_3_1_5_1_1_2_7_1_1_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E8:34_2_1_1_1_3_1_5_1_1_2_7_1_1_2"/><span class="kw">then </span>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="#E17:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A257</font></span></a>, <a class="txt" href="#E26:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E61">A244</font></span></a>, <a class="txt" href="#E9:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E94">A357</font></span></a>, <a class="txt" href="#E6:34_2_1_1_1_3_1_5_1_1_2_7_1_1_2"><span class="lab"><font color="Green" title="E133">A428</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_3_1_5_1_1_2_7_1_1_2"><span class="lab"><font color="Green" title="E131">A426</font></span></a>, <a class="txt" href="#E5:34_2_1_1_1_3_1_5_1_1_2_7_1_1_2"><span class="lab"><font color="Green" title="E132">A427</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>;<br/><span class="kw">hence </span><a NAME="E9:34_2_1_1_1_3_1_5_1_1_2_7_1_1_2"/>
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)</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="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E122">A415</font></span></a>, <a class="txt" href="#E17:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E126">A419</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E125">A418</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_2_7_1_1_2"><span class="lab"><font color="Green" title="E130">A425</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 NAME="E20:34_2_1_1_1_3_1_5_1_1_2_7"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">MD1</font>,<font color="Maroon" title="c16">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><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="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)</span>)</span></span>)</span>,<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="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T4" title="INTEGRA1:th.4">INTEGRA1:4</a></span>;<br/>
<a NAME="E21:34_2_1_1_1_3_1_5_1_1_2_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E129">A429</font></span>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E128">A421</font></span></a>, <a class="ref" href="integra1.html#T4" title="INTEGRA1:th.4">INTEGRA1:4</a></span>;<br/>
<a NAME="E22:34_2_1_1_1_3_1_5_1_1_2_7"/><span class="lab"><font color="Green" title="E130">A430</font></span>: 
<font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">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="c5">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E124">A417</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="lab"><font color="Green" title="E131">A431</font></span>: <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">k</font> = 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>) <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="#E17:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A257</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E61">A244</font></span></a>, <a class="txt" href="#E19:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E102">A371</font></span></a>, <a class="txt" href="#E35:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E114">A407</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E116">A409</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E117">A410</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E118">A411</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>
<br/>.= 
<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"><font color="Maroon" title="c2">f</font> <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="c5">D1</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span>)</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="c5">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E122">A415</font></span></a>, <a class="txt" href="#E22:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E130">A430</font></span></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>
;<br/>
<a NAME="E24:34_2_1_1_1_3_1_5_1_1_2_7"/>
<font color="Maroon" title="c16">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="c13">MD1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E119">A412</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="E25:34_2_1_1_1_3_1_5_1_1_2_7"/><span class="kw">then </span>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E129">A429</font></span></a>, <a class="ref" href="integra1.html#T8" title="INTEGRA1:th.8">INTEGRA1:8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E26:34_2_1_1_1_3_1_5_1_1_2_7"/>
<span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c13">MD1</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E121">A414</font></span></a>, <a class="txt" href="#E23:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E131">A431</font></span></a>, <a class="txt" href="#E21:34_2_1_1_1_3_1_5_1_1_2_7"><span class="lab"><font color="Green" title="E129">A429</font></span></a>, <a class="ref" href="funct_1.html#T51" title="FUNCT_1:th.51">FUNCT_1:51</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="E37:34_2_1_1_1_3_1_5_1_1_2"/><span class="lab"><font color="Green" title="E116">A432</font></span>: 
<font color="Maroon" title="c15">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c12">B</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="34_2_1_1_1_3_1_5_1_1_2_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c16">a</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> <span class="kw"> such that </span><br/><a NAME="E2:34_2_1_1_1_3_1_5_1_1_2_8"/><span class="lab"><font color="Green" title="E117">A433</font></span>: 
 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#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> holds <br/><font color="Maroon" title="c16">a</font> <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> <font color="Olive" title="b1">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="rfunct_1.html#T71" title="RFUNCT_1:th.71">RFUNCT_1:71</a></span>;<br/>
<a NAME="E3:34_2_1_1_1_3_1_5_1_1_2_8"/>
 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="c12">B</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">g</font></span>)</span> holds <br/><font color="Maroon" title="c16">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_2_8_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">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="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">B</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">g</font></span>)</span> implies <font color="Maroon" title="c16">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">x</font> )</span><br/>

<a NAME="E1:34_2_1_1_1_3_1_5_1_1_2_8_1"/><span class="lab"><font color="Green" title="E118">A434</font></span>: 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="c12">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="txt" href="#E17:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E101">A364</font></span></a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:34_2_1_1_1_3_1_5_1_1_2_8_1"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">B</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">g</font></span>)</span>
 ; <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="c16">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">x</font></span><br/>

<a NAME="E3:34_2_1_1_1_3_1_5_1_1_2_8_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">A435</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E4:34_2_1_1_1_3_1_5_1_1_2_8_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="c12">B</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/>
<a NAME="E5:34_2_1_1_1_3_1_5_1_1_2_8_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">a</font> <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> <font color="Maroon" title="c17">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_2_1_1_1_3_1_5_1_1_2_8"><span class="lab"><font color="Green" title="E117">A433</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_8_1"><span class="lab"><font color="Green" title="E118">A434</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:34_2_1_1_1_3_1_5_1_1_2_8_1"/>
<font color="Maroon" title="c16">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_2_8_1"><span class="lab"><font color="Green" title="E119">A435</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</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="E4:34_2_1_1_1_3_1_5_1_1_2_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E118">A436</font></span>: 
<font color="Maroon" title="c15">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c12">B</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/>
<span class="kw">consider </span><font color="Maroon" title="c17">a</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> <span class="kw"> such that </span><br/><a NAME="E6:34_2_1_1_1_3_1_5_1_1_2_8"/><span class="lab"><font color="Green" title="E119">A437</font></span>: 
 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#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> holds <br/><font color="Maroon" title="c2">f</font> <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> <font color="Maroon" title="c17">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="rfunct_1.html#T70" title="RFUNCT_1:th.70">RFUNCT_1:70</a></span>;<br/>
<a NAME="E7:34_2_1_1_1_3_1_5_1_1_2_8"/>
 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="c12">B</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">g</font></span>)</span> holds <br/><font color="Maroon" title="c15">g</font> <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> <font color="Maroon" title="c17">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_2_8_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">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="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">B</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">g</font></span>)</span> implies <font color="Maroon" title="c15">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">a</font> )</span><br/>

<a NAME="E1:34_2_1_1_1_3_1_5_1_1_2_8_2"/><span class="lab"><font color="Green" title="E120">A438</font></span>: 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="c12">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="txt" href="#E17:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E101">A364</font></span></a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:34_2_1_1_1_3_1_5_1_1_2_8_2"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">B</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">g</font></span>)</span>
 ; <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="c15">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">a</font></span><br/>

<a NAME="E3:34_2_1_1_1_3_1_5_1_1_2_8_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E121">A439</font></span>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E4:34_2_1_1_1_3_1_5_1_1_2_8_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="c12">B</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/>
<a NAME="E5:34_2_1_1_1_3_1_5_1_1_2_8_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:34_2_1_1_1_3_1_5_1_1_2_8"><span class="lab"><font color="Green" title="E119">A437</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_8_2"><span class="lab"><font color="Green" title="E120">A438</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:34_2_1_1_1_3_1_5_1_1_2_8_2"/>
<font color="Maroon" title="c15">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_2_8_2"><span class="lab"><font color="Green" title="E121">A439</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</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="E8:34_2_1_1_1_3_1_5_1_1_2_8"/><span class="kw">then </span>
<font color="Maroon" title="c15">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c12">B</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="E9:34_2_1_1_1_3_1_5_1_1_2_8"/>
<font color="Maroon" title="c15">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c12">B</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="#E4:34_2_1_1_1_3_1_5_1_1_2_8"><span class="lab"><font color="Green" title="E118">A436</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="E38:34_2_1_1_1_3_1_5_1_1_2"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="integra1.html#T11" title="INTEGRA1:th.11">INTEGRA1:11</a></span>;<br/><a NAME="E39:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">A440</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> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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> <font color="Maroon" title="c15">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T70" title="RELAT_1:th.70">RELAT_1:70</a>, <a class="ref" href="seq_4.html#T47" title="SEQ_4:th.47">SEQ_4:47</a></span>;<br/><a NAME="E40:34_2_1_1_1_3_1_5_1_1_2"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="integra1.html#T13" title="INTEGRA1:th.13">INTEGRA1:13</a></span>;<br/><a NAME="E41:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span>
 <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> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <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> <font color="Maroon" title="c15">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T70" title="RELAT_1:th.70">RELAT_1:70</a>, <a class="ref" href="seq_4.html#T48" title="SEQ_4:th.48">SEQ_4:48</a></span>;<br/><a NAME="E42:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span>
<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> <font color="Maroon" title="c2">f</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_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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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> <font color="Maroon" title="c15">g</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_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> <font color="Maroon" title="c15">g</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E117">A440</font></span></a>, <a class="ref" href="xreal_1.html#T13" title="XREAL_1:th.13">XREAL_1:13</a></span>;<br/><a NAME="E43:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E118">A441</font></span>: 
<span class="p1">(<span class="default"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></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="c13">MD1</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c15">g</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c15">g</font></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="c13">MD1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E96">A359</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/><a NAME="E44:34_2_1_1_1_3_1_5_1_1_2"/><span class="lab"><font color="Green" title="E119">A442</font></span>: 
<font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">j</font> <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="#E1:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E88">A351</font></span></a>, <a class="ref" href="xreal_1.html#T20" title="XREAL_1:th.20">XREAL_1:20</a></span>;<br/><a NAME="E45:34_2_1_1_1_3_1_5_1_1_2"/><span class="lab"><font color="Green" title="E120">A443</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</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">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E69">A263</font></span></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/><a NAME="E46:34_2_1_1_1_3_1_5_1_1_2"/><span class="lab"><font color="Green" title="E121">A444</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">MD2</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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></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="#E33:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E78">A272</font></span></a>, <a class="txt" href="#E30:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E76">A270</font></span></a>, <a class="txt" href="#E41:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E83">A277</font></span></a>, <a class="txt" href="#E22:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E68">A262</font></span></a>, <a class="txt" href="#E23:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E69">A263</font></span></a>, <a class="txt" href="#E10:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E95">A358</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>;<br/><a NAME="E47:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E122">A445</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">MD2</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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></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="#E33:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E78">A272</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/><a NAME="E48:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E123">A446</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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c14">MD2</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"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="#E34:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E113">A390</font></span></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/><a NAME="E49:34_2_1_1_1_3_1_5_1_1_2"/>
 for <font color="Olive" title="b1">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c13">MD1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">MD2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_2_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">x1</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="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c13">MD1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> implies <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">MD2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:34_2_1_1_1_3_1_5_1_1_2_9"/><span class="lab"><font color="Green" title="E124">A447</font></span>: 
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c13">MD1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 ; <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="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">MD2</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">x1</font> = <font color="Maroon" title="c16">x1</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_2_9_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_2_9_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="E1:34_2_1_1_1_3_1_5_1_1_2_9_1_1"/>
( <font color="Maroon" title="c17">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c13">MD1</font> or <font color="Maroon" title="c17">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_9"><span class="lab"><font color="Green" title="E124">A447</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"/><span class="lab"><font color="Green" title="E125">A448</font></span>: 
<font color="Maroon" title="c17">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c13">MD1</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="c17">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">MD2</font></span><br/><div class="add"><a NAME="E2:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c13">MD1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E3:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"/><span class="kw">then </span>
1 <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="c13">MD1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T32" title="FINSEQ_3:th.32">FINSEQ_3:32</a></span>;<br/><a NAME="E4:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E126">A449</font></span>: 
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="c13">MD1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E5:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c13">MD1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E94">A357</font></span></a>, <a class="ref" href="finseq_6.html#T119" title="FINSEQ_6:th.119">FINSEQ_6:119</a></span>;<br/><a NAME="E6:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E127">A450</font></span>: 
<font color="Maroon" title="c17">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"><span class="lab"><font color="Green" title="E125">A448</font></span></a></span>;<br/><a NAME="E7:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c6">D2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="ref" href="integra1.html#D18" title="INTEGRA1:def.18">INTEGRA1:def 18</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">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> <span class="kw"> such that </span><br/><a NAME="E9:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"/><span class="lab"><font color="Green" title="E128">A451</font></span>: 
<font color="Maroon" title="c18">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="c6">D2</font>
 <span class="kw">and </span><br/><a NAME="E10:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"/><span class="lab"><font color="Green" title="E129">A452</font></span>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"><span class="lab"><font color="Green" title="E127">A450</font></span></a>, <a class="ref" href="partfun1.html#T3" title="PARTFUN1:th.3">PARTFUN1:3</a></span>;<br/><a NAME="E11:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"/>
<font color="Maroon" title="c13">MD1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E71">A265</font></span></a>, <a class="txt" href="#E26:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E61">A244</font></span></a>, <a class="txt" href="#E27:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E73">A267</font></span></a>, <a class="txt" href="#E9:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E94">A357</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>;<br/><a NAME="E12:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E77">A271</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"><span class="lab"><font color="Green" title="E125">A448</font></span></a>, <a class="ref" href="integra3.html#T16" target="_self" title="INTEGRA3:th.16">Th16</a></span>;<br/><a NAME="E13:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E130">A453</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E75">A269</font></span></a>, <a class="txt" href="#E9:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"><span class="lab"><font color="Green" title="E128">A451</font></span></a>, <a class="txt" href="#E10:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"><span class="lab"><font color="Green" title="E129">A452</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E15:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"/><span class="lab"><font color="Green" title="E131">A454</font></span>: 
<font color="Maroon" title="c18">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">n</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#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E16:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"/><span class="lab"><font color="Green" title="E132">A455</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="#E17:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A257</font></span></a>, <a class="txt" href="#E25:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E71">A265</font></span></a>, <a class="txt" href="#E26:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E61">A244</font></span></a>, <a class="txt" href="#E27:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E73">A267</font></span></a>, <a class="txt" href="#E9:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E94">A357</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>;<br/><span class="kw">then </span><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="c13">MD1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="#E17:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A257</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>
<br/>.= 
<font color="Maroon" title="c10">j</font>

;<br/><a NAME="E18:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">MD1</font> <a href="seq_1.html#K1" title="SEQ_1:func.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="c13">MD1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A257</font></span></a>, <a class="txt" href="#E26:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E61">A244</font></span></a>, <a class="txt" href="#E9:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E94">A357</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"><span class="lab"><font color="Green" title="E126">A449</font></span></a>, <a class="txt" href="#E16:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"><span class="lab"><font color="Green" title="E132">A455</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>;<br/><a NAME="E19:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E55">A238</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"><span class="lab"><font color="Green" title="E125">A448</font></span></a>, <a class="ref" href="integra3.html#T16" target="_self" title="INTEGRA3:th.16">Th16</a></span>;<br/><a NAME="E20:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E67">A261</font></span></a>, <a class="txt" href="#E9:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"><span class="lab"><font color="Green" title="E128">A451</font></span></a>, <a class="txt" href="#E10:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"><span class="lab"><font color="Green" title="E129">A452</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/><a NAME="E21:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E22:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E133">A456</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></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="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E23:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</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="c18">k</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="#E13:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"><span class="lab"><font color="Green" title="E130">A453</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E24:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E134">A457</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span>
 <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="E25:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E135">A458</font></span>: 
<font color="Maroon" title="c19">n</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="c14">MD2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E122">A445</font></span></a>, <a class="txt" href="#E22:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"><span class="lab"><font color="Green" title="E133">A456</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"><span class="lab"><font color="Green" title="E131">A454</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E26:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"/>
<font color="Maroon" title="c19">n</font> <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>;<br/><span class="kw">then </span><font color="Maroon" title="c14">MD2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">n</font> = 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></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="#E33:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E78">A272</font></span></a>, <a class="txt" href="#E30:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E76">A270</font></span></a>, <a class="txt" href="#E23:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E69">A263</font></span></a>, <a class="txt" href="#E10:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E95">A358</font></span></a>, <a class="txt" href="#E24:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"><span class="lab"><font color="Green" title="E134">A457</font></span></a>, <a class="txt" href="#E22:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"><span class="lab"><font color="Green" title="E133">A456</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"><span class="lab"><font color="Green" title="E131">A454</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>
<br/>.= 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"><span class="lab"><font color="Green" title="E131">A454</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E28:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"/>
<font color="Maroon" title="c17">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">MD2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"><span class="lab"><font color="Green" title="E129">A452</font></span></a>, <a class="txt" href="#E25:34_2_1_1_1_3_1_5_1_1_2_9_1_1_1"><span class="lab"><font color="Green" title="E135">A458</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</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="34_2_1_1_1_3_1_5_1_1_2_9_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_3_1_5_1_1_2_9_1_1_2"/>
<font color="Maroon" title="c17">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 ; <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="c17">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">MD2</font></span><br/><div class="add"><a NAME="E2:34_2_1_1_1_3_1_5_1_1_2_9_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E125">A459</font></span>: 
<font color="Maroon" title="c17">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">j1</font> = <font color="Maroon" title="c10">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</font></span></a>, <a class="txt" href="#E16:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A245</font></span></a>, <a class="txt" href="#E51:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E86">A280</font></span></a>, <a class="ref" href="integra1.html#T7" title="INTEGRA1:th.7">INTEGRA1:7</a></span>;<br/><a NAME="E4:34_2_1_1_1_3_1_5_1_1_2_9_1_1_2"/><span class="lab"><font color="Green" title="E126">A460</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c6">D2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_2_1"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="ref" href="integra1.html#D18" title="INTEGRA1:def.18">INTEGRA1:def 18</a></span>;<br/><a NAME="E5:34_2_1_1_1_3_1_5_1_1_2_9_1_1_2"/>
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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> <font color="Maroon" title="c4">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E52">A235</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">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> <span class="kw"> such that </span><br/><a NAME="E7:34_2_1_1_1_3_1_5_1_1_2_9_1_1_2"/><span class="lab"><font color="Green" title="E127">A461</font></span>: 
<font color="Maroon" title="c19">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="c6">D2</font>
 <span class="kw">and </span><br/><a NAME="E8:34_2_1_1_1_3_1_5_1_1_2_9_1_1_2"/><span class="lab"><font color="Green" title="E128">A462</font></span>: 
<font color="Maroon" title="c17">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_2_1_1_1_3_1_5_1_1_2_9_1_1_2"><span class="lab"><font color="Green" title="E125">A459</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_3_1_5_1_1_2_9_1_1_2"><span class="lab"><font color="Green" title="E126">A460</font></span></a>, <a class="ref" href="partfun1.html#T3" title="PARTFUN1:th.3">PARTFUN1:3</a></span>;<br/><a NAME="E9:34_2_1_1_1_3_1_5_1_1_2_9_1_1_2"/>
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E54">A237</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/><a NAME="E10:34_2_1_1_1_3_1_5_1_1_2_9_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</font></span></a>, <a class="txt" href="#E16:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A245</font></span></a>, <a class="txt" href="#E51:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E86">A280</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_3_1_5_1_1_2_9_1_1_2"><span class="lab"><font color="Green" title="E125">A459</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E11:34_2_1_1_1_3_1_5_1_1_2_9_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E129">A463</font></span>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</font></span></a>, <a class="txt" href="#E8:34_2_1_1_1_3_1_5_1_1_2_9_1_1_2"><span class="lab"><font color="Green" title="E128">A462</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/><a NAME="E12:34_2_1_1_1_3_1_5_1_1_2_9_1_1_2"/>
<font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E88">A351</font></span></a>, <a class="ref" href="xreal_1.html#T20" title="XREAL_1:th.20">XREAL_1:20</a></span>;<br/><a NAME="E13:34_2_1_1_1_3_1_5_1_1_2_9_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E130">A464</font></span>: 
<font color="Maroon" title="c11">n1</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> <font color="Maroon" title="c18">j1</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="E14:34_2_1_1_1_3_1_5_1_1_2_9_1_1_2"/>
<font color="Maroon" title="c10">j</font> <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="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</font></span></a>, <a class="txt" href="#E16:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A245</font></span></a>, <a class="txt" href="#E51:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E86">A280</font></span></a>, <a class="ref" href="integra1.html#T7" title="INTEGRA1:th.7">INTEGRA1:7</a></span>;<br/><a NAME="E15:34_2_1_1_1_3_1_5_1_1_2_9_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E131">A465</font></span>: 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <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="#E28:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E74">A268</font></span></a>, <a class="txt" href="#E13:34_2_1_1_1_3_1_5_1_1_2_9_1_1_2"><span class="lab"><font color="Green" title="E130">A464</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/><a NAME="E16:34_2_1_1_1_3_1_5_1_1_2_9_1_1_2"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E54">A237</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/><a NAME="E17:34_2_1_1_1_3_1_5_1_1_2_9_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</font></span></a>, <a class="txt" href="#E16:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A245</font></span></a>, <a class="txt" href="#E51:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E86">A280</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_3_1_5_1_1_2_9_1_1_2"><span class="lab"><font color="Green" title="E125">A459</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E18:34_2_1_1_1_3_1_5_1_1_2_9_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E77">A271</font></span></a>, <a class="txt" href="#E8:34_2_1_1_1_3_1_5_1_1_2_9_1_1_2"><span class="lab"><font color="Green" title="E128">A462</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_3_1_5_1_1_2_9_1_1_2"><span class="lab"><font color="Green" title="E131">A465</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E19:34_2_1_1_1_3_1_5_1_1_2_9_1_1_2"/>
<font color="Maroon" title="c17">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">MD2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E75">A269</font></span></a>, <a class="txt" href="#E21:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E67">A261</font></span></a>, <a class="txt" href="#E10:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E95">A358</font></span></a>, <a class="txt" href="#E7:34_2_1_1_1_3_1_5_1_1_2_9_1_1_2"><span class="lab"><font color="Green" title="E127">A461</font></span></a>, <a class="txt" href="#E8:34_2_1_1_1_3_1_5_1_1_2_9_1_1_2"><span class="lab"><font color="Green" title="E128">A462</font></span></a>, <a class="txt" href="#E11:34_2_1_1_1_3_1_5_1_1_2_9_1_1_2"><span class="lab"><font color="Green" title="E129">A463</font></span></a>, <a class="ref" href="integra3.html#T17" target="_self" title="INTEGRA3:th.17">Th17</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="E4:34_2_1_1_1_3_1_5_1_1_2_9"/>
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">MD2</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="E50:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A466</font></span>: 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c13">MD1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">MD2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E51:34_2_1_1_1_3_1_5_1_1_2"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">MD2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E52:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span>
1 <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="c14">MD2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T32" title="FINSEQ_3:th.32">FINSEQ_3:32</a></span>;<br/><a NAME="E53:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E125">A467</font></span>: 
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="c14">MD2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E54:34_2_1_1_1_3_1_5_1_1_2"/><span class="lab"><font color="Green" title="E126">A468</font></span>: 
<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="c14">MD2</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> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E122">A445</font></span></a></span>;<br/><a NAME="E55:34_2_1_1_1_3_1_5_1_1_2"/>
 for <font color="Olive" title="b1">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">MD2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c13">MD1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_2_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">x1</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="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">MD2</font> implies <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c13">MD1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:34_2_1_1_1_3_1_5_1_1_2_10"/><span class="lab"><font color="Green" title="E127">A469</font></span>: 
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">MD2</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="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c13">MD1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">x1</font> = <font color="Maroon" title="c16">x1</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E3:34_2_1_1_1_3_1_5_1_1_2_10"/>
<font color="Maroon" title="c14">MD2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E76">A270</font></span></a>, <a class="txt" href="#E41:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E83">A277</font></span></a>, <a class="txt" href="#E22:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E68">A262</font></span></a>, <a class="txt" href="#E23:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E69">A263</font></span></a>, <a class="txt" href="#E10:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E95">A358</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>;<br/>
<a NAME="E4:34_2_1_1_1_3_1_5_1_1_2_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E128">A470</font></span>: 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E77">A271</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_10"><span class="lab"><font color="Green" title="E127">A469</font></span></a>, <a class="ref" href="integra3.html#T16" target="_self" title="INTEGRA3:th.16">Th16</a></span>;<br/>
<a NAME="E5:34_2_1_1_1_3_1_5_1_1_2_10"/>
<font color="Maroon" title="c14">MD2</font> <a href="seq_1.html#K1" title="SEQ_1:func.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="c14">MD2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E78">A272</font></span></a>, <a class="txt" href="#E30:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E76">A270</font></span></a>, <a class="txt" href="#E23:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E69">A263</font></span></a>, <a class="txt" href="#E10:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E95">A358</font></span></a>, <a class="txt" href="#E53:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E125">A467</font></span></a>, <a class="txt" href="#E46:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E121">A444</font></span></a>, <a class="txt" href="#E54:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E126">A468</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>;<br/>
<a NAME="E6:34_2_1_1_1_3_1_5_1_1_2_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E129">A471</font></span>: 
<font color="Maroon" title="c17">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E55">A238</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_10"><span class="lab"><font color="Green" title="E127">A469</font></span></a>, <a class="ref" href="integra3.html#T16" target="_self" title="INTEGRA3:th.16">Th16</a></span>;<br/>
<a NAME="E7:34_2_1_1_1_3_1_5_1_1_2_10"/><span class="lab"><font color="Green" title="E130">A472</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">MD2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c6">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E95">A358</font></span></a>, <a class="ref" href="finseq_6.html#T119" title="FINSEQ_6:th.119">FINSEQ_6:119</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_2_10_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_2_10_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="E1:34_2_1_1_1_3_1_5_1_1_2_10_1_1"/>
( <font color="Maroon" title="c17">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font> or <font color="Maroon" title="c17">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1"><span class="lab"><font color="Green" title="E41">A14</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_10"><span class="lab"><font color="Green" title="E127">A469</font></span></a>, <a class="txt" href="#E7:34_2_1_1_1_3_1_5_1_1_2_10"><span class="lab"><font color="Green" title="E130">A472</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_2_10_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_3_1_5_1_1_2_10_1_1_1"/>
<font color="Maroon" title="c17">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</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="c17">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c13">MD1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">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> <span class="kw"> such that </span><br/><a NAME="E3:34_2_1_1_1_3_1_5_1_1_2_10_1_1_1"/><span class="lab"><font color="Green" title="E131">A473</font></span>: 
<font color="Maroon" title="c18">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="c5">D1</font>
 <span class="kw">and </span><br/><a NAME="E4:34_2_1_1_1_3_1_5_1_1_2_10_1_1_1"/><span class="lab"><font color="Green" title="E132">A474</font></span>: 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T3" title="PARTFUN1:th.3">PARTFUN1:3</a></span>;<br/><a NAME="E5:34_2_1_1_1_3_1_5_1_1_2_10_1_1_1"/><span class="lab"><font color="Green" title="E133">A475</font></span>: 
<font color="Maroon" title="c11">n1</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> <font color="Maroon" title="c18">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E74">A268</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_3_1_5_1_1_2_10"><span class="lab"><font color="Green" title="E128">A470</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_2_10_1_1_1"><span class="lab"><font color="Green" title="E131">A473</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_3_1_5_1_1_2_10_1_1_1"><span class="lab"><font color="Green" title="E132">A474</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/><a NAME="E6:34_2_1_1_1_3_1_5_1_1_2_10_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E134">A476</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">n1</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="E7:34_2_1_1_1_3_1_5_1_1_2_10_1_1_1"/>
<font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n1</font> <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#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E9:34_2_1_1_1_3_1_5_1_1_2_10_1_1_1"/><span class="lab"><font color="Green" title="E135">A477</font></span>: 
<font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_3_1_5_1_1_2_10_1_1_1"><span class="lab"><font color="Green" title="E133">A475</font></span></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E10:34_2_1_1_1_3_1_5_1_1_2_10_1_1_1"/><span class="lab"><font color="Green" title="E136">A478</font></span>: 
<font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</font></span></a>, <a class="txt" href="#E6:34_2_1_1_1_3_1_5_1_1_2_10"><span class="lab"><font color="Green" title="E129">A471</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_2_10_1_1_1"><span class="lab"><font color="Green" title="E131">A473</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_3_1_5_1_1_2_10_1_1_1"><span class="lab"><font color="Green" title="E132">A474</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/><a NAME="E11:34_2_1_1_1_3_1_5_1_1_2_10_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">n1</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="c13">MD1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E100">A363</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E12:34_2_1_1_1_3_1_5_1_1_2_10_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">n</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="c13">MD1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:34_2_1_1_1_3_1_5_1_1_2_10_1_1_1"><span class="lab"><font color="Green" title="E134">A476</font></span></a>, <a class="txt" href="#E9:34_2_1_1_1_3_1_5_1_1_2_10_1_1_1"><span class="lab"><font color="Green" title="E135">A477</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E13:34_2_1_1_1_3_1_5_1_1_2_10_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E137">A479</font></span>: 
<font color="Maroon" title="c13">MD1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c13">MD1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E14:34_2_1_1_1_3_1_5_1_1_2_10_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">n1</font>
 ;<br/><a NAME="E15:34_2_1_1_1_3_1_5_1_1_2_10_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E138">A480</font></span>: 
<font color="Maroon" title="c18">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="#E10:34_2_1_1_1_3_1_5_1_1_2_10_1_1_1"><span class="lab"><font color="Green" title="E136">A478</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E16:34_2_1_1_1_3_1_5_1_1_2_10_1_1_1"/>
<font color="Maroon" title="c19">n</font> <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>;<br/><span class="kw">then </span><font color="Maroon" title="c13">MD1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">n</font> = 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">n1</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> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A257</font></span></a>, <a class="txt" href="#E26:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E61">A244</font></span></a>, <a class="txt" href="#E9:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E94">A357</font></span></a>, <a class="txt" href="#E6:34_2_1_1_1_3_1_5_1_1_2_10_1_1_1"><span class="lab"><font color="Green" title="E134">A476</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_3_1_5_1_1_2_10_1_1_1"><span class="lab"><font color="Green" title="E138">A480</font></span></a>, <a class="txt" href="#E9:34_2_1_1_1_3_1_5_1_1_2_10_1_1_1"><span class="lab"><font color="Green" title="E135">A477</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>
<br/>.= 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">k</font>

;<br/><span class="kw">hence </span><a NAME="E18:34_2_1_1_1_3_1_5_1_1_2_10_1_1_1"/>
<font color="Maroon" title="c17">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c13">MD1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_3_1_5_1_1_2_10_1_1_1"><span class="lab"><font color="Green" title="E132">A474</font></span></a>, <a class="txt" href="#E13:34_2_1_1_1_3_1_5_1_1_2_10_1_1_1"><span class="lab"><font color="Green" title="E137">A479</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</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="34_2_1_1_1_3_1_5_1_1_2_10_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2"/>
<font color="Maroon" title="c17">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</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="c17">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c13">MD1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">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="E3:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2"/><span class="lab"><font color="Green" title="E131">A481</font></span>: 
<font color="Maroon" title="c18">n</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">D</font>
 <span class="kw">and </span><br/><a NAME="E4:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2"/><span class="lab"><font color="Green" title="E132">A482</font></span>: 
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T3" title="PARTFUN1:th.3">PARTFUN1:3</a></span>;<br/><a NAME="E5:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2"/><span class="lab"><font color="Green" title="E133">A483</font></span>: 
 not <font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_2_10_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2_1"/><span class="lab"><font color="Green" title="E134">A484</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="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_2_10_1_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_2_10_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="E1:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2_1_1_1"/>
( <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">j</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="34_2_1_1_1_3_1_5_1_1_2_10_1_1_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c10">j</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="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">j</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:34_2_1_1_1_3_1_5_1_1_2_10_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="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</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="34_2_1_1_1_3_1_5_1_1_2_10_1_1_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2_1_1_1_2"/>
<font color="Maroon" title="c10">j</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="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">j</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:34_2_1_1_1_3_1_5_1_1_2_10_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="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</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">consider </span><font color="Maroon" title="c19">y1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E3:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2_1"/><span class="lab"><font color="Green" title="E135">A485</font></span>: 
<font color="Maroon" title="c19">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/>
<a NAME="E4:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2_1"/><span class="lab"><font color="Green" title="E136">A486</font></span>: 
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2"><span class="lab"><font color="Green" title="E131">A481</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2_1"/>
<font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E6:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E137">A487</font></span>: 
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E52">A235</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2"><span class="lab"><font color="Green" title="E131">A481</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/>
<a NAME="E7:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2_1"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E54">A237</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/>
<a NAME="E8:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2_1"/><span class="kw">then </span>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2_1"><span class="lab"><font color="Green" title="E137">A487</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E9:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:34_2_1_1_1_3_1_5_1_1_2_10"><span class="lab"><font color="Green" title="E129">A471</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2"><span class="lab"><font color="Green" title="E132">A482</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2_1"><span class="lab"><font color="Green" title="E134">A484</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/>
<a NAME="E10:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E138">A488</font></span>: 
<font color="Maroon" title="c17">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</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="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2"><span class="lab"><font color="Green" title="E132">A482</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2_1"><span class="lab"><font color="Green" title="E136">A486</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E11:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2_1"/>
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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> <font color="Maroon" title="c4">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E52">A235</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="E12:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</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="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E54">A237</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2_1"><span class="lab"><font color="Green" title="E135">A485</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2"><span class="lab"><font color="Green" title="E132">A482</font></span></a>, <a class="txt" href="#E6:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2_1"><span class="lab"><font color="Green" title="E137">A487</font></span></a>, <a class="txt" href="#E10:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2_1"><span class="lab"><font color="Green" title="E138">A488</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2_1"><span class="lab"><font color="Green" title="E135">A485</font></span></a>, <a class="ref" href="integra3.html#T5" target="_self" title="INTEGRA3:th.5">Th5</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="E6:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2"/><span class="lab"><font color="Green" title="E134">A489</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="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_2_10_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_2_10_1_1_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="E1:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2_2_1"/>
( <font color="Maroon" title="c11">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c11">n1</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="34_2_1_1_1_3_1_5_1_1_2_10_1_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2_2_1_1"/>
<font color="Maroon" title="c11">n1</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="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n1</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2_2_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="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E58">A241</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="34_2_1_1_1_3_1_5_1_1_2_10_1_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2_2_1_2"/>
<font color="Maroon" title="c11">n1</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="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n1</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2_2_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="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E58">A241</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 NAME="E7:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2"/>
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</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> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E59">A242</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/><a NAME="E8:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E90">A353</font></span></a>, <a class="txt" href="#E6:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2"><span class="lab"><font color="Green" title="E134">A489</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E9:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_3_1_5_1_1_2_10"><span class="lab"><font color="Green" title="E128">A470</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2"><span class="lab"><font color="Green" title="E132">A482</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E10:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E57">A240</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2"><span class="lab"><font color="Green" title="E131">A481</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/><a NAME="E11:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">n</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="E12:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">n</font> or <font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E13:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2"><span class="lab"><font color="Green" title="E132">A482</font></span></a>, <a class="txt" href="#E5:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2"><span class="lab"><font color="Green" title="E133">A483</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E14:34_2_1_1_1_3_1_5_1_1_2_10_1_1_2"/>
<font color="Maroon" title="c17">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c13">MD1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</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="E9:34_2_1_1_1_3_1_5_1_1_2_10"/>
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c13">MD1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.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><a NAME="E56:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">MD2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c13">MD1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E57:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E127">A490</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">MD2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c13">MD1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E124">A466</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><a NAME="E58:34_2_1_1_1_3_1_5_1_1_2"/>
 <a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c13">MD1</font> <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="c12">B</font>,<font color="Maroon" title="c12">B</font>)</span>)</span>,<font color="Maroon" title="c13">MD1</font>)</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/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">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> <span class="kw"> such that </span><br/><a NAME="E60:34_2_1_1_1_3_1_5_1_1_2"/><span class="lab"><font color="Green" title="E128">A491</font></span>: 
<font color="Maroon" title="c16">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="c12">B</font>,<font color="Maroon" title="c12">B</font>)</span>)</span>,<font color="Maroon" title="c13">MD1</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E61:34_2_1_1_1_3_1_5_1_1_2"/><span class="lab"><font color="Green" title="E129">A492</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"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> (<font color="Maroon" title="c12">B</font>,<font color="Maroon" title="c12">B</font>)</span>)</span>,<font color="Maroon" title="c13">MD1</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c13">MD1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T3" title="PARTFUN1:th.3">PARTFUN1:3</a></span>;<br/><a NAME="E62:34_2_1_1_1_3_1_5_1_1_2"/><span class="lab"><font color="Green" title="E130">A493</font></span>: 
<font color="Maroon" title="c16">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="c12">B</font>,<font color="Maroon" title="c12">B</font>)</span>)</span>,<font color="Maroon" title="c13">MD1</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E128">A491</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="E63:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E131">A494</font></span>: 
<font color="Maroon" title="c16">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="c13">MD1</font></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="E64:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E132">A495</font></span>: 
<font color="Maroon" title="c16">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="c13">MD1</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="E65:34_2_1_1_1_3_1_5_1_1_2"/><span class="lab"><font color="Green" title="E133">A496</font></span>: 
<font color="Maroon" title="c16">k</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="c13">MD1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E131">A494</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E66:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E100">A363</font></span></a>, <a class="ref" href="xreal_1.html#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/><a NAME="E67:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E134">A497</font></span>: 
<font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n1</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="c5">D1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E68:34_2_1_1_1_3_1_5_1_1_2"/><span class="lab"><font color="Green" title="E135">A498</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E62:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E130">A493</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E69:34_2_1_1_1_3_1_5_1_1_2"/><span class="lab"><font color="Green" title="E136">A499</font></span>: 
<font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="#E51:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E86">A280</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E70:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <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="E71:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E137">A500</font></span>: 
<font color="Maroon" title="c16">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" title="XREAL_1:th.29">XREAL_1:29</a></span>;<br/><a NAME="E72:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E68:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E135">A498</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E73:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E138">A501</font></span>: 
<font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n1</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="c5">D1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E67:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E134">A497</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E74:34_2_1_1_1_3_1_5_1_1_2"/>
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n1</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="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n1</font></span>)</span>)</span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_2_11"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_2_11_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="E1:34_2_1_1_1_3_1_5_1_1_2_11_1"/>
( <font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c16">k</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="34_2_1_1_1_3_1_5_1_1_2_11_1_1"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_3_1_5_1_1_2_11_1_1"/><span class="lab"><font color="Green" title="E139">A502</font></span>: 
<font color="Maroon" title="c16">k</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#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n1</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="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n1</font></span>)</span>)</span>)</span> )</span><br/><div class="add"><a NAME="E2:34_2_1_1_1_3_1_5_1_1_2_11_1_1"/><span class="kw">then </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="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">MD1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E132">A495</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E3:34_2_1_1_1_3_1_5_1_1_2_11_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E140">A503</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="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="#E17:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A257</font></span></a>, <a class="txt" href="#E26:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E61">A244</font></span></a>, <a class="txt" href="#E9:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E94">A357</font></span></a>, <a class="txt" href="#E12:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E97">A360</font></span></a>, <a class="txt" href="#E68:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E135">A498</font></span></a>, <a class="txt" href="#E65:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E133">A496</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>;<br/><a NAME="E4:34_2_1_1_1_3_1_5_1_1_2_11_1_1"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n1</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n1</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="#E68:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E135">A498</font></span></a>, <a class="txt" href="#E71:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E137">A500</font></span></a>, <a class="txt" href="#E73:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E138">A501</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="E5:34_2_1_1_1_3_1_5_1_1_2_11_1_1"/>
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n1</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="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n1</font></span>)</span>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E91">A354</font></span></a>, <a class="txt" href="#E69:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E136">A499</font></span></a>, <a class="txt" href="#E64:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E132">A495</font></span></a>, <a class="txt" href="#E73:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E138">A501</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_11_1_1"><span class="lab"><font color="Green" title="E139">A502</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_2_11_1_1"><span class="lab"><font color="Green" title="E140">A503</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="34_2_1_1_1_3_1_5_1_1_2_11_1_2"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_3_1_5_1_1_2_11_1_2"/><span class="lab"><font color="Green" title="E139">A504</font></span>: 
<font color="Maroon" title="c16">k</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#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n1</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="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n1</font></span>)</span>)</span>)</span> )</span><br/><div class="add"><a NAME="E2:34_2_1_1_1_3_1_5_1_1_2_11_1_2"/><span class="kw">then </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="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">MD1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E132">A495</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E3:34_2_1_1_1_3_1_5_1_1_2_11_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E140">A505</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="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="#E17:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A257</font></span></a>, <a class="txt" href="#E26:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E61">A244</font></span></a>, <a class="txt" href="#E9:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E94">A357</font></span></a>, <a class="txt" href="#E12:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E97">A360</font></span></a>, <a class="txt" href="#E68:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E135">A498</font></span></a>, <a class="txt" href="#E65:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E133">A496</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>;<br/><a NAME="E4:34_2_1_1_1_3_1_5_1_1_2_11_1_2"/><span class="lab"><font color="Green" title="E141">A506</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">MD1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">k</font> <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="#E64:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E132">A495</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_11_1_2"><span class="lab"><font color="Green" title="E139">A504</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E5:34_2_1_1_1_3_1_5_1_1_2_11_1_2"/><span class="lab"><font color="Green" title="E142">A507</font></span>: 
<font color="Maroon" title="c16">k</font> <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="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">MD1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E132">A495</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_11_1_2"><span class="lab"><font color="Green" title="E139">A504</font></span></a>, <a class="ref" href="integra1.html#T7" title="INTEGRA1:th.7">INTEGRA1:7</a></span>;<br/><a NAME="E6:34_2_1_1_1_3_1_5_1_1_2_11_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E143">A508</font></span>: 
<font color="Maroon" title="c16">k</font> <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="c13">MD1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E7:34_2_1_1_1_3_1_5_1_1_2_11_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">k</font> <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="#E5:34_2_1_1_1_3_1_5_1_1_2_11_1_2"><span class="lab"><font color="Green" title="E142">A507</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E8:34_2_1_1_1_3_1_5_1_1_2_11_1_2"/><span class="kw">then </span>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">k</font> <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> <span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="#E17:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A257</font></span></a>, <a class="txt" href="#E26:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E61">A244</font></span></a>, <a class="txt" href="#E9:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E94">A357</font></span></a>, <a class="txt" href="#E12:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E97">A360</font></span></a>, <a class="txt" href="#E5:34_2_1_1_1_3_1_5_1_1_2_11_1_2"><span class="lab"><font color="Green" title="E142">A507</font></span></a>, <a class="txt" href="#E6:34_2_1_1_1_3_1_5_1_1_2_11_1_2"><span class="lab"><font color="Green" title="E143">A508</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_3_1_5_1_1_2_11_1_2"><span class="lab"><font color="Green" title="E141">A506</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>;<br/><span class="kw">hence </span><a NAME="E9:34_2_1_1_1_3_1_5_1_1_2_11_1_2"/>
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n1</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="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n1</font></span>)</span>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E68:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E135">A498</font></span></a>, <a class="txt" href="#E71:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E137">A500</font></span></a>, <a class="txt" href="#E73:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E138">A501</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_2_11_1_2"><span class="lab"><font color="Green" title="E140">A505</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 NAME="E75:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">MD1</font>,<font color="Maroon" title="c16">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><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="c5">D1</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n1</font></span>)</span>)</span>)</span></span>)</span>,<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="c5">D1</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n1</font></span>)</span>)</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T4" title="INTEGRA1:th.4">INTEGRA1:4</a></span>;<br/><a NAME="E76:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E139">A509</font></span>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">MD1</font>,<font color="Maroon" title="c16">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n1</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T4" title="INTEGRA1:th.4">INTEGRA1:4</a></span>;<br/><a NAME="E77:34_2_1_1_1_3_1_5_1_1_2"/>
<font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n1</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="c5">D1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E73:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E138">A501</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="E78:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n1</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="c5">D1</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="E79:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E140">A510</font></span>: 
<font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n1</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="c5">D1</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="E80:34_2_1_1_1_3_1_5_1_1_2"/>
<font color="Maroon" title="c16">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="c13">MD1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E131">A494</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="E81:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span>
 <a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c13">MD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c13">MD1</font>,<font color="Maroon" title="c16">k</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E61:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E129">A492</font></span></a>, <a class="ref" href="integra1.html#T20" title="INTEGRA1:th.20">INTEGRA1:20</a></span>;<br/><a NAME="E82:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span>
 <a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c13">MD1</font> <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="c5">D1</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E73:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E138">A501</font></span></a>, <a class="txt" href="#E76:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E139">A509</font></span></a>, <a class="ref" href="integra1.html#T20" title="INTEGRA1:th.20">INTEGRA1:20</a></span>;<br/><a NAME="E83:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span>
 <a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c13">MD1</font> <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="c5">D1</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E79:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E140">A510</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="E84:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span>
 <a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c13">MD1</font> <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="c5">D1</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="E85:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E141">A511</font></span>: 
 <a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c13">MD1</font> <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="c5">D1</font>
 ;<br/><a NAME="E86:34_2_1_1_1_3_1_5_1_1_2"/>
<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> <font color="Maroon" title="c2">f</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_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> <font color="Maroon" title="c2">f</font></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="#E1:34"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="integra3.html#E14"><span class="lab"><font color="Green" title="E12">Lm3</font></span></a>, <a class="ref" href="xreal_1.html#T48" title="XREAL_1:th.48">XREAL_1:48</a></span>;<br/><a NAME="E87:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E142">A512</font></span>: 
<span class="p1">(<span class="default"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></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="c13">MD1</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"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></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="c5">D1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E85:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E141">A511</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/><a NAME="E88:34_2_1_1_1_3_1_5_1_1_2"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E54">A237</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/><a NAME="E89:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E143">A513</font></span>: 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</font></span></a>, <a class="txt" href="#E16:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A245</font></span></a>, <a class="txt" href="#E51:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E86">A280</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E90:34_2_1_1_1_3_1_5_1_1_2"/><span class="lab"><font color="Green" title="E144">A514</font></span>: 
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E54">A237</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/><a NAME="E91:34_2_1_1_1_3_1_5_1_1_2"/><span class="lab"><font color="Green" title="E145">A515</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</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>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E78">A272</font></span></a>, <a class="txt" href="#E27:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E108">A385</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E92:34_2_1_1_1_3_1_5_1_1_2"/><span class="lab"><font color="Green" title="E146">A516</font></span>: 
 for <font color="Olive" title="b1">k</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">k</font> &amp; <font color="Olive" title="b1">k</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> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c14">MD2</font>)</span>)</span> holds <br/><span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c14">MD2</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</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="34_2_1_1_1_3_1_5_1_1_2_12"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">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"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">k</font> &amp; <font color="Maroon" title="c17">k</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> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c14">MD2</font>)</span>)</span> implies <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c14">MD2</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">k</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:34_2_1_1_1_3_1_5_1_1_2_12"/><span class="lab"><font color="Green" title="E147">A517</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">k</font>
 <span class="kw">and </span><br/><a NAME="E2:34_2_1_1_1_3_1_5_1_1_2_12"/><span class="lab"><font color="Green" title="E148">A518</font></span>: 
<font color="Maroon" title="c17">k</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> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c14">MD2</font>)</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"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c14">MD2</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">k</font></span><br/>

<a NAME="E3:34_2_1_1_1_3_1_5_1_1_2_12"/><span class="lab"><font color="Green" title="E149">A519</font></span>: 
<font color="Maroon" title="c17">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c14">MD2</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_12"><span class="lab"><font color="Green" title="E147">A517</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_3_1_5_1_1_2_12"><span class="lab"><font color="Green" title="E148">A518</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E4:34_2_1_1_1_3_1_5_1_1_2_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E150">A520</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="#E32:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E111">A388</font></span></a>, <a class="txt" href="#E48:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E123">A446</font></span></a>, <a class="txt" href="#E45:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E120">A443</font></span></a>, <a class="txt" href="#E91:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E145">A515</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_12"><span class="lab"><font color="Green" title="E147">A517</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_3_1_5_1_1_2_12"><span class="lab"><font color="Green" title="E148">A518</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>;<br/>
<a NAME="E5:34_2_1_1_1_3_1_5_1_1_2_12"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</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#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E6:34_2_1_1_1_3_1_5_1_1_2_12"/><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="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_12"><span class="lab"><font color="Green" title="E147">A517</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<a NAME="E7:34_2_1_1_1_3_1_5_1_1_2_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E151">A521</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="ref" href="xreal_1.html#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">k2</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="E9:34_2_1_1_1_3_1_5_1_1_2_12"/><span class="lab"><font color="Green" title="E152">A522</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="Maroon" title="c18">k2</font>
 ;<br/>
<a NAME="E10:34_2_1_1_1_3_1_5_1_1_2_12"/>
<font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</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"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="#E47:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E122">A445</font></span></a>, <a class="txt" href="#E34:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E113">A390</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_3_1_5_1_1_2_12"><span class="lab"><font color="Green" title="E148">A518</font></span></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/>
<a NAME="E11:34_2_1_1_1_3_1_5_1_1_2_12"/><span class="kw">then </span>
<font color="Maroon" title="c17">k</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</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="E12:34_2_1_1_1_3_1_5_1_1_2_12"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E120">A443</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E13:34_2_1_1_1_3_1_5_1_1_2_12"/><span class="kw">then </span>
<font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c18">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">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:34_2_1_1_1_3_1_5_1_1_2_12"><span class="lab"><font color="Green" title="E151">A521</font></span></a>, <a class="txt" href="#E9:34_2_1_1_1_3_1_5_1_1_2_12"><span class="lab"><font color="Green" title="E152">A522</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E14:34_2_1_1_1_3_1_5_1_1_2_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E153">A523</font></span>: 
<font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c18">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="c6">D2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/>
<a NAME="E15:34_2_1_1_1_3_1_5_1_1_2_12"/><span class="kw">then </span>
<font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c18">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="c6">D2</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="E16:34_2_1_1_1_3_1_5_1_1_2_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E154">A524</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">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"><font color="Maroon" title="c2">f</font> <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="c6">D2</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c18">k2</font></span>)</span>)</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="c6">D2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c18">k2</font></span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_3_1_5_1_1_2_12"><span class="lab"><font color="Green" title="E150">A520</font></span></a>, <a class="txt" href="#E9:34_2_1_1_1_3_1_5_1_1_2_12"><span class="lab"><font color="Green" title="E152">A522</font></span></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/>
<a NAME="E17:34_2_1_1_1_3_1_5_1_1_2_12"/><span class="lab"><font color="Green" title="E155">A525</font></span>: 
<font color="Maroon" title="c17">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="c14">MD2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_2_12"><span class="lab"><font color="Green" title="E149">A519</font></span></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/>
<a NAME="E18:34_2_1_1_1_3_1_5_1_1_2_12"/><span class="lab"><font color="Green" title="E156">A526</font></span>: 
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c14">MD2</font>,<font color="Maroon" title="c17">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c18">k2</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="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c14">MD2</font>,<font color="Maroon" title="c17">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c6">D2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c18">k2</font></span>)</span>)</span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_3_1_5_1_1_2_12_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:34_2_1_1_1_3_1_5_1_1_2_12_1"/>
<font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c18">k2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <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="#E20:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E66">A260</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_12"><span class="lab"><font color="Green" title="E147">A517</font></span></a>, <a class="txt" href="#E9:34_2_1_1_1_3_1_5_1_1_2_12"><span class="lab"><font color="Green" title="E152">A522</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<a NAME="E2:34_2_1_1_1_3_1_5_1_1_2_12_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E157">A527</font></span>: 
<font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c18">k2</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="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E3:34_2_1_1_1_3_1_5_1_1_2_12_1"/><span class="lab"><font color="Green" title="E158">A528</font></span>: 
<font color="Maroon" title="c17">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="c14">MD2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:34_2_1_1_1_3_1_5_1_1_2_12"><span class="lab"><font color="Green" title="E155">A525</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="E4:34_2_1_1_1_3_1_5_1_1_2_12_1"/><span class="lab"><font color="Green" title="E159">A529</font></span>: 
<font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c18">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="c6">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:34_2_1_1_1_3_1_5_1_1_2_12"><span class="lab"><font color="Green" title="E153">A523</font></span></a>, <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="34_2_1_1_1_3_1_5_1_1_2_12_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="E1:34_2_1_1_1_3_1_5_1_1_2_12_1_1"/>
( <font color="Maroon" title="c17">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c17">k</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="34_2_1_1_1_3_1_5_1_1_2_12_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_3_1_5_1_1_2_12_1_1_1"/><span class="lab"><font color="Green" title="E160">A530</font></span>: 
<font color="Maroon" title="c17">k</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#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c14">MD2</font>,<font color="Maroon" title="c17">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c18">k2</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="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c14">MD2</font>,<font color="Maroon" title="c17">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c6">D2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c18">k2</font></span>)</span>)</span>)</span> )</span><br/><div class="add"><a NAME="E2:34_2_1_1_1_3_1_5_1_1_2_12_1_1_1"/><span class="kw">then </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="c14">MD2</font>,<font color="Maroon" title="c17">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">MD2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_2_12_1"><span class="lab"><font color="Green" title="E158">A528</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E3:34_2_1_1_1_3_1_5_1_1_2_12_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E161">A531</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="c14">MD2</font>,<font color="Maroon" title="c17">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="#E33:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E78">A272</font></span></a>, <a class="txt" href="#E23:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E69">A263</font></span></a>, <a class="txt" href="#E10:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E95">A358</font></span></a>, <a class="txt" href="#E32:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E111">A388</font></span></a>, <a class="txt" href="#E34:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E113">A390</font></span></a>, <a class="txt" href="#E48:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E123">A446</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_12"><span class="lab"><font color="Green" title="E147">A517</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_3_1_5_1_1_2_12"><span class="lab"><font color="Green" title="E148">A518</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_2_12"><span class="lab"><font color="Green" title="E149">A519</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>;<br/><a NAME="E4:34_2_1_1_1_3_1_5_1_1_2_12_1_1_1"/><span class="lab"><font color="Green" title="E162">A532</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c18">k2</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c18">k2</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="#E2:34_2_1_1_1_3_1_5_1_1_2_12_1"><span class="lab"><font color="Green" title="E157">A527</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_3_1_5_1_1_2_12_1"><span class="lab"><font color="Green" title="E159">A529</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E5:34_2_1_1_1_3_1_5_1_1_2_12_1_1_1"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c14">MD2</font>,<font color="Maroon" title="c17">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E91">A354</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_2_12_1"><span class="lab"><font color="Green" title="E158">A528</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_12_1_1_1"><span class="lab"><font color="Green" title="E160">A530</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="E6:34_2_1_1_1_3_1_5_1_1_2_12_1_1_1"/>
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c14">MD2</font>,<font color="Maroon" title="c17">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c18">k2</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="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c14">MD2</font>,<font color="Maroon" title="c17">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c6">D2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c18">k2</font></span>)</span>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E12:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E58">A241</font></span></a>, <a class="txt" href="#E9:34_2_1_1_1_3_1_5_1_1_2_12"><span class="lab"><font color="Green" title="E152">A522</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_3_1_5_1_1_2_12_1"><span class="lab"><font color="Green" title="E157">A527</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_3_1_5_1_1_2_12_1"><span class="lab"><font color="Green" title="E159">A529</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_12_1_1_1"><span class="lab"><font color="Green" title="E160">A530</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_2_12_1_1_1"><span class="lab"><font color="Green" title="E161">A531</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_3_1_5_1_1_2_12_1_1_1"><span class="lab"><font color="Green" title="E162">A532</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</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="34_2_1_1_1_3_1_5_1_1_2_12_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_1_3_1_5_1_1_2_12_1_1_2"/><span class="lab"><font color="Green" title="E160">A533</font></span>: 
<font color="Maroon" title="c17">k</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#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c14">MD2</font>,<font color="Maroon" title="c17">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c18">k2</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="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c14">MD2</font>,<font color="Maroon" title="c17">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c6">D2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c18">k2</font></span>)</span>)</span>)</span> )</span><br/><div class="add"><a NAME="E2:34_2_1_1_1_3_1_5_1_1_2_12_1_1_2"/><span class="kw">then </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="c14">MD2</font>,<font color="Maroon" title="c17">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">MD2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_2_12_1"><span class="lab"><font color="Green" title="E158">A528</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E3:34_2_1_1_1_3_1_5_1_1_2_12_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E161">A534</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="c14">MD2</font>,<font color="Maroon" title="c17">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="#E33:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E78">A272</font></span></a>, <a class="txt" href="#E23:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E69">A263</font></span></a>, <a class="txt" href="#E10:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E95">A358</font></span></a>, <a class="txt" href="#E32:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E111">A388</font></span></a>, <a class="txt" href="#E34:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E113">A390</font></span></a>, <a class="txt" href="#E48:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E123">A446</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_12"><span class="lab"><font color="Green" title="E147">A517</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_3_1_5_1_1_2_12"><span class="lab"><font color="Green" title="E148">A518</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_2_12"><span class="lab"><font color="Green" title="E149">A519</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>;<br/><a NAME="E4:34_2_1_1_1_3_1_5_1_1_2_12_1_1_2"/><span class="lab"><font color="Green" title="E162">A535</font></span>: 
<font color="Maroon" title="c17">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="#E48:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E123">A446</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_3_1_5_1_1_2_12"><span class="lab"><font color="Green" title="E148">A518</font></span></a>, <a class="ref" href="xreal_1.html#T146" title="XREAL_1:th.146">XREAL_1:146</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E5:34_2_1_1_1_3_1_5_1_1_2_12_1_1_2"/><span class="lab"><font color="Green" title="E163">A536</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c14">MD2</font>,<font color="Maroon" title="c17">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">MD2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</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="#E3:34_2_1_1_1_3_1_5_1_1_2_12_1"><span class="lab"><font color="Green" title="E158">A528</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_12_1_1_2"><span class="lab"><font color="Green" title="E160">A533</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E6:34_2_1_1_1_3_1_5_1_1_2_12_1_1_2"/><span class="lab"><font color="Green" title="E164">A537</font></span>: 
<font color="Maroon" title="c17">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <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="c14">MD2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_2_12_1"><span class="lab"><font color="Green" title="E158">A528</font></span></a>, <a class="txt" href="#E1:34_2_1_1_1_3_1_5_1_1_2_12_1_1_2"><span class="lab"><font color="Green" title="E160">A533</font></span></a>, <a class="ref" href="integra1.html#T7" title="INTEGRA1:th.7">INTEGRA1:7</a></span>;<br/><a NAME="E7:34_2_1_1_1_3_1_5_1_1_2_12_1_1_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E8:34_2_1_1_1_3_1_5_1_1_2_12_1_1_2"/><span class="kw">then </span>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c14">MD2</font>,<font color="Maroon" title="c17">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</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="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="#E33:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E78">A272</font></span></a>, <a class="txt" href="#E23:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E69">A263</font></span></a>, <a class="txt" href="#E10:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E95">A358</font></span></a>, <a class="txt" href="#E32:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E111">A388</font></span></a>, <a class="txt" href="#E34:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E113">A390</font></span></a>, <a class="txt" href="#E6:34_2_1_1_1_3_1_5_1_1_2_12_1_1_2"><span class="lab"><font color="Green" title="E164">A537</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_3_1_5_1_1_2_12_1_1_2"><span class="lab"><font color="Green" title="E162">A535</font></span></a>, <a class="txt" href="#E5:34_2_1_1_1_3_1_5_1_1_2_12_1_1_2"><span class="lab"><font color="Green" title="E163">A536</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>;<br/><span class="kw">hence </span><a NAME="E9:34_2_1_1_1_3_1_5_1_1_2_12_1_1_2"/>
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c14">MD2</font>,<font color="Maroon" title="c17">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c18">k2</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="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c14">MD2</font>,<font color="Maroon" title="c17">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c6">D2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c18">k2</font></span>)</span>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:34_2_1_1_1_3_1_5_1_1_2_12"><span class="lab"><font color="Green" title="E152">A522</font></span></a>, <a class="txt" href="#E2:34_2_1_1_1_3_1_5_1_1_2_12_1"><span class="lab"><font color="Green" title="E157">A527</font></span></a>, <a class="txt" href="#E4:34_2_1_1_1_3_1_5_1_1_2_12_1"><span class="lab"><font color="Green" title="E159">A529</font></span></a>, <a class="txt" href="#E3:34_2_1_1_1_3_1_5_1_1_2_12_1_1_2"><span class="lab"><font color="Green" title="E161">A534</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 NAME="E19:34_2_1_1_1_3_1_5_1_1_2_12"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c14">MD2</font>,<font color="Maroon" title="c17">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><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="c14">MD2</font>,<font color="Maroon" title="c17">k</font>)</span>)</span></span>)</span>,<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="c14">MD2</font>,<font color="Maroon" title="c17">k</font>)</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T4" title="INTEGRA1:th.4">INTEGRA1:4</a></span>;<br/>
<a NAME="E20:34_2_1_1_1_3_1_5_1_1_2_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E157">A538</font></span>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c14">MD2</font>,<font color="Maroon" title="c17">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c18">k2</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:34_2_1_1_1_3_1_5_1_1_2_12"><span class="lab"><font color="Green" title="E156">A526</font></span></a>, <a class="ref" href="integra1.html#T4" title="INTEGRA1:th.4">INTEGRA1:4</a></span>;<br/>
<a NAME="E21:34_2_1_1_1_3_1_5_1_1_2_12"/>
<font color="Maroon" title="c17">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="c14">MD2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:34_2_1_1_1_3_1_5_1_1_2_12"><span class="lab"><font color="Green" title="E155">A525</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="E22:34_2_1_1_1_3_1_5_1_1_2_12"/><span class="kw">then </span>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c6">D2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c18">k2</font></span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:34_2_1_1_1_3_1_5_1_1_2_12"><span class="lab"><font color="Green" title="E157">A538</font></span></a>, <a class="ref" href="integra1.html#T8" title="INTEGRA1:th.8">INTEGRA1:8</a></span>;<br/>
<a NAME="E23:34_2_1_1_1_3_1_5_1_1_2_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E158">A539</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="c6">D2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c18">k2</font></span>)</span>)</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"><font color="Maroon" title="c15">g</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="c6">D2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c18">k2</font></span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T51" title="FUNCT_1:th.51">FUNCT_1:51</a></span>;<br/>
<a NAME="E24:34_2_1_1_1_3_1_5_1_1_2_12"/>
<font color="Maroon" title="c17">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="c14">MD2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:34_2_1_1_1_3_1_5_1_1_2_12"><span class="lab"><font color="Green" title="E155">A525</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">hence </span><a NAME="E25:34_2_1_1_1_3_1_5_1_1_2_12"/>
<span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c14">MD2</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:34_2_1_1_1_3_1_5_1_1_2_12"><span class="lab"><font color="Green" title="E154">A524</font></span></a>, <a class="txt" href="#E20:34_2_1_1_1_3_1_5_1_1_2_12"><span class="lab"><font color="Green" title="E157">A538</font></span></a>, <a class="txt" href="#E23:34_2_1_1_1_3_1_5_1_1_2_12"><span class="lab"><font color="Green" title="E158">A539</font></span></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1: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><a NAME="E93:34_2_1_1_1_3_1_5_1_1_2"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E54">A237</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/><a NAME="E94:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E147">A540</font></span>: 
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c13">MD1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD1</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E105">A374</font></span></a>, <a class="txt" href="#E90:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E144">A514</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/><a NAME="E95:34_2_1_1_1_3_1_5_1_1_2"/>
<font color="Maroon" title="c10">j</font> <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="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</font></span></a>, <a class="txt" href="#E16:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A245</font></span></a>, <a class="txt" href="#E51:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E86">A280</font></span></a>, <a class="ref" href="integra1.html#T7" title="INTEGRA1:th.7">INTEGRA1:7</a></span>;<br/><a NAME="E96:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <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="#E12:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E58">A241</font></span></a>, <a class="txt" href="#E44:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E119">A442</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/><a NAME="E97:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c12">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E91">A354</font></span></a>, <a class="txt" href="#E89:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E143">A513</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E98:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c14">MD2</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> (<font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c13">MD1</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"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c15">g</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c15">g</font></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="c13">MD1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E93">A356</font></span></a>, <a class="txt" href="#E37:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E116">A432</font></span></a>, <a class="txt" href="#E57:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E127">A490</font></span></a>, <a class="txt" href="#E94:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E147">A540</font></span></a>, <a class="txt" href="#E31:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E110">A387</font></span></a>, <a class="ref" href="integra3.html#T13" target="_self" title="INTEGRA3:th.13">Th13</a></span>;<br/><a NAME="E99:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E148">A541</font></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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c14">MD2</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> (<font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c13">MD1</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"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></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="c13">MD1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E118">A441</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E100:34_2_1_1_1_3_1_5_1_1_2"/>
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E109">A386</font></span></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/><a NAME="E101:34_2_1_1_1_3_1_5_1_1_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"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>)</span>)</span> <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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="#E22:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E68">A262</font></span></a>, <a class="txt" href="#E32:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E111">A388</font></span></a>, <a class="txt" href="#E45:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E120">A443</font></span></a>, <a class="txt" href="#E91:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E145">A515</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>;<br/><a NAME="E102:34_2_1_1_1_3_1_5_1_1_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"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c14">MD2</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> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E78">A272</font></span></a>, <a class="txt" href="#E27:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E108">A385</font></span></a>, <a class="txt" href="#E48:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E123">A446</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E103:34_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E149">A542</font></span>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c14">MD2</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E92:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E146">A516</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</a></span>;<br/><a NAME="E104:34_2_1_1_1_3_1_5_1_1_2"/>
<font color="Maroon" title="c11">n1</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">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E73">A267</font></span></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/><span class="kw">then </span> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c10">j</font>)</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="#E17:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A257</font></span></a>, <a class="txt" href="#E25:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E71">A265</font></span></a>, <a class="txt" href="#E15:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E61">A244</font></span></a>, <a class="txt" href="#E35:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E114">A407</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="#E17:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A257</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>
;<br/><a NAME="E106:34_2_1_1_1_3_1_5_1_1_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"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c13">MD1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c10">j</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E102">A371</font></span></a>, <a class="txt" href="#E36:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E115">A408</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</a></span>;<br/><span class="kw">hence </span><a NAME="E107:34_2_1_1_1_3_1_5_1_1_2"/>
<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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</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"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c10">j</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"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></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="c5">D1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E99:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E148">A541</font></span></a>, <a class="txt" href="#E87:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E142">A512</font></span></a>, <a class="txt" href="#E103:34_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E149">A542</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:34_2_1_1_1_3_1_5"/>
<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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</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"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c10">j</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"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></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="c5">D1</font></span>)</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="E54:34_2_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A543</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)) <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</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="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</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"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p4">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c10">j</font>)</span>)</span></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="c9">i</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c5">D1</font></span>)</span></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"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c5">D1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E60">A243</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<a NAME="E55:34_2_1_1_1_3_1"/>
<font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">n1</font> <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="E56:34_2_1_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E58">A241</font></span></a>, <a class="txt" href="#E28:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E74">A268</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/>
<a NAME="E57:34_2_1_1_1_3_1"/><span class="kw">then </span>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E65">A259</font></span></a>, <a class="txt" href="#E37:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E80">A274</font></span></a>, <a class="txt" href="#E29:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E75">A269</font></span></a>, <a class="txt" href="#E31:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E77">A271</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/>
<a NAME="E58:34_2_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A544</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E78">A272</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E59:34_2_1_1_1_3_1"/>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</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="c6">D2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E65">A259</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="E60:34_2_1_1_1_3_1"/><span class="kw">then </span>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</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">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/>
<a NAME="E61:34_2_1_1_1_3_1"/><span class="kw">then </span>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</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">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</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><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)) = 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D20" title="INTEGRA1:def.20">INTEGRA1:def 20</a></span>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),1,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E66">A260</font></span></a>, <a class="ref" href="finseq_6.html#T116" title="FINSEQ_6:th.116">FINSEQ_6:116</a></span>
;<br/>
<span class="kw">then </span><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)) <a href="real_1.html#K7" title="REAL_1:func.7">+</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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>)</span>)</span></span>)</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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),1,<span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T75" title="RVSUM_1:th.75">RVSUM_1:75</a></span>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),1,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E66">A260</font></span></a>, <a class="txt" href="#E58:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E89">A544</font></span></a>, <a class="txt" href="#E24:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E70">A264</font></span></a>, <a class="ref" href="integra2.html#T4" title="INTEGRA2:th.4">INTEGRA2:4</a></span>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E68">A262</font></span></a>, <a class="ref" href="finseq_6.html#T116" title="FINSEQ_6:th.116">FINSEQ_6:116</a></span>
;<br/>
<a NAME="E64:34_2_1_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)) <a href="real_1.html#K7" title="REAL_1:func.7">+</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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)</span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E85">A279</font></span></a>, <a class="ref" href="integra1.html#D20" title="INTEGRA1:def.20">INTEGRA1:def 20</a></span>;<br/>
<a NAME="E65:34_2_1_1_1_3_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)) <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</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="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">n1</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</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"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p4">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c10">j</font>)</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)) <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">j</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E84">A278</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E66:34_2_1_1_1_3_1"/>
 ex <font color="Olive" title="b1">j</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/>( <font color="Olive" title="b1">j</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="c5">D1</font> &amp; <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Olive" title="b1">j</font>) &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Olive" title="b1">j</font>)) <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Olive" title="b1">j</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="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</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="c5">D1</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A236</font></span></a>, <a class="txt" href="#E5:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E54">A237</font></span></a>, <a class="txt" href="#E54:34_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E88">A543</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="E4:34_2_1_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <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="E7:34_2_1_1_1"/>
 for <font color="Olive" title="b1">k</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S10" title="NAT_1:sch.10">NAT_1:sch 10</a>(<a class="txt" href="#E4:34_2_1_1_1"><span class="lab"><font color="Green" title="E47">A40</font></span></a>, <a class="txt" href="#E6:34_2_1_1_1"><span class="lab"><font color="Green" title="E48">A231</font></span></a>);<br/></span>
<a NAME="E8:34_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">i</font>]
 
;<br/>
<span class="kw">hence </span><a NAME="E9:34_2_1_1_1"/>
 ex <font color="Olive" title="b1">j</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/>( <font color="Olive" title="b1">j</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="c5">D1</font> &amp; <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Olive" title="b1">j</font>) &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Olive" title="b1">j</font>)) <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Olive" title="b1">j</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</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="c5">D1</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2_1_1_1"><span class="lab"><font color="Green" title="E44">A17</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="E3:34_2_1_1"/><span class="lab"><font color="Green" title="E44">A545</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</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="c5">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></span>;<br/>
<a NAME="E4:34_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.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="c5">D1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E5:34_2_1_1"/><span class="kw">then </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#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font></span>)</span>)</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/>
<a NAME="E6:34_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A546</font></span>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.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="c6">D2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font></span>)</span>)</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/>
<a NAME="E7:34_2_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</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">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">j</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="E9:34_2_1_1"/><span class="lab"><font color="Green" title="E46">A547</font></span>: 
<font color="Maroon" title="c7">j</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="c5">D1</font>
 <span class="kw">and </span><br/><a NAME="E10:34_2_1_1"/><span class="lab"><font color="Green" title="E47">A548</font></span>: 
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.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="c4">D</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c7">j</font>)
 <span class="kw">and </span><br/><a NAME="E11:34_2_1_1"/><span class="lab"><font color="Green" title="E48">A549</font></span>: 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c7">j</font>)) <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c7">j</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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</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="c5">D1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_2_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a></span>;<br/>
<a NAME="E12:34_2_1_1"/><span class="lab"><font color="Green" title="E49">A550</font></span>: 
<font color="Maroon" title="c7">j</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="c5">D1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:34_2_1_1_2"/><span class="lab"><font color="Green" title="E50">A551</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:34_2_1_1_2"/>
<font color="Maroon" title="c7">j</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="c5">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:34_2_1_1"><span class="lab"><font color="Green" title="E46">A547</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E3:34_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2_1_1_2"><span class="lab"><font color="Green" title="E50">A551</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E4:34_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.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="c5">D1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:34_2_1_1"><span class="lab"><font color="Green" title="E46">A547</font></span></a>, <a class="txt" href="#E3:34_2_1_1"><span class="lab"><font color="Green" title="E44">A545</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/>
<a NAME="E5:34_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A552</font></span>: 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="integra1.html#D2" title="INTEGRA1:def.2">INTEGRA1:def 2</a></span>;<br/>
<a NAME="E6:34_2_1_1_2"/><span class="lab"><font color="Green" title="E52">A553</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="c5">D1</font>,<font color="Maroon" title="c7">j</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_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="E1:34_2_1_1_2_1_1"/>
( <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c7">j</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="34_2_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c7">j</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="c5">D1</font>,<font color="Maroon" title="c7">j</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:34_2_1_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="c5">D1</font>,<font color="Maroon" title="c7">j</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="#E9:34_2_1_1"><span class="lab"><font color="Green" title="E46">A547</font></span></a>, <a class="txt" href="#E5:34_2_1_1_2"><span class="lab"><font color="Green" title="E51">A552</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="34_2_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:34_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c7">j</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="c5">D1</font>,<font color="Maroon" title="c7">j</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:34_2_1_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="c5">D1</font>,<font color="Maroon" title="c7">j</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="#E9:34_2_1_1"><span class="lab"><font color="Green" title="E46">A547</font></span></a>, <a class="txt" href="#E5:34_2_1_1_2"><span class="lab"><font color="Green" title="E51">A552</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 NAME="E7:34_2_1_1_2"/>
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.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="c4">D</font></span>)</span> <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> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c7">j</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:34_2_1_1"><span class="lab"><font color="Green" title="E47">A548</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:34_2_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:34_2_1_1_2"><span class="lab"><font color="Green" title="E52">A553</font></span></a>, <a class="ref" href="integra1.html#D2" title="INTEGRA1:def.2">INTEGRA1: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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E13:34_2_1_1"/>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font></span>)</span>) <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">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E3:34_2_1_1"><span class="lab"><font color="Green" title="E44">A545</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E14:34_2_1_1"/><span class="kw">then </span>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</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="c6">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E6:34_2_1_1"><span class="lab"><font color="Green" title="E45">A546</font></span></a>, <a class="ref" href="seq_4.html#T138" title="SEQ_4:th.138">SEQ_4:138</a></span>;<br/>
<a NAME="E15:34_2_1_1"/><span class="kw">then </span>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font>) <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> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</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="c5">D1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:34_2_1_1"><span class="lab"><font color="Green" title="E48">A549</font></span></a>, <a class="txt" href="#E12:34_2_1_1"><span class="lab"><font color="Green" title="E49">A550</font></span></a>, <a class="ref" href="integra1.html#T43" title="INTEGRA1:th.43">INTEGRA1:43</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:34_2_1_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</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> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</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="c5">D1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T43" title="INTEGRA1:th.43">INTEGRA1:43</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="E8:34_2_1"/>
 ex <font color="Olive" title="b1">D2</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Maroon" title="c4">D</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b1">D2</font> &amp; <font color="Maroon" title="c5">D1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b1">D2</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b1">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">D2</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> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</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="c5">D1</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_2_1"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="txt" href="#E3:34_2_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E4:34_2_1"><span class="lab"><font color="Green" title="E41">A14</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="E3:34_2"/>
 ex <font color="Olive" title="b1">D2</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Maroon" title="c4">D</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b1">D2</font> &amp; <font color="Maroon" title="c5">D1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b1">D2</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b1">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">D2</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> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</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="c5">D1</font></span>)</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="E7:34"/><span class="lab"><font color="Green" title="E38">A554</font></span>: 
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="fdiff_1.html#D1" title="FDIFF_1:def.1">FDIFF_1:def 1</a></span>;<br/>
<a NAME="E8:34"/><span class="lab"><font color="Green" title="E39">A555</font></span>: 
 <a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c3">T</font> is  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="fdiff_1.html#D1" title="FDIFF_1:def.1">FDIFF_1:def 1</a></span>;<br/>
<a NAME="E9:34"/><span class="lab"><font color="Green" title="E40">A556</font></span>: 
 for <font color="Olive" title="b1">e</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  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="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">m</font> &amp; <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">e</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">e</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.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="c4">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  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="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">m</font> &amp; <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">e</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:34_3"/>
<font color="Maroon" title="c4">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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">  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="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">m</font> &amp; <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">e</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">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="E3:34_3"/><span class="lab"><font color="Green" title="E41">A557</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="c5">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="seq_1.html#K1" title="SEQ_1:func.1">.</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="c4">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E7:34"><span class="lab"><font color="Green" title="E38">A554</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="E4:34_3"/>
 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="c5">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="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">m</font> &amp; <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="seq_1.html#K1" title="SEQ_1:func.1">.</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="c4">e</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3_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"> ( <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font> implies (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">m</font> &amp; <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">e</font> ) )</span><br/>

<a NAME="E1:34_3_1"/><span class="lab"><font color="Green" title="E42">A558</font></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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c6">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra3.html#D2" target="_self" title="INTEGRA3:def.2">Def2</a></span>;<br/>
<a NAME="E2:34_3_1"/>
 <a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c6">m</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>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c6">m</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/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">i</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="E4:34_3_1"/><span class="lab"><font color="Green" title="E43">A559</font></span>: 
<font color="Maroon" title="c7">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> <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>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c6">m</font></span>)</span>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:34_3_1"/><span class="lab"><font color="Green" title="E44">A560</font></span>: 
 <a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c6">m</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>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c6">m</font></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T3" title="PARTFUN1:th.3">PARTFUN1:3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">D</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E7:34_3_1"/><span class="lab"><font color="Green" title="E45">A561</font></span>: 
<font color="Maroon" title="c8">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c6">m</font>
 ;<br/>
<a NAME="E8:34_3_1"/>
<font color="Maroon" title="c7">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> <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>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c6">m</font></span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_3_1"><span class="lab"><font color="Green" title="E43">A559</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="E9:34_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">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="c8">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:34_3_1"><span class="lab"><font color="Green" title="E45">A561</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/>
<a NAME="E10:34_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">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="c8">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="E11:34_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A562</font></span>: 
 <a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c7">i</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_3_1"><span class="lab"><font color="Green" title="E44">A560</font></span></a>, <a class="txt" href="#E7:34_3_1"><span class="lab"><font color="Green" title="E45">A561</font></span></a>, <a class="ref" href="integra1.html#T20" title="INTEGRA1:th.20">INTEGRA1:20</a></span>;<br/>
<span class="kw">assume </span><a NAME="E12:34_3_1"/>
<font color="Maroon" title="c5">n</font> <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="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">m</font> &amp; <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">e</font> )</span><br/>

<a NAME="E13:34_3_1"/><span class="kw">then </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="seq_1.html#K1" title="SEQ_1:func.1">.</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="c4">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_3"><span class="lab"><font color="Green" title="E41">A557</font></span></a></span>;<br/>
<a NAME="E14:34_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A563</font></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="seq_1.html#K1" title="SEQ_1:func.1">.</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="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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="seq_1.html#K1" title="SEQ_1:func.1">.</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></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">e</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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="seq_1.html#K1" title="SEQ_1:func.1">.</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></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#T4" title="ABSVALUE:th.4">ABSVALUE:4</a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E15:34_3_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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="#E8:34"><span class="lab"><font color="Green" title="E39">A555</font></span></a>, <a class="ref" href="seq_1.html#T5" title="SEQ_1:th.5">SEQ_1:5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:34_3_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">m</font> &amp; <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">e</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:34_3_1"><span class="lab"><font color="Green" title="E47">A563</font></span></a>, <a class="txt" href="#E1:34_3_1"><span class="lab"><font color="Green" title="E42">A558</font></span></a>, <a class="txt" href="#E11:34_3_1"><span class="lab"><font color="Green" title="E46">A562</font></span></a>, <a class="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</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="E5:34_3"/>
 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="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">m</font> &amp; <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">e</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="E10:34"/><span class="lab"><font color="Green" title="E41">A564</font></span>: 
 for <font color="Olive" title="b1">e</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 <font color="Olive" title="b1">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  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"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">m</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#K11" title="INTEGRA1:func.11">lower_integral</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">e</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c4">h</font> =  <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> <font color="Maroon" title="c2">f</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">H</font> =  <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> <font color="Maroon" title="c2">f</font></span>)</span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">e</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"> ( <font color="Maroon" title="c6">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  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"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">m</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#K11" title="INTEGRA1:func.11">lower_integral</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">e</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:34_4"/><span class="lab"><font color="Green" title="E42">A565</font></span>: 
<font color="Maroon" title="c6">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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">  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"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">m</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#K11" title="INTEGRA1:func.11">lower_integral</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">e</font></span><br/>

<a NAME="E2:34_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A566</font></span>: 
<font color="Maroon" title="c6">e</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 <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#T139" title="XREAL_1:th.139">XREAL_1:139</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">e</font> = <font color="Maroon" title="c6">e</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<a NAME="E4:34_4"/><span class="lab"><font color="Green" title="E44">A567</font></span>: 
<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> <font color="Maroon" title="c2">f</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_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> <font color="Maroon" title="c2">f</font></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="#E1:34"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="integra3.html#E14"><span class="lab"><font color="Green" title="E12">Lm3</font></span></a>, <a class="ref" href="xreal_1.html#T48" title="XREAL_1:th.48">XREAL_1:48</a></span>;<br/>
<a NAME="E5:34_4"/><span class="lab"><font color="Green" title="E45">A569</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="integra2.html#T36" title="INTEGRA2:th.36">INTEGRA2:36</a></span>;<br/>
<a NAME="E6:34_4"/>
 <a href="integra1.html#K11" title="INTEGRA1:func.11">lower_integral</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D15" title="INTEGRA1:def.15">INTEGRA1:def 15</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">y</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> <span class="kw"> such that </span><br/><a NAME="E8:34_4"/><span class="lab"><font color="Green" title="E46">A570</font></span>: 
<font color="Maroon" title="c8">y</font> <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#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E9:34_4"/><span class="lab"><font color="Green" title="E47">A571</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K11" title="INTEGRA1:func.11">lower_integral</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_4"><span class="lab"><font color="Green" title="E43">A566</font></span></a>, <a class="txt" href="#E5:34_4"><span class="lab"><font color="Green" title="E45">A569</font></span></a>, <a class="ref" href="seq_4.html#D1" title="SEQ_4:def.1">SEQ_4:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">D</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E11:34_4"/>
<font color="Maroon" title="c9">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E12:34_4"/><span class="lab"><font color="Green" title="E48">A573</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">D</font>
 <span class="kw">and </span><br/><a NAME="E13:34_4"/><span class="lab"><font color="Green" title="E49">A574</font></span>: 
<font color="Maroon" title="c9">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E8:34_4"><span class="lab"><font color="Green" title="E46">A570</font></span></a>, <a class="txt" href="integra3.html#E20"><span class="lab"><font color="Green" title="E18">Lm7</font></span></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>  =  <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="c9">D</font>,$1)</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">p</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">D</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">v</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="E15:34_4"/><span class="lab"><font color="Green" title="E50">A575</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">v</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="c9">D</font> &amp; (  for <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</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="c11">v</font> holds <br/><font color="Maroon" title="c11">v</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">j</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">consider </span><font color="Maroon" title="c12">v1</font> being   <a href="integra2.html#V1" title="INTEGRA2:attr.1">non-decreasing</a>   <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="E17:34_4"/><span class="lab"><font color="Green" title="E51">A576</font></span>: 
<font color="Maroon" title="c11">v</font>,<font color="Maroon" title="c12">v1</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra2.html#T3" title="INTEGRA2:th.3">INTEGRA2:3</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( $1 <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="c12">v1</font> &amp; <font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  );<br/>
<a NAME="E18:34_4"/><span class="lab"><font color="Green" title="E52">A577</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">v</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="c9">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:34_4"><span class="lab"><font color="Green" title="E50">A575</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="E19:34_4"/><span class="lab"><font color="Green" title="E53">A578</font></span>: 
 ex <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c13">H</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E2:34_4_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">v</font>
 <span class="kw">and </span><br/><a NAME="E3:34_4_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">v1</font>
 <span class="kw">and </span><br/><a NAME="E4:34_4_1"/>
<font color="Maroon" title="c13">H</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">and </span><br/><a NAME="E5:34_4_1"/><span class="lab"><font color="Green" title="E54">A579</font></span>: 
<font color="Maroon" title="c11">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">v1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:34_4"><span class="lab"><font color="Green" title="E51">A576</font></span></a>, <a class="ref" href="classes1.html#T77" title="CLASSES1:th.77">CLASSES1:77</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">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> <span class="kw"> such that </span><br/><a NAME="E7:34_4_1"/><span class="lab"><font color="Green" title="E55">A580</font></span>: 
<font color="Maroon" title="c14">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="c9">D</font>
 <span class="kw">and </span><br/><a NAME="E8:34_4_1"/><span class="lab"><font color="Green" title="E56">A581</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="c9">D</font>,<font color="Maroon" title="c14">k</font>)</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="#E3:34"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="ref" href="integra3.html#T2" target="_self" title="INTEGRA3:th.2">Th2</a></span>;<br/>
<a NAME="E9:34_4_1"/><span class="lab"><font color="Green" title="E57">A582</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">D</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="c11">v</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:34_4"><span class="lab"><font color="Green" title="E50">A575</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="E10:34_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">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="c12">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:34_4"><span class="lab"><font color="Green" title="E50">A575</font></span></a>, <a class="txt" href="#E18:34_4"><span class="lab"><font color="Green" title="E52">A577</font></span></a>, <a class="txt" href="#E5:34_4_1"><span class="lab"><font color="Green" title="E54">A579</font></span></a>, <a class="txt" href="#E7:34_4_1"><span class="lab"><font color="Green" title="E55">A580</font></span></a>, <a class="ref" href="funct_1.html#T11" title="FUNCT_1:th.11">FUNCT_1:11</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">Hk</font> = <font color="Maroon" title="c13">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/>
<a NAME="E12:34_4_1"/>
<font color="Maroon" title="c11">v</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">k</font> <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="#E15:34_4"><span class="lab"><font color="Green" title="E50">A575</font></span></a>, <a class="txt" href="#E18:34_4"><span class="lab"><font color="Green" title="E52">A577</font></span></a>, <a class="txt" href="#E7:34_4_1"><span class="lab"><font color="Green" title="E55">A580</font></span></a>, <a class="txt" href="#E8:34_4_1"><span class="lab"><font color="Green" title="E56">A581</font></span></a>, <a class="txt" href="#E9:34_4_1"><span class="lab"><font color="Green" title="E57">A582</font></span></a></span>;<br/>
<a NAME="E13:34_4_1"/><span class="kw">then </span>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">Hk</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:34_4"><span class="lab"><font color="Green" title="E50">A575</font></span></a>, <a class="txt" href="#E18:34_4"><span class="lab"><font color="Green" title="E52">A577</font></span></a>, <a class="txt" href="#E5:34_4_1"><span class="lab"><font color="Green" title="E54">A579</font></span></a>, <a class="txt" href="#E7:34_4_1"><span class="lab"><font color="Green" title="E55">A580</font></span></a>, <a class="txt" href="#E9:34_4_1"><span class="lab"><font color="Green" title="E57">A582</font></span></a>, <a class="ref" href="funct_1.html#T11" title="FUNCT_1:th.11">FUNCT_1:11</a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:34_4_1"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</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>
<span class="kw">consider </span><font color="Maroon" title="c13">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E21:34_4"/><span class="lab"><font color="Green" title="E54">A583</font></span>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">k</font>] &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="#E19:34_4"><span class="lab"><font color="Green" title="E53">A578</font></span></a>);<br/></span>
<a NAME="E22:34_4"/><span class="lab"><font color="Green" title="E55">A584</font></span>: 
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">D</font></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#T129" title="XREAL_1:th.129">XREAL_1:129</a></span>;<br/>
<a NAME="E23:34_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A585</font></span>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">D</font></span>)</span></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"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</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="#E4:34_4"><span class="lab"><font color="Green" title="E44">A567</font></span></a>, <a class="ref" href="xreal_1.html#T129" title="XREAL_1:th.129">XREAL_1:129</a></span>;<br/>
<a NAME="E24:34_4"/>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_4_2"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_4_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="E1:34_4_2_1"/>
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k</font> or  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_4_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:34_4_2_1_1"/>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</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><br/><div class="add"><span class="kw">hence </span><a NAME="E2:34_4_2_1_1"/>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</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="#E21:34_4"><span class="lab"><font color="Green" title="E54">A583</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_4_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:34_4_2_1_2"/>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</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="square_1.html#K1" title="SQUARE_1:func.1">min</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</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><br/><div class="add"><span class="kw">hence </span><a NAME="E2:34_4_2_1_2"/>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</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="#E1:34_4"><span class="lab"><font color="Green" title="E42">A565</font></span></a>, <a class="txt" href="#E23:34_4"><span class="lab"><font color="Green" title="E56">A585</font></span></a>, <a class="ref" href="xreal_1.html#T139" title="XREAL_1:th.139">XREAL_1:139</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">then </span><span class="kw">consider </span><font color="Maroon" title="c14">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="E26:34_4"/><span class="lab"><font color="Green" title="E57">A586</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="c14">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="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">m</font> &amp; <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:34"><span class="lab"><font color="Green" title="E40">A556</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c14">n</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">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="c14">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="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">m</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#K11" title="INTEGRA1:func.11">lower_integral</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">e</font></span><br/>

<a NAME="E27:34_4"/><span class="lab"><font color="Green" title="E58">A587</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c9">D</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:34_4"><span class="lab"><font color="Green" title="E48">A573</font></span></a>, <a class="ref" href="integra1.html#D11" title="INTEGRA1:def.11">INTEGRA1:def 11</a></span>;<br/>
<a NAME="E28:34_4"/>
 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="c14">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="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">m</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#K11" title="INTEGRA1:func.11">lower_integral</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">e</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_4_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:34_4_3"/><span class="lab"><font color="Green" title="E59">A588</font></span>: 
<font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_4_3_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:34_4_3_1"/><span class="lab"><font color="Green" title="E60">A589</font></span>: 
 for <font color="Olive" title="b1">n1</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">n1</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="c9">D</font> holds <br/> <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="c9">D</font>,<font color="Olive" title="b1">n1</font>)</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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_4_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">n1</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="c15">n1</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="c9">D</font> implies  <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="c9">D</font>,<font color="Maroon" title="c15">n1</font>)</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><br/>

<span class="kw">assume </span><a NAME="E1:34_4_3_1_1"/><span class="lab"><font color="Green" title="E61">A590</font></span>: 
<font color="Maroon" title="c15">n1</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="c9">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="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="c9">D</font>,<font color="Maroon" title="c15">n1</font>)</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><br/>

<a NAME="E2:34_4_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A591</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">n1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_4_3_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="E1:34_4_3_1_1_1"/>
( <font color="Maroon" title="c15">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c15">n1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_4_3_1_1"><span class="lab"><font color="Green" title="E62">A591</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="34_4_3_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:34_4_3_1_1_1_1"/><span class="lab"><font color="Green" title="E63">A592</font></span>: 
<font color="Maroon" title="c15">n1</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="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="c9">D</font>,<font color="Maroon" title="c15">n1</font>)</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><br/><div class="add"><a NAME="E2:34_4_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A593</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="c9">D</font>,<font color="Maroon" title="c15">n1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_4_3_1_1"><span class="lab"><font color="Green" title="E61">A590</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E3:34_4_3_1_1_1_1"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c9">D</font>,<font color="Maroon" title="c15">n1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_4_3_1_1"><span class="lab"><font color="Green" title="E61">A590</font></span></a>, <a class="txt" href="#E1:34_4_3_1_1_1_1"><span class="lab"><font color="Green" title="E63">A592</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E4:34_4_3_1_1_1_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="c9">D</font>,<font color="Maroon" title="c15">n1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="#E2:34_4_3_1_1_1_1"><span class="lab"><font color="Green" title="E64">A593</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E5:34_4_3_1_1_1_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="c9">D</font>,<font color="Maroon" title="c15">n1</font>)</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="#E13:34_4"><span class="lab"><font color="Green" title="E49">A574</font></span></a>, <a class="txt" href="#E1:34_4_3_1_1_1_1"><span class="lab"><font color="Green" title="E63">A592</font></span></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</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="34_4_3_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:34_4_3_1_1_1_2"/><span class="lab"><font color="Green" title="E63">A594</font></span>: 
<font color="Maroon" title="c15">n1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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="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="c9">D</font>,<font color="Maroon" title="c15">n1</font>)</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><br/><div class="add"><a NAME="E2:34_4_3_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A595</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="c9">D</font>,<font color="Maroon" title="c15">n1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_4_3_1_1"><span class="lab"><font color="Green" title="E61">A590</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E3:34_4_3_1_1_1_2"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c9">D</font>,<font color="Maroon" title="c15">n1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">n1</font> <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="#E1:34_4_3_1_1"><span class="lab"><font color="Green" title="E61">A590</font></span></a>, <a class="txt" href="#E1:34_4_3_1_1_1_2"><span class="lab"><font color="Green" title="E63">A594</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E4:34_4_3_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A596</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="c9">D</font>,<font color="Maroon" title="c15">n1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">n1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_4_3_1_1_1_2"><span class="lab"><font color="Green" title="E64">A595</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><a NAME="E5:34_4_3_1_1_1_2"/>
<font color="Maroon" title="c15">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">n1</font> <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="xreal_1.html#T29" title="XREAL_1:th.29">XREAL_1:29</a></span>;<br/><a NAME="E6:34_4_3_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A597</font></span>: 
<font color="Maroon" title="c15">n1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">n1</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="E7:34_4_3_1_1_1_2"/>
<font color="Maroon" title="c15">n1</font> <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="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_4_3_1_1"><span class="lab"><font color="Green" title="E61">A590</font></span></a>, <a class="txt" href="#E1:34_4_3_1_1_1_2"><span class="lab"><font color="Green" title="E63">A594</font></span></a>, <a class="ref" href="integra1.html#T7" title="INTEGRA1:th.7">INTEGRA1:7</a></span>;<br/><a NAME="E8:34_4_3_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">n1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_4_3_1_1"><span class="lab"><font color="Green" title="E61">A590</font></span></a>, <a class="txt" href="#E6:34_4_3_1_1_1_2"><span class="lab"><font color="Green" title="E66">A597</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E9:34_4_3_1_1_1_2"/>
 <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="c9">D</font>,<font color="Maroon" title="c15">n1</font>)</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="#E4:34_4_3_1_1_1_2"><span class="lab"><font color="Green" title="E65">A596</font></span></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</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 NAME="E2:34_4_3_1"/><span class="lab"><font color="Green" title="E61">A598</font></span>: 
<font color="Maroon" title="c13">k</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="c12">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:34_4"><span class="lab"><font color="Green" title="E54">A583</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E3:34_4_3_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:34_4"><span class="lab"><font color="Green" title="E54">A583</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E4:34_4_3_1"/><span class="kw">then </span>
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="c12">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_4_3_1"><span class="lab"><font color="Green" title="E61">A598</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E5:34_4_3_1"/><span class="kw">then </span>
1 <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="c12">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E6:34_4_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A599</font></span>: 
<font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c12">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E7:34_4_3_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c11">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c12">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:34_4"><span class="lab"><font color="Green" title="E51">A576</font></span></a>, <a class="ref" href="classes1.html#T75" title="CLASSES1:th.75">CLASSES1:75</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">n1</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="E9:34_4_3_1"/><span class="lab"><font color="Green" title="E63">A600</font></span>: 
<font color="Maroon" title="c15">n1</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="c11">v</font>
 <span class="kw">and </span><br/><a NAME="E10:34_4_3_1"/><span class="lab"><font color="Green" title="E64">A601</font></span>: 
<font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">v</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:34_4_3_1"><span class="lab"><font color="Green" title="E62">A599</font></span></a>, <a class="ref" href="partfun1.html#T3" title="PARTFUN1:th.3">PARTFUN1:3</a></span>;<br/>
<a NAME="E11:34_4_3_1"/>
<font color="Maroon" title="c15">n1</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="c9">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:34_4"><span class="lab"><font color="Green" title="E50">A575</font></span></a>, <a class="txt" href="#E9:34_4_3_1"><span class="lab"><font color="Green" title="E63">A600</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="E12:34_4_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A602</font></span>: 
<font color="Maroon" title="c15">n1</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="c9">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="E13:34_4_3_1"/>
<font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c9">D</font>,<font color="Maroon" title="c15">n1</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:34_4"><span class="lab"><font color="Green" title="E50">A575</font></span></a>, <a class="txt" href="#E9:34_4_3_1"><span class="lab"><font color="Green" title="E63">A600</font></span></a>, <a class="txt" href="#E10:34_4_3_1"><span class="lab"><font color="Green" title="E64">A601</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:34_4_3_1"/>
<font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <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="#E1:34_4_3_1"><span class="lab"><font color="Green" title="E60">A589</font></span></a>, <a class="txt" href="#E12:34_4_3_1"><span class="lab"><font color="Green" title="E65">A602</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="E2:34_4_3"/><span class="lab"><font color="Green" title="E60">A603</font></span>: 
<font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">min</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="c9">D</font>)</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_4_3_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:34_4_3_2"/><span class="lab"><font color="Green" title="E61">A604</font></span>: 
<font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_4_3_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:34_4_3_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">v1</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="c11">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:34_4"><span class="lab"><font color="Green" title="E51">A576</font></span></a>, <a class="ref" href="rfinseq.html#T3" title="RFINSEQ:th.3">RFINSEQ:3</a></span>;<br/>
<a NAME="E2:34_4_3_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">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="c11">v</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:34_4"><span class="lab"><font color="Green" title="E54">A583</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="E3:34_4_3_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A605</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E4:34_4_3_2_1"/>
<font color="Maroon" title="c13">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="c12">v1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:34_4"><span class="lab"><font color="Green" title="E54">A583</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="E5:34_4_3_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">k</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="c12">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E6:34_4_3_2_1"/><span class="kw">then </span>
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="c12">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_4_3_2_1"><span class="lab"><font color="Green" title="E62">A605</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E7:34_4_3_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A606</font></span>: 
1 <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="c12">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">assume </span><a NAME="E8:34_4_3_2_1"/>
<font color="Maroon" title="c13">k</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"> contradiction</span><br/>

<a NAME="E9:34_4_3_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">k</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="#E3:34_4_3_2_1"><span class="lab"><font color="Green" title="E62">A605</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:34_4_3_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:34_4"><span class="lab"><font color="Green" title="E54">A583</font></span></a>, <a class="txt" href="#E1:34_4_3"><span class="lab"><font color="Green" title="E59">A588</font></span></a>, <a class="txt" href="#E7:34_4_3_2_1"><span class="lab"><font color="Green" title="E63">A606</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="E2:34_4_3_2"/>
 <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">min</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="c9">D</font>)</span>)</span></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="c9">D</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D7" title="XXREAL_2:def.7">XXREAL_2:def 7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">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> <span class="kw"> such that </span><br/><a NAME="E4:34_4_3_2"/><span class="lab"><font color="Green" title="E62">A607</font></span>: 
<font color="Maroon" title="c15">m</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="c9">D</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:34_4_3_2"/><span class="lab"><font color="Green" title="E63">A608</font></span>: 
 <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">min</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="c9">D</font>)</span>)</span></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="c9">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T3" title="PARTFUN1:th.3">PARTFUN1:3</a></span>;<br/>
<a NAME="E6:34_4_3_2"/>
<font color="Maroon" title="c15">m</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="c9">D</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_4_3_2"><span class="lab"><font color="Green" title="E62">A607</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="E7:34_4_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A609</font></span>: 
<font color="Maroon" title="c15">m</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="c9">D</font></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="E8:34_4_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">m</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="c9">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="E9:34_4_3_2"/><span class="kw">then </span>
 <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">min</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="c9">D</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c9">D</font>,<font color="Maroon" title="c15">m</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_4_3_2"><span class="lab"><font color="Green" title="E63">A608</font></span></a>, <a class="ref" href="integra1.html#T20" title="INTEGRA1:th.20">INTEGRA1:20</a></span>;<br/>
<a NAME="E10:34_4_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A610</font></span>: 
<font color="Maroon" title="c11">v</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">min</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="c9">D</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:34_4"><span class="lab"><font color="Green" title="E50">A575</font></span></a>, <a class="txt" href="#E18:34_4"><span class="lab"><font color="Green" title="E52">A577</font></span></a>, <a class="txt" href="#E7:34_4_3_2"><span class="lab"><font color="Green" title="E64">A609</font></span></a></span>;<br/>
<a NAME="E11:34_4_3_2"/><span class="lab"><font color="Green" title="E66">A611</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c11">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c12">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:34_4"><span class="lab"><font color="Green" title="E51">A576</font></span></a>, <a class="ref" href="classes1.html#T75" title="CLASSES1:th.75">CLASSES1:75</a></span>;<br/>
<a NAME="E12:34_4_3_2"/>
<font color="Maroon" title="c15">m</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="c11">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:34_4"><span class="lab"><font color="Green" title="E50">A575</font></span></a>, <a class="txt" href="#E7:34_4_3_2"><span class="lab"><font color="Green" title="E64">A609</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="E13:34_4_3_2"/><span class="kw">then </span>
 <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">min</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="c9">D</font>)</span>)</span></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> <font color="Maroon" title="c11">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:34_4_3_2"><span class="lab"><font color="Green" title="E65">A610</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">m1</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="E15:34_4_3_2"/><span class="lab"><font color="Green" title="E67">A612</font></span>: 
<font color="Maroon" title="c16">m1</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="c12">v1</font>
 <span class="kw">and </span><br/><a NAME="E16:34_4_3_2"/><span class="lab"><font color="Green" title="E68">A613</font></span>: 
 <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">min</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="c9">D</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">m1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:34_4_3_2"><span class="lab"><font color="Green" title="E66">A611</font></span></a>, <a class="ref" href="partfun1.html#T3" title="PARTFUN1:th.3">PARTFUN1:3</a></span>;<br/>
<a NAME="E17:34_4_3_2"/>
<font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c12">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:34_4"><span class="lab"><font color="Green" title="E54">A583</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">k2</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="E19:34_4_3_2"/><span class="lab"><font color="Green" title="E69">A614</font></span>: 
<font color="Maroon" title="c17">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="c11">v</font>
 <span class="kw">and </span><br/><a NAME="E20:34_4_3_2"/><span class="lab"><font color="Green" title="E70">A615</font></span>: 
<font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">v</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:34_4_3_2"><span class="lab"><font color="Green" title="E66">A611</font></span></a>, <a class="ref" href="partfun1.html#T3" title="PARTFUN1:th.3">PARTFUN1:3</a></span>;<br/>
<a NAME="E21:34_4_3_2"/><span class="lab"><font color="Green" title="E71">A616</font></span>: 
<font color="Maroon" title="c17">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="c9">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:34_4"><span class="lab"><font color="Green" title="E50">A575</font></span></a>, <a class="txt" href="#E19:34_4_3_2"><span class="lab"><font color="Green" title="E69">A614</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="E22:34_4_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A617</font></span>: 
<font color="Maroon" title="c17">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="c9">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="E23:34_4_3_2"/>
<font color="Maroon" title="c17">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> <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="c9">D</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:34_4_3_2"><span class="lab"><font color="Green" title="E71">A616</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/>
<a NAME="E24:34_4_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A618</font></span>: 
<font color="Maroon" title="c17">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> <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="c9">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="E25:34_4_3_2"/>
<font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c9">D</font>,<font color="Maroon" title="c17">k2</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:34_4"><span class="lab"><font color="Green" title="E50">A575</font></span></a>, <a class="txt" href="#E19:34_4_3_2"><span class="lab"><font color="Green" title="E69">A614</font></span></a>, <a class="txt" href="#E20:34_4_3_2"><span class="lab"><font color="Green" title="E70">A615</font></span></a></span>;<br/>
<a NAME="E26:34_4_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k</font> <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="c9">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">k2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:34_4_3_2"><span class="lab"><font color="Green" title="E72">A617</font></span></a>, <a class="ref" href="integra1.html#T20" title="INTEGRA1:th.20">INTEGRA1:20</a></span>;<br/>
<a NAME="E27:34_4_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k</font> <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="c9">D</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:34_4_3_2"><span class="lab"><font color="Green" title="E73">A618</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="E28:34_4_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A619</font></span>: 
<font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">min</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="c9">D</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D7" title="XXREAL_2:def.7">XXREAL_2:def 7</a></span>;<br/>
<a NAME="E29:34_4_3_2"/>
<font color="Maroon" title="c16">m1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:34_4_3_2"><span class="lab"><font color="Green" title="E67">A612</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E30:34_4_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">min</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="c9">D</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:34_4"><span class="lab"><font color="Green" title="E54">A583</font></span></a>, <a class="txt" href="#E1:34_4_3_2"><span class="lab"><font color="Green" title="E61">A604</font></span></a>, <a class="txt" href="#E15:34_4_3_2"><span class="lab"><font color="Green" title="E67">A612</font></span></a>, <a class="txt" href="#E16:34_4_3_2"><span class="lab"><font color="Green" title="E68">A613</font></span></a>, <a class="ref" href="integra2.html#T2" title="INTEGRA2:th.2">INTEGRA2:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E31:34_4_3_2"/>
<font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">min</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="c9">D</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_4_3_2"><span class="lab"><font color="Green" title="E61">A604</font></span></a>, <a class="txt" href="#E28:34_4_3_2"><span class="lab"><font color="Green" title="E74">A619</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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>
<a NAME="E3:34_4_3"/>
<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> <font color="Maroon" title="c2">f</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_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> <font color="Maroon" title="c2">f</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"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></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="ref" href="xreal_1.html#T29" title="XREAL_1:th.29">XREAL_1:29</a></span>;<br/>
<a NAME="E4:34_4_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A620</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">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"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <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="c9">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"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">sD</font> =  <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c9">D</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c16">s</font> =  <a href="integra1.html#K11" title="INTEGRA1:func.11">lower_integral</a> <font color="Maroon" title="c2">f</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c17">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="c14">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">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"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</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#K11" title="INTEGRA1:func.11">lower_integral</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">e</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c18">D1</font> = <font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c17">m</font> as    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font> ;<br/>
<a NAME="E6:34_4_3"/><span class="lab"><font color="Green" title="E62">A621</font></span>: 
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></span>;<br/>
<span class="kw">assume </span><a NAME="E7:34_4_3"/><span class="lab"><font color="Green" title="E63">A622</font></span>: 
<font color="Maroon" title="c14">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">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"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</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#K11" title="INTEGRA1:func.11">lower_integral</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">e</font></span><br/>

<a NAME="E8:34_4_3"/><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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:34_4"><span class="lab"><font color="Green" title="E57">A586</font></span></a></span>;<br/>
<a NAME="E9:34_4_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A623</font></span>: 
 <a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c18">D1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra3.html#D2" target="_self" title="INTEGRA3:def.2">Def2</a></span>;<br/>
<a NAME="E10:34_4_3"/>
<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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:34_4"><span class="lab"><font color="Green" title="E57">A586</font></span></a>, <a class="txt" href="#E7:34_4_3"><span class="lab"><font color="Green" title="E63">A622</font></span></a></span>;<br/>
<a NAME="E11:34_4_3"/><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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:34_4_3"><span class="lab"><font color="Green" title="E62">A621</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E12:34_4_3"/><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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">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">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:34_4"><span class="lab"><font color="Green" title="E55">A584</font></span></a>, <a class="txt" href="#E4:34_4"><span class="lab"><font color="Green" title="E44">A567</font></span></a>, <a class="ref" href="xreal_1.html#T79" title="XREAL_1:th.79">XREAL_1:79</a>, <a class="ref" href="xreal_1.html#T129" title="XREAL_1:th.129">XREAL_1:129</a></span>;<br/>
<a NAME="E13:34_4_3"/><span class="kw">then </span>
<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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">D</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">e</font>
 
;<br/>
<a NAME="E14:34_4_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A624</font></span>: 
<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="c9">D</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></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"><a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T81" title="XREAL_1:th.81">XREAL_1:81</a></span>;<br/>
<a NAME="E15:34_4_3"/>
<font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c17">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K1" title="INTEGRA1:func.1">divs</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D3" title="INTEGRA1:def.3">INTEGRA1:def 3</a></span>;<br/>
<a NAME="E16:34_4_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A625</font></span>: 
<font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c17">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E17:34_4_3"/>
<span class="p1">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c17">m</font></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="E18:34_4_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c17">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D11" title="INTEGRA1:def.11">INTEGRA1:def 11</a></span>;<br/>
<a NAME="E19:34_4_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font> <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#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:34_4_3"><span class="lab"><font color="Green" title="E66">A625</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="E20:34_4_3"/><span class="kw">then </span>
 <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"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_4"><span class="lab"><font color="Green" title="E45">A569</font></span></a>, <a class="ref" href="seq_4.html#D1" title="SEQ_4:def.1">SEQ_4:def 1</a></span>;<br/>
<a NAME="E21:34_4_3"/><span class="kw">then </span>
 <a href="integra1.html#K11" title="INTEGRA1:func.11">lower_integral</a> <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D15" title="INTEGRA1:def.15">INTEGRA1:def 15</a></span>;<br/>
<a NAME="E22:34_4_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A626</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K11" title="INTEGRA1:func.11">lower_integral</a> <font color="Maroon" title="c2">f</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="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</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="xreal_1.html#T48" title="XREAL_1:th.48">XREAL_1:48</a></span>;<br/>
<a NAME="E23:34_4_3"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:34_4"><span class="lab"><font color="Green" title="E57">A586</font></span></a>, <a class="txt" href="#E7:34_4_3"><span class="lab"><font color="Green" title="E63">A622</font></span></a></span>;<br/>
<a NAME="E24:34_4_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A627</font></span>: 
<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="c9">D</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</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"><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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">D</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></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"><a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_4_3"><span class="lab"><font color="Green" title="E61">A620</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c19">sD1</font> =  <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c17">m</font></span>)</span>);<br/>
<span class="kw">consider </span><font color="Maroon" title="c20">D2</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E26:34_4_3"/><span class="lab"><font color="Green" title="E69">A628</font></span>: 
<font color="Maroon" title="c9">D</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Maroon" title="c20">D2</font>
 <span class="kw">and </span><br/><a NAME="E27:34_4_3"/>
<font color="Maroon" title="c18">D1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Maroon" title="c20">D2</font>
 <span class="kw">and </span><br/><a NAME="E28:34_4_3"/><span class="lab"><font color="Green" title="E70">A629</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c20">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c18">D1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c9">D</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E29:34_4_3"/>
 <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"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c20">D2</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> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c9">D</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E30:34_4_3"/>
 <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"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c20">D2</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> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c18">D1</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34"><span class="lab"><font color="Green" title="E36">A5</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c21">sD2</font> =  <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c20">D2</font>);<br/>
<a NAME="E31:34_4_3"/><span class="lab"><font color="Green" title="E71">A630</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c9">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> (<font color="Maroon" title="c2">f</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c17">m</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"><span class="p2">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c20">D2</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> (<font color="Maroon" title="c2">f</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c17">m</font></span>)</span>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c9">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> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c20">D2</font>)</span>)</span>
 
;<br/>
<a NAME="E32:34_4_3"/>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></span>;<br/>
<a NAME="E33:34_4_3"/><span class="kw">then </span>
 <a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c18">D1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:34_4_3"><span class="lab"><font color="Green" title="E64">A623</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E34:34_4_3"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">D3</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Maroon" title="c9">D</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b1">D3</font> &amp; <font color="Maroon" title="c18">D1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b1">D3</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b1">D3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c18">D1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c9">D</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">D3</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> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c18">D1</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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">D</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</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="c18">D1</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:34"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E2:34_4_3"><span class="lab"><font color="Green" title="E60">A603</font></span></a></span>;<br/>
<a NAME="E35:34_4_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A631</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c20">D2</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> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c18">D1</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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">D</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</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="c18">D1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:34_4_3"><span class="lab"><font color="Green" title="E70">A629</font></span></a>, <a class="ref" href="integra3.html#T6" target="_self" title="INTEGRA3:th.6">Th6</a></span>;<br/>
<a NAME="E36:34_4_3"/>
<span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c9">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> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c20">D2</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E26:34_4_3"><span class="lab"><font color="Green" title="E69">A628</font></span></a>, <a class="ref" href="integra1.html#T46" title="INTEGRA1:th.46">INTEGRA1:46</a>, <a class="ref" href="xreal_1.html#T47" title="XREAL_1:th.47">XREAL_1:47</a></span>;<br/>
<a NAME="E37:34_4_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A632</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c9">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> (<font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c17">m</font></span>)</span>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c20">D2</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> (<font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c17">m</font></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:34_4_3"><span class="lab"><font color="Green" title="E71">A630</font></span></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/>
<a NAME="E38:34_4_3"/>
 <a href="integra3.html#K1" title="INTEGRA3:func.1">delta</a> <font color="Maroon" title="c18">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra3.html#D2" target="_self" title="INTEGRA3:def.2">Def2</a></span>;<br/>
<a NAME="E39:34_4_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c20">D2</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> (<font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c17">m</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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">D</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></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"><a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:34_4_3"><span class="lab"><font color="Green" title="E72">A631</font></span></a>, <a class="txt" href="#E24:34_4_3"><span class="lab"><font color="Green" title="E68">A627</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E40:34_4_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c9">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> (<font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c17">m</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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">D</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></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"><a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:34_4_3"><span class="lab"><font color="Green" title="E73">A632</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E41:34_4_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c9">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> (<font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c17">m</font></span>)</span>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:34_4_3"><span class="lab"><font color="Green" title="E65">A624</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E42:34_4_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A633</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c9">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> (<font color="Maroon" title="c2">f</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c17">m</font></span>)</span>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E43:34_4_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K11" title="INTEGRA1:func.11">lower_integral</a> <font color="Maroon" title="c2">f</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> (<font color="Maroon" title="c2">f</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c17">m</font></span>)</span>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c17">m</font></span>)</span>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c9">D</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:34_4"><span class="lab"><font color="Green" title="E47">A571</font></span></a>, <a class="txt" href="#E27:34_4"><span class="lab"><font color="Green" title="E58">A587</font></span></a>, <a class="ref" href="xreal_1.html#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/>
<a NAME="E44:34_4_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="integra1.html#K11" title="INTEGRA1:func.11">lower_integral</a> <font color="Maroon" title="c2">f</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> (<font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c17">m</font></span>)</span>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c9">D</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></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> (<font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c17">m</font></span>)</span>)</span>)</span>
 
<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="E45:34_4_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="integra1.html#K11" title="INTEGRA1:func.11">lower_integral</a> <font color="Maroon" title="c2">f</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> (<font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c17">m</font></span>)</span>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:34_4_3"><span class="lab"><font color="Green" title="E74">A633</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E46:34_4_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="integra1.html#K11" title="INTEGRA1:func.11">lower_integral</a> <font color="Maroon" title="c2">f</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="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">e</font>
 
<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="E47:34_4_3"/><span class="kw">then </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#K11" title="INTEGRA1:func.11">lower_integral</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:34_4_3"><span class="lab"><font color="Green" title="E67">A626</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/>
<a NAME="E48:34_4_3"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="integra1.html#K11" title="INTEGRA1:func.11">lower_integral</a> <font color="Maroon" title="c2">f</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> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T52" title="COMPLEX1:th.52">COMPLEX1:52</a></span>;<br/>
<span class="kw">hence </span><a NAME="E49:34_4_3"/>
 <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="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</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#K11" title="INTEGRA1:func.11">lower_integral</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">e</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>
<span class="kw">hence </span><a NAME="E29:34_4"/>
 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="c14">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="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">m</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#K11" title="INTEGRA1:func.11">lower_integral</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">e</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>
<span class="kw">hence </span><a NAME="E11:34"/>
 <a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<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="ref" href="seq_2.html#D6" title="SEQ_2:def.6">SEQ_2:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K11" title="INTEGRA1:func.11">lower_integral</a> <font color="Maroon" title="c2">f</font></span><br/>

<span class="kw">hence </span><a NAME="E12:34"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K11" title="INTEGRA1:func.11">lower_integral</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:34"><span class="lab"><font color="Green" title="E41">A564</font></span></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2: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>
