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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#K3" title="INTEGRA2:func.3">upper_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#K3" title="INTEGRA2:func.3">upper_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#K10" title="INTEGRA1:func.10">upper_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#K3" title="INTEGRA2:func.3">upper_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#K3" title="INTEGRA2:func.3">upper_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#K10" title="INTEGRA1:func.10">upper_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#K3" title="INTEGRA2:func.3">upper_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#K3" title="INTEGRA2:func.3">upper_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#K10" title="INTEGRA1:func.10">upper_integral</a> <font color="Maroon" title="c2">f</font> ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:35"/><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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not  <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>  or  not  <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>  or (  <a href="integra2.html#K3" title="INTEGRA2:func.3">upper_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#K3" title="INTEGRA2:func.3">upper_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#K10" title="INTEGRA1:func.10">upper_integral</a> <font color="Maroon" title="c2">f</font> ) )</span><br/>

<a NAME="E2:35"/><span class="lab"><font color="Green" title="E33">A2</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#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">D</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b3">D2</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#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">D1</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b3">D2</font>)</span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_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#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">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#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">D2</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#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">D2</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="E2:35_1"/><span class="lab"><font color="Green" title="E34">A3</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="E3:35_1"/><span class="lab"><font color="Green" title="E35">A4</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="E4:35_1"/><span class="lab"><font color="Green" title="E36">A5</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="E5:35_1"/><span class="lab"><font color="Green" title="E37">A6</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</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="#E1:35"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E3:35_1"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="ref" href="integra1.html#T45" title="INTEGRA1:th.45">INTEGRA1:45</a>, <a class="ref" href="xreal_1.html#T48" title="XREAL_1:th.48">XREAL_1:48</a></span>;<br/>
<a NAME="E6:35_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">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#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</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="#E1:35"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E2:35_1"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="ref" href="integra1.html#T45" title="INTEGRA1:th.45">INTEGRA1:45</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="E7:35_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#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">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#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">D2</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#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">D2</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E3:35_1"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E4:35_1"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="txt" href="#E5:35_1"><span class="lab"><font color="Green" title="E37">A6</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="E3:35"/><span class="lab"><font color="Green" title="E34">A7</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#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">D1</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b3">D2</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="35_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#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">D2</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="E1:35_2"/><span class="lab"><font color="Green" title="E35">A8</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#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">D2</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="E2:35_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#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">D2</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="35_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="E2:35_2_1"/><span class="lab"><font color="Green" title="E36">A9</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="E3:35_2_1"/><span class="lab"><font color="Green" title="E37">A10</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="E4:35_2_1"/><span class="lab"><font color="Green" title="E38">A11</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="E5:35_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#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">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#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:35_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#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</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="#E2:35"><span class="lab"><font color="Green" title="E33">A2</font></span></a></span>;<br/>
<a NAME="E7:35_2_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</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="35_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#K4" title="INTEGRA1:func.4">upper_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#K4" title="INTEGRA1:func.4">upper_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="E1:35_2_1_1"/><span class="lab"><font color="Green" title="E39">A12</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="E2:35_2_1_1"/><span class="lab"><font color="Green" title="E40">A13</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="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="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="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="35_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="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="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="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="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="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="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="E1:35_2_1_1_1"/><span class="lab"><font color="Green" title="E41">A14</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="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="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="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="E2:35_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A15</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="E3:35_2_1_1_1"/><span class="lab"><font color="Green" title="E43">A16</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="35_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="E1:35_2_1_1_1_1"/><span class="lab"><font color="Green" title="E44">A17</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="E2:35_2_1_1_1_1"/><span class="lab"><font color="Green" title="E45">A18</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="E3:35_2_1_1_1_1"/><span class="lab"><font color="Green" title="E46">A19</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="E4:35_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="E5:35_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="E6:35_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A20</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="E7:35_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="#E2:35_2_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</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="E8:35_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="E9:35_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="#E6:35_2_1_1_1_1"><span class="lab"><font color="Green" title="E47">A20</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E10:35_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A21</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="#E2:35_2_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/>
<a NAME="E11:35_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="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="#E3:35_2_1_1_1_1"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/>
<a NAME="E12:35_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="35_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="35_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="E1:35_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="35_2_1_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E50">A23</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="E2:35_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="E3:35_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="E4:35_2_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A24</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/><a NAME="E5:35_2_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A25</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="E6:35_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> <font color="Maroon" title="c4">D</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_2_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E51">A24</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="E7:35_2_1_1_1_1_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#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="E8:35_2_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A26</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="E9:35_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="#E4:35_2_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E51">A24</font></span></a>, <a class="ref" href="integra1.html#T20" title="INTEGRA1:th.20">INTEGRA1:20</a></span>;<br/><a NAME="E10:35_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="#E8:35_2_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E53">A26</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="E11:35_2_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A27</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="E12:35_2_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E55">A28</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="#E4:35_2_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E51">A24</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E13:35_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="#E11:35_2_1_1_1_1"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E50">A23</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E14:35_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="#E10:35_2_1_1_1_1"><span class="lab"><font color="Green" title="E48">A21</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E52">A25</font></span></a>, <a class="txt" href="#E12:35_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#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><a NAME="E15:35_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="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="E16:35_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="#E2:35_2_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</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="E17:35_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="#E15:35_2_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E56">A29</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="E18:35_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="#E11:35_2_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E54">A27</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="35_2_1_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E50">A30</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="E2:35_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="#E1:35_2_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="ref" href="integra1.html#T7" title="INTEGRA1:th.7">INTEGRA1:7</a></span>;<br/><a NAME="E3:35_2_1_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A31</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="E4:35_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="#E1:35_2_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E50">A30</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E5:35_2_1_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A32</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="#E3:35_2_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E51">A31</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E6:35_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="#E1:35_2_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E50">A30</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E7:35_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="#E11:35_2_1_1_1_1"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="txt" href="#E10:35_2_1_1_1_1"><span class="lab"><font color="Green" title="E48">A21</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E8:35_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="#E5:35_2_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E52">A32</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E9:35_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="E10:35_2_1_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">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>,<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="E11:35_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="#E1:35_2_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</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="E12:35_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="E13:35_2_1_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A34</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="E14:35_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="#E1:35_2_1_1_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="ref" href="integra1.html#T20" title="INTEGRA1:th.20">INTEGRA1:20</a></span>;<br/><a NAME="E15:35_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="#E13:35_2_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E54">A34</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="E16:35_2_1_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A35</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="E17:35_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="#E2:35_2_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</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="E18:35_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="#E10:35_2_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E53">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="E19:35_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="#E16:35_2_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E55">A35</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="E13:35_2_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_2"><span class="lab"><font color="Green" title="E35">A8</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="E4:35_2_1_1_1"/><span class="lab"><font color="Green" title="E44">A36</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:35_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="E2:35_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="E3:35_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A37</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="E5:35_2_1_1_1_2"/><span class="lab"><font color="Green" title="E46">A38</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="E6:35_2_1_1_1_2"/><span class="lab"><font color="Green" title="E47">A39</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="E7:35_2_1_1_1_2"/>
<font color="Maroon">H<sub>2</sub></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="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="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="35_2_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:35_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E48">A40</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="#E3:35_2_1_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E45">A37</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="txt" href="#E6:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E47">A39</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="#E5:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="ref" href="integra1.html#T7" title="INTEGRA1:th.7">INTEGRA1:7</a></span>;<br/>
<a NAME="E3:35_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E49">A41</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="#E5:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E48">A40</font></span></a>, <a class="ref" href="integra1.html#T7" title="INTEGRA1:th.7">INTEGRA1:7</a></span>;<br/>
<a NAME="E4:35_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="E5:35_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#K4" title="INTEGRA1:func.4">upper_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#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/>
<a NAME="E6:35_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A42</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#K4" title="INTEGRA1:func.4">upper_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="E7:35_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E51">A43</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="#E5:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E48">A40</font></span></a>, <a class="ref" href="integra1.html#T7" title="INTEGRA1:th.7">INTEGRA1:7</a></span>;<br/>
<a NAME="E8:35_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A44</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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E9:35_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A45</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="E10:35_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A46</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="E11:35_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="#E8:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E52">A44</font></span></a>, <a class="ref" href="integra1.html#T35" title="INTEGRA1:th.35">INTEGRA1:35</a></span>;<br/>
<a NAME="E12:35_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A47</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="#E10:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E54">A46</font></span></a>, <a class="ref" href="finseq_6.html#T116" title="FINSEQ_6:th.116">FINSEQ_6:116</a></span>;<br/>
<a NAME="E13:35_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="E14:35_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="E15:35_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">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="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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E49">A41</font></span></a>, <a class="ref" href="integra3.html#T8" target="_self" title="INTEGRA3:th.8">Th8</a></span>;<br/>
<a NAME="E16:35_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A49</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="E17:35_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E58">A50</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#K4" title="INTEGRA1:func.4">upper_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="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#K4" title="INTEGRA1:func.4">upper_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="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="35_2_1_1_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:35_2_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E59">A51</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="35_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="E3:35_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="E4:35_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A52</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="E5:35_2_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E61">A53</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="E6:35_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="E7:35_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A54</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="#E10:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E54">A46</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E8:35_2_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E63">A55</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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E9:35_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A56</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="E10:35_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="#E4:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E60">A52</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E11:35_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A57</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="txt" href="#E7:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E62">A54</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E12:35_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A58</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="#E4:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E60">A52</font></span></a>, <a class="txt" href="#E8:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E63">A55</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="E13:35_2_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E67">A59</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="#E10:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E54">A46</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E14:35_2_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E68">A60</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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E15:35_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="#E4:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E60">A52</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E61">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="E16:35_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="#E9:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E64">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="E17:35_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A61</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="txt" href="#E13:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E67">A59</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E18:35_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">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="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="#E5:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E61">A53</font></span></a>, <a class="txt" href="#E11:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E65">A57</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="E19:35_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="E20:35_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A63</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="#E8:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E52">A44</font></span></a>, <a class="txt" href="#E17:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E69">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="E21:35_2_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E72">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="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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E49">A41</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E22:35_2_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E73">A65</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="35_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="E1:35_2_1_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E74">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="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="35_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="E1:35_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="#E1:35_2_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E74">A66</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_1_1_2_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_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="E3:35_2_1_1_1_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E75">A67</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="E4:35_2_1_1_1_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E76">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="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="E5:35_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="#E3:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E49">A41</font></span></a>, <a class="txt" href="#E20:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A63</font></span></a>, <a class="txt" href="#E21:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A64</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E75">A67</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E76">A68</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/><a NAME="E6:35_2_1_1_1_2_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A69</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="E7:35_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="#E18:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E70">A62</font></span></a>, <a class="txt" href="#E12:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E66">A58</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E8:35_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="#E5:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="txt" href="#E14:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E68">A60</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E75">A67</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E76">A68</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="E9:35_2_1_1_1_2_1_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_2_1_1_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E77">A69</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="35_2_1_1_1_2_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_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="E3:35_2_1_1_1_2_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E75">A70</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="E4:35_2_1_1_1_2_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E76">A71</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="E5:35_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="#E20:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A63</font></span></a>, <a class="txt" href="#E18:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E70">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="E6:35_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="#E3:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E49">A41</font></span></a>, <a class="txt" href="#E21:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A64</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E75">A70</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E76">A71</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/><a NAME="E7:35_2_1_1_1_2_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A72</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="E8:35_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="#E5:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="txt" href="#E12:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E66">A58</font></span></a>, <a class="txt" href="#E14:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E68">A60</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E75">A70</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E76">A71</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="E9:35_2_1_1_1_2_1_1_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_2_1_1_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E77">A72</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="E23:35_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="#E5:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E48">A40</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/>
<a NAME="E24:35_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A73</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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E25:35_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="#E5:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E48">A40</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/>
<a NAME="E26:35_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A74</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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E49">A41</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E27:35_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>  <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="#E11:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E65">A57</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="E28:35_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A75</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="#E4:35_2_1"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="txt" href="#E22:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E73">A65</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="E29:35_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>  <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="#E17:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E69">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="E30:35_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A76</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="#E4:35_2_1"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="txt" href="#E22:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E73">A65</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="E31:35_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="#E20:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A63</font></span></a>, <a class="txt" href="#E18:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E70">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="E32:35_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="#E12:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E66">A58</font></span></a>, <a class="txt" href="#E26:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E75">A74</font></span></a>, <a class="txt" href="#E24:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A73</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/>
<a NAME="E33:35_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A77</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="#E28:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E76">A75</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="E34:35_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="#E18:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E70">A62</font></span></a>, <a class="txt" href="#E12:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E66">A58</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E35:35_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="#E20:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A63</font></span></a>, <a class="txt" href="#E26:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E75">A74</font></span></a>, <a class="txt" href="#E24:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A73</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/>
<a NAME="E36:35_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="#E30:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E77">A76</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="E37:35_2_1_1_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_2"><span class="lab"><font color="Green" title="E35">A8</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E61">A53</font></span></a>, <a class="txt" href="#E17:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E69">A61</font></span></a>, <a class="txt" href="#E11:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E65">A57</font></span></a>, <a class="txt" href="#E33:35_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E78">A77</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="E2:35_2_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E60">A78</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="#E10:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E54">A46</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E3:35_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="#E5:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E4:35_2_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A79</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#K4" title="INTEGRA1:func.4">upper_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#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/>
<a NAME="E5:35_2_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E62">A80</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="#E5:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E6:35_2_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A81</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#K4" title="INTEGRA1:func.4">upper_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#K4" title="INTEGRA1:func.4">upper_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="#E4:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E61">A79</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>;<br/>
<a NAME="E7:35_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="E8:35_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#K4" title="INTEGRA1:func.4">upper_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="#E5:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E62">A80</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E61">A79</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>;<br/>
<a NAME="E9:35_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#K4" title="INTEGRA1:func.4">upper_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#K4" title="INTEGRA1:func.4">upper_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="#E6:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E63">A81</font></span></a>, <a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>;<br/>
<a NAME="E10:35_2_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A82</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#K4" title="INTEGRA1:func.4">upper_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#K4" title="INTEGRA1:func.4">upper_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="E11:35_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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E12:35_2_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A83</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="E13:35_2_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A84</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="E14:35_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#K4" title="INTEGRA1:func.4">upper_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="#E12:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E65">A83</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/>
<a NAME="E15:35_2_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A85</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#K4" title="INTEGRA1:func.4">upper_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="E16:35_2_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A86</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#K4" title="INTEGRA1:func.4">upper_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="#E16:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E57">A49</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E17:35_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#K4" title="INTEGRA1:func.4">upper_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="#E2:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E60">A78</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E18:35_2_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A87</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#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/>
<a NAME="E19:35_2_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A88</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="E20:35_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="#E16:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E57">A49</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/>
<a NAME="E21:35_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="#E1:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E59">A51</font></span></a></span>;<br/>
<a NAME="E22:35_2_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A89</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#K4" title="INTEGRA1:func.4">upper_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="#E16:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E57">A49</font></span></a>, <a class="txt" href="#E13:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E66">A84</font></span></a>, <a class="txt" href="#E15:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E67">A85</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E60">A78</font></span></a>, <a class="txt" href="#E16:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E68">A86</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>;<br/>
<a NAME="E23:35_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="E24:35_2_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A90</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#K4" title="INTEGRA1:func.4">upper_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="#E16:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E57">A49</font></span></a>, <a class="txt" href="#E13:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E66">A84</font></span></a>, <a class="txt" href="#E15:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E67">A85</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E60">A78</font></span></a>, <a class="txt" href="#E16:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E68">A86</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="35_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="35_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="E1:35_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#K4" title="INTEGRA1:func.4">upper_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#K4" title="INTEGRA1:func.4">upper_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="#E24:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E72">A90</font></span></a>, <a class="txt" href="#E22:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E71">A89</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="35_2_1_1_1_2_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_1_1_2_1_1_2_1_1"/><span class="lab"><font color="Green" title="E73">A91</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#K4" title="INTEGRA1:func.4">upper_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#K4" title="INTEGRA1:func.4">upper_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="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#K4" title="INTEGRA1:func.4">upper_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="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="E2:35_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="#E5:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E48">A40</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E3:35_2_1_1_1_2_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A92</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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/><a NAME="E4:35_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="#E5:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E48">A40</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E5:35_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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E49">A41</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/><a NAME="E6:35_2_1_1_1_2_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A93</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="#E3:35_2_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E74">A92</font></span></a>, <a class="ref" href="integra1.html#T4" title="INTEGRA1:th.4">INTEGRA1:4</a></span>;<br/><a NAME="E7:35_2_1_1_1_2_1_1_2_1_1"/><span class="lab"><font color="Green" title="E76">A94</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="E8:35_2_1_1_1_2_1_1_2_1_1"/><span class="lab"><font color="Green" title="E77">A95</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="#E1:35"><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="E9:35_2_1_1_1_2_1_1_2_1_1"/><span class="lab"><font color="Green" title="E78">A96</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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/><a NAME="E10:35_2_1_1_1_2_1_1_2_1_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#K4" title="INTEGRA1:func.4">upper_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#K13" 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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E57">A49</font></span></a>, <a class="txt" href="#E13:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E66">A84</font></span></a>, <a class="txt" href="#E15:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E67">A85</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E60">A78</font></span></a>, <a class="txt" href="#E16:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E68">A86</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>;<br/><a NAME="E11:35_2_1_1_1_2_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A97</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="#E16:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E57">A49</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E73">A91</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/><a NAME="E12:35_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="#E9:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E53">A45</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E13:35_2_1_1_1_2_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A98</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="#E9:35_2_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E78">A96</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E14:35_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="#E11:35_2_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E79">A97</font></span></a></span>;<br/><a NAME="E15:35_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="#E10:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E54">A46</font></span></a>, <a class="txt" href="#E11:35_2_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E79">A97</font></span></a>, <a class="txt" href="#E9:35_2_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E78">A96</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E16:35_2_1_1_1_2_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A99</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="#E6:35_2_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E75">A93</font></span></a>, <a class="txt" href="#E13:35_2_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E80">A98</font></span></a>, <a class="ref" href="integra1.html#T4" title="INTEGRA1:th.4">INTEGRA1:4</a></span>;<br/><a NAME="E17:35_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#K4" title="INTEGRA1:func.4">upper_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#K4" title="INTEGRA1:func.4">upper_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="#E13:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E66">A84</font></span></a>, <a class="txt" href="#E15:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E67">A85</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E60">A78</font></span></a>, <a class="txt" href="#E16:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E68">A86</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>;<br/><a NAME="E18:35_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#K4" title="INTEGRA1:func.4">upper_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#K4" title="INTEGRA1:func.4">upper_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="#E1:35_2_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E73">A91</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#K4" title="INTEGRA1:func.4">upper_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#K4" title="INTEGRA1:func.4">upper_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#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><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="#E19:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E70">A88</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</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#K4" title="INTEGRA1:func.4">upper_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="#E5:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="txt" href="#E10:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E64">A82</font></span></a>, <a class="txt" href="#E11:35_2_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E79">A97</font></span></a>, <a class="txt" href="#E16:35_2_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E81">A99</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>
;<br/><span class="kw">hence </span><a NAME="E20:35_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#K4" title="INTEGRA1:func.4">upper_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="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#K4" title="INTEGRA1:func.4">upper_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="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="#E7:35_2_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E76">A94</font></span></a>, <a class="txt" href="#E8:35_2_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E77">A95</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="35_2_1_1_1_2_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_1_1_2_1_1_2_1_2"/><span class="lab"><font color="Green" title="E73">A100</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#K4" title="INTEGRA1:func.4">upper_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#K4" title="INTEGRA1:func.4">upper_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="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#K4" title="INTEGRA1:func.4">upper_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="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="E2:35_2_1_1_1_2_1_1_2_1_2"/><span class="lab"><font color="Green" title="E74">A101</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#K4" title="INTEGRA1:func.4">upper_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#K4" title="INTEGRA1:func.4">upper_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="#E13:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E66">A84</font></span></a>, <a class="txt" href="#E15:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E67">A85</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E60">A78</font></span></a>, <a class="txt" href="#E16:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E68">A86</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>;<br/><a NAME="E3:35_2_1_1_1_2_1_1_2_1_2"/><span class="lab"><font color="Green" title="E75">A102</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#K4" title="INTEGRA1:func.4">upper_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="#E16:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E57">A49</font></span></a>, <a class="txt" href="#E13:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E66">A84</font></span></a>, <a class="txt" href="#E15:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E67">A85</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E60">A78</font></span></a>, <a class="txt" href="#E16:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E68">A86</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E73">A100</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="#E3:35_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E75">A102</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="E5:35_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#K4" title="INTEGRA1:func.4">upper_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#K4" title="INTEGRA1:func.4">upper_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#K4" title="INTEGRA1:func.4">upper_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="#E1:35_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E73">A100</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E74">A101</font></span></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/><a NAME="E6:35_2_1_1_1_2_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A103</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#K4" title="INTEGRA1:func.4">upper_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#K4" title="INTEGRA1:func.4">upper_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#K4" title="INTEGRA1:func.4">upper_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="E7:35_2_1_1_1_2_1_1_2_1_2"/><span class="lab"><font color="Green" title="E77">A104</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="E8:35_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="#E5:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E48">A40</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E9:35_2_1_1_1_2_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A105</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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/><a NAME="E10:35_2_1_1_1_2_1_1_2_1_2"/><span class="lab"><font color="Green" title="E79">A106</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="E11:35_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="#E16:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E57">A49</font></span></a>, <a class="txt" href="#E13:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E66">A84</font></span></a>, <a class="txt" href="#E15:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E67">A85</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E60">A78</font></span></a>, <a class="txt" href="#E16:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E68">A86</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E73">A100</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="E80">A107</font></span>: 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="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
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E57">A49</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="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>

;<br/><a NAME="E13:35_2_1_1_1_2_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A108</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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/><a NAME="E14:35_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="#E5:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E48">A40</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E15:35_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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E49">A41</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/><a NAME="E16:35_2_1_1_1_2_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A109</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="#E9:35_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E78">A105</font></span></a>, <a class="txt" href="#E12:35_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E80">A107</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><a NAME="E17:35_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#K4" title="INTEGRA1:func.4">upper_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="#E2:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E60">A78</font></span></a>, <a class="txt" href="#E16:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E68">A86</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E18:35_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#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/><a NAME="E19:35_2_1_1_1_2_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A110</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="E20:35_2_1_1_1_2_1_1_2_1_2"/><span class="lab"><font color="Green" title="E84">A111</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="#E10:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E54">A46</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E21:35_2_1_1_1_2_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A112</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="#E19:35_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E83">A110</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E22:35_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="E23:35_2_1_1_1_2_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A113</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="#E19:35_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E83">A110</font></span></a>, <a class="txt" href="#E20:35_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E84">A111</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E24:35_2_1_1_1_2_1_1_2_1_2"/><span class="lab"><font color="Green" title="E87">A114</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="#E2:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E60">A78</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E25:35_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="E26:35_2_1_1_1_2_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A115</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="#E13:35_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E81">A108</font></span></a>, <a class="txt" href="#E24:35_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E87">A114</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E27:35_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="#E13:35_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E81">A108</font></span></a>, <a class="txt" href="#E24:35_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E87">A114</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="#E26:35_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E88">A115</font></span></a>, <a class="txt" href="#E16:35_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E82">A109</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="#E23:35_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E86">A113</font></span></a>, <a class="txt" href="#E21:35_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E85">A112</font></span></a></span>
;<br/><a NAME="E29:35_2_1_1_1_2_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A116</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="E30:35_2_1_1_1_2_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A117</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_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#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><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="#E5:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/><a NAME="E31:35_2_1_1_1_2_1_1_2_1_2"/><span class="lab"><font color="Green" title="E91">A118</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"><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="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="35_2_1_1_1_2_1_1_2_1_2_4"><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">SR</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> <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>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">VR</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="c14">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="c15">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="E1:35_2_1_1_1_2_1_1_2_1_2_4"/>
 <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="#E5:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="ref" href="integra1.html#T8" title="INTEGRA1:th.8">INTEGRA1:8</a></span>;<br/>
<a NAME="E2:35_2_1_1_1_2_1_1_2_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A119</font></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"><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#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="#E1:35"><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="E3:35_2_1_1_1_2_1_1_2_1_2_4"/>
<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="#E18:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E69">A87</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="E4:35_2_1_1_1_2_1_1_2_1_2_4"/><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="E5:35_2_1_1_1_2_1_1_2_1_2_4"/><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"><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#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="#E1:35"><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="E6:35_2_1_1_1_2_1_1_2_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A120</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> <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#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="#E7:35_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E77">A104</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/>
<a NAME="E7:35_2_1_1_1_2_1_1_2_1_2_4"/>
<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="#E16:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E57">A49</font></span></a>, <a class="txt" href="#E13:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E66">A84</font></span></a>, <a class="txt" href="#E15:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E67">A85</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E60">A78</font></span></a>, <a class="txt" href="#E16:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E68">A86</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E73">A100</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="E94">A121</font></span>: 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="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
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E57">A49</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="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>

;<br/>
<a NAME="E9:35_2_1_1_1_2_1_1_2_1_2_4"/><span class="lab"><font color="Green" title="E95">A122</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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E10:35_2_1_1_1_2_1_1_2_1_2_4"/><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="#E8:35_2_1_1_1_2_1_1_2_1_2_4"><span class="lab"><font color="Green" title="E94">A121</font></span></a>, <a class="ref" href="integra1.html#T8" title="INTEGRA1:th.8">INTEGRA1:8</a></span>;<br/>
<a NAME="E11:35_2_1_1_1_2_1_1_2_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A123</font></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"><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#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="#E1:35"><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="c16">A</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> <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="E13:35_2_1_1_1_2_1_1_2_1_2_4"/><span class="lab"><font color="Green" title="E97">A124</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<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="c16">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="c16">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="#E30:35_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E90">A117</font></span></a></span>;<br/>
<a NAME="E14:35_2_1_1_1_2_1_1_2_1_2_4"/>
<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><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#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="#E10:35_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E79">A106</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_2_1_1_2_1_2_4"><span class="lab"><font color="Green" title="E92">A119</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/>
<a NAME="E15:35_2_1_1_1_2_1_1_2_1_2_4"/><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#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> <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#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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E64">A82</font></span></a>, <a class="txt" href="#E13:35_2_1_1_1_2_1_1_2_1_2_4"><span class="lab"><font color="Green" title="E97">A124</font></span></a>, <a class="ref" href="xreal_1.html#T20" title="XREAL_1:th.20">XREAL_1:20</a></span>;<br/>
<a NAME="E16:35_2_1_1_1_2_1_1_2_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A125</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#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> <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#T20" title="XREAL_1:th.20">XREAL_1:20</a></span>;<br/>
<a NAME="E17:35_2_1_1_1_2_1_1_2_1_2_4"/>
<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><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#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="#E7:35_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E77">A104</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_2_1_1_2_1_2_4"><span class="lab"><font color="Green" title="E92">A119</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/>
<a NAME="E18:35_2_1_1_1_2_1_1_2_1_2_4"/><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#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="#E16:35_2_1_1_1_2_1_1_2_1_2_4"><span class="lab"><font color="Green" title="E98">A125</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E19:35_2_1_1_1_2_1_1_2_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A126</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#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 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#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> <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="#E6:35_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E76">A103</font></span></a>, <a class="txt" href="#E9:35_2_1_1_1_2_1_1_2_1_2_4"><span class="lab"><font color="Green" title="E95">A122</font></span></a>, <a class="txt" href="#E8:35_2_1_1_1_2_1_1_2_1_2_4"><span class="lab"><font color="Green" title="E94">A121</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>
<br/>.= 
<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> <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#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> <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="#E19:35_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E70">A88</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>
;<br/>
<a NAME="E21:35_2_1_1_1_2_1_1_2_1_2_4"/><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> <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#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="#E10:35_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E79">A106</font></span></a>, <a class="txt" href="#E11:35_2_1_1_1_2_1_1_2_1_2_4"><span class="lab"><font color="Green" title="E96">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="E22:35_2_1_1_1_2_1_1_2_1_2_4"/><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#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#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> <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#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/>
<a NAME="E23:35_2_1_1_1_2_1_1_2_1_2_4"/><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> <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#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><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#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/>
<a NAME="E24:35_2_1_1_1_2_1_1_2_1_2_4"/><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> <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="#E6:35_2_1_1_1_2_1_1_2_1_2_4"><span class="lab"><font color="Green" title="E93">A120</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E25:35_2_1_1_1_2_1_1_2_1_2_4"/><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#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 NAME="E26:35_2_1_1_1_2_1_1_2_1_2_4"/><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"><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="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="#E19:35_2_1_1_1_2_1_1_2_1_2_4"><span class="lab"><font color="Green" title="E99">A126</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="E27:35_2_1_1_1_2_1_1_2_1_2_4"/>
<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"><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="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="E32:35_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="#E1:35"><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="E33:35_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="#E5:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A38</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="E34:35_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#K4" title="INTEGRA1:func.4">upper_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="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#K4" title="INTEGRA1:func.4">upper_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="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="#E29:35_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E89">A116</font></span></a>, <a class="txt" href="#E31:35_2_1_1_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E91">A118</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="E26:35_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#K4" title="INTEGRA1:func.4">upper_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="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#K4" title="INTEGRA1:func.4">upper_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="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="E18:35_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="E19:35_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A127</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="E20:35_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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E21:35_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A128</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="E22:35_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A129</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="E23:35_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E62">A130</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="#E9:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E53">A45</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E24:35_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A131</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="E25:35_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E64">A132</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="#E7:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E51">A43</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="E26:35_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A133</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="E27:35_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="35_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="E1:35_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="E3:35_2_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E66">A134</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="E4:35_2_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E67">A135</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="E5:35_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="#E3:35_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E66">A134</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="E6:35_2_1_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A136</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="#E26:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E65">A133</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>;<br/>
<a NAME="E7:35_2_1_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A137</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="#E3:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E49">A41</font></span></a>, <a class="ref" href="rfinseq.html#T6" title="RFINSEQ:th.6">RFINSEQ:6</a></span>;<br/>
<a NAME="E8:35_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="#E6:35_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E68">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="E9:35_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="#E7:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E51">A43</font></span></a>, <a class="txt" href="#E7:35_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E69">A137</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/>
<a NAME="E10:35_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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E7:35_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E69">A137</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E11:35_2_1_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A138</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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E7:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E51">A43</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E12:35_2_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E71">A139</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="#E3:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E49">A41</font></span></a>, <a class="txt" href="#E6:35_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E68">A136</font></span></a>, <a class="ref" href="rfinseq.html#T6" title="RFINSEQ:th.6">RFINSEQ:6</a></span>;<br/>
<a NAME="E13:35_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="#E7:35_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E69">A137</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="E14:35_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="#E4:35_2_1"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E67">A135</font></span></a>, <a class="txt" href="#E12:35_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E71">A139</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="E16:35_2_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E72">A140</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="E17:35_2_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E73">A141</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="E18:35_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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E67">A135</font></span></a>, <a class="txt" href="#E12:35_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E71">A139</font></span></a>, <a class="txt" href="#E7:35_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E69">A137</font></span></a>, <a class="txt" href="#E17:35_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E73">A141</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E19:35_2_1_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A142</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="#E8:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E52">A44</font></span></a>, <a class="txt" href="#E16:35_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E72">A140</font></span></a>, <a class="txt" href="#E11:35_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E70">A138</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="E20:35_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="#E16:35_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E72">A140</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E21:35_2_1_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A143</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="#E19:35_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E74">A142</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E22:35_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="#E23:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E62">A130</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>;<br/>
<a NAME="E23:35_2_1_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">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> <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="E24:35_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="#E8:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E52">A44</font></span></a>, <a class="txt" href="#E21:35_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E75">A143</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="E25:35_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="#E17:35_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E73">A141</font></span></a>, <a class="txt" href="#E23:35_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E76">A144</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="E28:35_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A145</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="E29:35_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E67">A146</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="#E25:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E64">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="E30:35_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="#E6:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E47">A39</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/>
<a NAME="E31:35_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A147</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="#E5:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E48">A40</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/>
<a NAME="E32:35_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="35_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="E1:35_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="E3:35_2_1_1_1_2_1_3"/><span class="lab"><font color="Green" title="E69">A148</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="E4:35_2_1_1_1_2_1_3"/><span class="lab"><font color="Green" title="E70">A149</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="E5:35_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="#E3:35_2_1_1_1_2_1_3"><span class="lab"><font color="Green" title="E69">A148</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="E6:35_2_1_1_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A150</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="#E23:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E62">A130</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>;<br/>
<a NAME="E7:35_2_1_1_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A151</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="#E8:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E52">A44</font></span></a>, <a class="ref" href="rfinseq.html#T6" title="RFINSEQ:th.6">RFINSEQ:6</a></span>;<br/>
<a NAME="E8:35_2_1_1_1_2_1_3"/><span class="lab"><font color="Green" title="E73">A152</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="#E26:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E65">A133</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>;<br/>
<a NAME="E9:35_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="#E6:35_2_1_1_1_2_1_3"><span class="lab"><font color="Green" title="E71">A150</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E10:35_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="#E8:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E52">A44</font></span></a>, <a class="txt" href="#E7:35_2_1_1_1_2_1_3"><span class="lab"><font color="Green" title="E72">A151</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/>
<a NAME="E11:35_2_1_1_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A153</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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E7:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E51">A43</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E12:35_2_1_1_1_2_1_3"/><span class="lab"><font color="Green" title="E75">A154</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="35_2_1_1_1_2_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:35_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="E3:35_2_1_1_1_2_1_3_1"/><span class="lab"><font color="Green" title="E76">A155</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="E4:35_2_1_1_1_2_1_3_1"/><span class="lab"><font color="Green" title="E77">A156</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="E5:35_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="#E3:35_2_1_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E76">A155</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="E6:35_2_1_1_1_2_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A157</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="E7:35_2_1_1_1_2_1_3_1"/><span class="lab"><font color="Green" title="E79">A158</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="#E3:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E49">A41</font></span></a>, <a class="txt" href="#E11:35_2_1_1_1_2_1_3"><span class="lab"><font color="Green" title="E74">A153</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E76">A155</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E77">A156</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="E8:35_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> <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="#E8:35_2_1_1_1_2_1_3"><span class="lab"><font color="Green" title="E73">A152</font></span></a>, <a class="txt" href="#E6:35_2_1_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E78">A157</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E9:35_2_1_1_1_2_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">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> <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="E10:35_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> <font color="Maroon" title="c9">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_2_1_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E78">A157</font></span></a>, <a class="txt" href="#E7:35_2_1_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E79">A158</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E11:35_2_1_1_1_2_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> <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="#E3:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E49">A41</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E77">A156</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="E12:35_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="#E9:35_2_1_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E80">A159</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="E13:35_2_1_1_1_2_1_3"/><span class="lab"><font color="Green" title="E76">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#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="35_2_1_1_1_2_1_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:35_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="E3:35_2_1_1_1_2_1_3_2"/><span class="lab"><font color="Green" title="E77">A161</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="E4:35_2_1_1_1_2_1_3_2"/><span class="lab"><font color="Green" title="E78">A162</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="E5:35_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="#E3:35_2_1_1_1_2_1_3_2"><span class="lab"><font color="Green" title="E77">A161</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E6:35_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="#E3:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E45">A37</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_2_1_3_2"><span class="lab"><font color="Green" title="E77">A161</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_2_1_3_2"><span class="lab"><font color="Green" title="E78">A162</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/>
<a NAME="E7:35_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="#E31:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E68">A147</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="E8:35_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="#E11:35_2_1_1_1_2_1_3"><span class="lab"><font color="Green" title="E74">A153</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="E14:35_2_1_1_1_2_1_3"/><span class="lab"><font color="Green" title="E77">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#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="35_2_1_1_1_2_1_3_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:35_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="#E29:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E67">A146</font></span></a>, <a class="txt" href="#E8:35_2_1_1_1_2_1_3"><span class="lab"><font color="Green" title="E73">A152</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E2:35_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="E3:35_2_1_1_1_2_1_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A164</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="E4:35_2_1_1_1_2_1_3_3"/><span class="lab"><font color="Green" title="E79">A165</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="E5:35_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="#E29:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E67">A146</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="E6:35_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="#E3:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E49">A41</font></span></a>, <a class="txt" href="#E13:35_2_1_1_1_2_1_3"><span class="lab"><font color="Green" title="E76">A160</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_2_1_3_3"><span class="lab"><font color="Green" title="E79">A165</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_2_1_3_3"><span class="lab"><font color="Green" title="E78">A164</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="E15:35_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="#E7:35_2_1_1_1_2_1_3"><span class="lab"><font color="Green" title="E72">A151</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="E16:35_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="#E4:35_2_1"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="txt" href="#E8:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E52">A44</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_2_1_3"><span class="lab"><font color="Green" title="E70">A149</font></span></a>, <a class="txt" href="#E6:35_2_1_1_1_2_1_3"><span class="lab"><font color="Green" title="E71">A150</font></span></a>, <a class="txt" href="#E14:35_2_1_1_1_2_1_3"><span class="lab"><font color="Green" title="E77">A163</font></span></a>, <a class="txt" href="#E12:35_2_1_1_1_2_1_3"><span class="lab"><font color="Green" title="E75">A154</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="E33:35_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="E34:35_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A166</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="#E28:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E66">A145</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="E35:35_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="#E5:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E48">A40</font></span></a>, <a class="txt" href="#E29:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E67">A146</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="E36:35_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A167</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="#E29:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E67">A146</font></span></a>, <a class="ref" href="finseq_6.html#T116" title="FINSEQ_6:th.116">FINSEQ_6:116</a></span>;<br/>
<a NAME="E37:35_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A168</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="#E12:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E55">A47</font></span></a>, <a class="txt" href="#E34:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E69">A166</font></span></a>, <a class="ref" href="integra3.html#T6" target="_self" title="INTEGRA3:th.6">Th6</a></span>;<br/>
<a NAME="E38:35_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E72">A169</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="35_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="E1:35_2_1_1_1_2_1_4"/><span class="lab"><font color="Green" title="E73">A170</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="E2:35_2_1_1_1_2_1_4"/><span class="lab"><font color="Green" title="E74">A171</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="E3:35_2_1_1_1_2_1_4"/><span class="lab"><font color="Green" title="E75">A172</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="35_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="35_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="E1:35_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="#E3:35_2_1_1_1_2_1_4"><span class="lab"><font color="Green" title="E75">A172</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="35_2_1_1_1_2_1_4_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_1_1_2_1_4_1_1_1"/><span class="lab"><font color="Green" title="E76">A173</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="E2:35_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="#E26:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E65">A133</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_2_1_4"><span class="lab"><font color="Green" title="E74">A171</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E3:35_2_1_1_1_2_1_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A174</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="#E1:35_2_1_1_1_2_1_4"><span class="lab"><font color="Green" title="E73">A170</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E4:35_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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/><a NAME="E5:35_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="E6:35_2_1_1_1_2_1_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A175</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="E7:35_2_1_1_1_2_1_4_1_1_1"/><span class="lab"><font color="Green" title="E79">A176</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="#E2:35_2_1_1_1_2_1_4"><span class="lab"><font color="Green" title="E74">A171</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_2_1_4_1_1_1"><span class="lab"><font color="Green" title="E76">A173</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E8:35_2_1_1_1_2_1_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A177</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="#E6:35_2_1_1_1_2_1_4_1_1_1"><span class="lab"><font color="Green" title="E78">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="E9:35_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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E49">A41</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_2_1_4"><span class="lab"><font color="Green" title="E74">A171</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_2_1_4_1_1_1"><span class="lab"><font color="Green" title="E77">A174</font></span></a>, <a class="ref" href="integra3.html#T7" target="_self" title="INTEGRA3:th.7">Th7</a></span>;<br/><a NAME="E10:35_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="#E6:35_2_1_1_1_2_1_4_1_1_1"><span class="lab"><font color="Green" title="E78">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="E11:35_2_1_1_1_2_1_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A178</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="#E8:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E52">A44</font></span></a>, <a class="ref" href="rfinseq.html#T6" title="RFINSEQ:th.6">RFINSEQ:6</a></span>;<br/><a NAME="E12:35_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="#E26:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E65">A133</font></span></a>, <a class="txt" href="#E7:35_2_1_1_1_2_1_4_1_1_1"><span class="lab"><font color="Green" title="E79">A176</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E13:35_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="#E6:35_2_1_1_1_2_1_4_1_1_1"><span class="lab"><font color="Green" title="E78">A175</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E14:35_2_1_1_1_2_1_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A179</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="#E1:35_2_1_1_1_2_1_4_1_1_1"><span class="lab"><font color="Green" title="E76">A173</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_2_1_4_1_1_1"><span class="lab"><font color="Green" title="E77">A174</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="E15:35_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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_2_1_4_1_1_1"><span class="lab"><font color="Green" title="E77">A174</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="E16:35_2_1_1_1_2_1_4_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E49">A41</font></span></a>, <a class="txt" href="#E37:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E71">A168</font></span></a>, <a class="txt" href="#E11:35_2_1_1_1_2_1_4_1_1_1"><span class="lab"><font color="Green" title="E81">A178</font></span></a>, <a class="txt" href="#E14:35_2_1_1_1_2_1_4_1_1_1"><span class="lab"><font color="Green" title="E82">A179</font></span></a>, <a class="txt" href="#E8:35_2_1_1_1_2_1_4_1_1_1"><span class="lab"><font color="Green" title="E80">A177</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="35_2_1_1_1_2_1_4_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_1_1_2_1_4_1_1_2"/><span class="lab"><font color="Green" title="E76">A180</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="E2:35_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="#E26:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E65">A133</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_2_1_4"><span class="lab"><font color="Green" title="E74">A171</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E3:35_2_1_1_1_2_1_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A181</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="#E1:35_2_1_1_1_2_1_4"><span class="lab"><font color="Green" title="E73">A170</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E4:35_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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E49">A41</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_2_1_4"><span class="lab"><font color="Green" title="E74">A171</font></span></a>, <a class="ref" href="integra3.html#T7" target="_self" title="INTEGRA3:th.7">Th7</a></span>;<br/><a NAME="E5:35_2_1_1_1_2_1_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A182</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="#E1:35_2_1_1_1_2_1_4_1_1_2"><span class="lab"><font color="Green" title="E76">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="E6:35_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="#E23:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E62">A130</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E7:35_2_1_1_1_2_1_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A183</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="#E1:35_2_1_1_1_2_1_4"><span class="lab"><font color="Green" title="E73">A170</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E8:35_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="#E1:35_2_1_1_1_2_1_4"><span class="lab"><font color="Green" title="E73">A170</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_2_1_4"><span class="lab"><font color="Green" title="E74">A171</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E9:35_2_1_1_1_2_1_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A184</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="#E7:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E51">A43</font></span></a>, <a class="ref" href="rfinseq.html#T6" title="RFINSEQ:th.6">RFINSEQ:6</a></span>;<br/><a NAME="E10:35_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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_2_1_4_1_1_2"><span class="lab"><font color="Green" title="E77">A181</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/><a NAME="E11:35_2_1_1_1_2_1_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A185</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="#E1:35_2_1_1_1_2_1_4_1_1_2"><span class="lab"><font color="Green" title="E76">A180</font></span></a>, <a class="txt" href="#E7:35_2_1_1_1_2_1_4_1_1_2"><span class="lab"><font color="Green" title="E79">A183</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="E12:35_2_1_1_1_2_1_4_1_1_2"/><span class="lab"><font color="Green" title="E82">A186</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="#E1:35_2_1_1_1_2_1_4"><span class="lab"><font color="Green" title="E73">A170</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1_2_1_4_1_1_2"><span class="lab"><font color="Green" title="E78">A182</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E13:35_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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_2_1_4_1_1_2"><span class="lab"><font color="Green" title="E77">A181</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="E14:35_2_1_1_1_2_1_4_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E52">A44</font></span></a>, <a class="txt" href="#E37:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E71">A168</font></span></a>, <a class="txt" href="#E9:35_2_1_1_1_2_1_4_1_1_2"><span class="lab"><font color="Green" title="E80">A184</font></span></a>, <a class="txt" href="#E11:35_2_1_1_1_2_1_4_1_1_2"><span class="lab"><font color="Green" title="E81">A185</font></span></a>, <a class="txt" href="#E12:35_2_1_1_1_2_1_4_1_1_2"><span class="lab"><font color="Green" title="E82">A186</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="E5:35_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="E39:35_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E73">A187</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#K4" title="INTEGRA1:func.4">upper_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#K4" title="INTEGRA1:func.4">upper_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#K4" title="INTEGRA1:func.4">upper_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="35_2_1_1_1_2_1_5"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:35_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="#E8:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E52">A44</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="E2:35_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#K4" title="INTEGRA1:func.4">upper_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#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/>
<a NAME="E3:35_2_1_1_1_2_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A188</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#K4" title="INTEGRA1:func.4">upper_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#K4" title="INTEGRA1:func.4">upper_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#K4" title="INTEGRA1:func.4">upper_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#K4" title="INTEGRA1:func.4">upper_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="E4:35_2_1_1_1_2_1_5"/><span class="lab"><font color="Green" title="E75">A189</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="E5:35_2_1_1_1_2_1_5"/><span class="lab"><font color="Green" title="E76">A190</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#K4" title="INTEGRA1:func.4">upper_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#K4" title="INTEGRA1:func.4">upper_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#K4" title="INTEGRA1:func.4">upper_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/>

<a NAME="E6:35_2_1_1_1_2_1_5"/><span class="lab"><font color="Green" title="E77">A191</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<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#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/>
<a NAME="E7:35_2_1_1_1_2_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A192</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E65">A133</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E76">A190</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>;<br/>
<a NAME="E8:35_2_1_1_1_2_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A193</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> <font color="Maroon" title="c9">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E75">A189</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E9:35_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="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="#E38:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E72">A169</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E75">A189</font></span></a>, <a class="txt" href="#E7:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E78">A192</font></span></a></span>;<br/>
<a NAME="E10:35_2_1_1_1_2_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A194</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> <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="#E29:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E67">A146</font></span></a>, <a class="txt" href="#E38:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E72">A169</font></span></a></span>;<br/>
<a NAME="E11:35_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="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="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E12:35_2_1_1_1_2_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A195</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#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="#E23:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E62">A130</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E13:35_2_1_1_1_2_1_5"/>
<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="#E26:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E65">A133</font></span></a>, <a class="txt" href="#E7:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E78">A192</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E14:35_2_1_1_1_2_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A196</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="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="#E4:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E75">A189</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E15:35_2_1_1_1_2_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A197</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="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E16:35_2_1_1_1_2_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A198</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#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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E17:35_2_1_1_1_2_1_5"/><span class="lab"><font color="Green" title="E85">A199</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> <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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E15:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E83">A197</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E18:35_2_1_1_1_2_1_5"/><span class="lab"><font color="Green" title="E86">A200</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">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="c10">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="c10">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="c10">k</font>)</span>)</span>)</span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_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="35_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="E1:35_2_1_1_1_2_1_5_1_1"/>
( <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">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="35_2_1_1_1_2_1_5_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_1_1_2_1_5_1_1_1"/><span class="lab"><font color="Green" title="E87">A201</font></span>: 
<font color="Maroon" title="c10">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="c10">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="c10">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="c10">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="c10">k</font>)</span>)</span>)</span>)</span> )</span><br/><div class="add"><a NAME="E2:35_2_1_1_1_2_1_5_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A202</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">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="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E83">A197</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E3:35_2_1_1_1_2_1_5_1_1_1"/><span class="lab"><font color="Green" title="E89">A203</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">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="#E15:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E83">A197</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_2_1_5_1_1_1"><span class="lab"><font color="Green" title="E87">A201</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E4:35_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="c10">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="#E29:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E67">A146</font></span></a>, <a class="txt" href="#E38:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E72">A169</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_2_1_5_1_1_1"><span class="lab"><font color="Green" title="E87">A201</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:35_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="c10">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="c10">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="c10">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="c10">k</font>)</span>)</span>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E84">A198</font></span></a>, <a class="txt" href="#E17:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E85">A199</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_2_1_5_1_1_1"><span class="lab"><font color="Green" title="E89">A203</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_2_1_5_1_1_1"><span class="lab"><font color="Green" title="E88">A202</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="35_2_1_1_1_2_1_5_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_1_1_2_1_5_1_1_2"/><span class="lab"><font color="Green" title="E87">A204</font></span>: 
<font color="Maroon" title="c10">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="c10">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="c10">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="c10">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="c10">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="c11">k1</font> = <font color="Maroon" title="c10">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E83">A197</font></span></a>, <a class="ref" href="integra1.html#T7" title="INTEGRA1:th.7">INTEGRA1:7</a></span>;<br/><a NAME="E3:35_2_1_1_1_2_1_5_1_1_2"/>
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</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/><a NAME="E4:35_2_1_1_1_2_1_5_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">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="E5:35_2_1_1_1_2_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A205</font></span>: 
<font color="Maroon" title="c11">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="#E7:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E78">A192</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E6:35_2_1_1_1_2_1_5_1_1_2"/><span class="lab"><font color="Green" title="E89">A206</font></span>: 
<font color="Maroon" title="c10">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="c5">D1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E83">A197</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_2_1_5_1_1_2"><span class="lab"><font color="Green" title="E87">A204</font></span></a>, <a class="ref" href="integra1.html#T7" title="INTEGRA1:th.7">INTEGRA1:7</a></span>;<br/><a NAME="E7:35_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="c11">k1</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="E8:35_2_1_1_1_2_1_5_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">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="c11">k1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E72">A169</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1_2_1_5_1_1_2"><span class="lab"><font color="Green" title="E88">A205</font></span></a></span>;<br/><a NAME="E9:35_2_1_1_1_2_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A207</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="c10">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="c11">k1</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E72">A169</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E75">A189</font></span></a>, <a class="txt" href="#E7:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E78">A192</font></span></a>, <a class="txt" href="#E8:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E79">A193</font></span></a></span>;<br/><a NAME="E10:35_2_1_1_1_2_1_5_1_1_2"/><span class="lab"><font color="Green" title="E91">A208</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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E72">A169</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E75">A189</font></span></a>, <a class="txt" href="#E7:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E78">A192</font></span></a>, <a class="txt" href="#E8:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E79">A193</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_2_1_5_1_1_2"><span class="lab"><font color="Green" title="E87">A204</font></span></a></span>;<br/><a NAME="E11:35_2_1_1_1_2_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A209</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="c10">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="c10">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="#E16:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E84">A198</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E12:35_2_1_1_1_2_1_5_1_1_2"/><span class="lab"><font color="Green" title="E93">A210</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="c10">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="c10">k</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E84">A198</font></span></a>, <a class="txt" href="#E10:35_2_1_1_1_2_1_5_1_1_2"><span class="lab"><font color="Green" title="E91">A208</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E13:35_2_1_1_1_2_1_5_1_1_2"/><span class="lab"><font color="Green" title="E94">A211</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">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="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E83">A197</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_2_1_5_1_1_2"><span class="lab"><font color="Green" title="E87">A204</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E14:35_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="c10">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="c10">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="#E15:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E83">A197</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_2_1_5_1_1_2"><span class="lab"><font color="Green" title="E87">A204</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="E15:35_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="c10">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="c10">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="c10">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="c10">k</font>)</span>)</span>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E15:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E83">A197</font></span></a>, <a class="txt" href="#E13:35_2_1_1_1_2_1_5_1_1_2"><span class="lab"><font color="Green" title="E94">A211</font></span></a>, <a class="txt" href="#E6:35_2_1_1_1_2_1_5_1_1_2"><span class="lab"><font color="Green" title="E89">A206</font></span></a>, <a class="txt" href="#E11:35_2_1_1_1_2_1_5_1_1_2"><span class="lab"><font color="Green" title="E92">A209</font></span></a>, <a class="txt" href="#E12:35_2_1_1_1_2_1_5_1_1_2"><span class="lab"><font color="Green" title="E93">A210</font></span></a>, <a class="txt" href="#E9:35_2_1_1_1_2_1_5_1_1_2"><span class="lab"><font color="Green" title="E90">A207</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="E19:35_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="c10">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="c10">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="c10">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="E20:35_2_1_1_1_2_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A212</font></span>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">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="c10">k</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E86">A200</font></span></a>, <a class="ref" href="integra1.html#T4" title="INTEGRA1:th.4">INTEGRA1:4</a></span>;<br/>
<a NAME="E21:35_2_1_1_1_2_1_5"/><span class="lab"><font color="Green" title="E88">A213</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="#E14:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E82">A196</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="E22:35_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#K4" title="INTEGRA1:func.4">upper_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="#E3:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E49">A41</font></span></a>, <a class="txt" href="#E6:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E77">A191</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="E23:35_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#K4" title="INTEGRA1:func.4">upper_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="E89">A214</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<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> = 
<span class="p1">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_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="c10">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E79">A193</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#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><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="c10">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="c10">k</font>)</span>)</span>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E88">A213</font></span></a>, <a class="txt" href="#E20:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E87">A212</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>
;<br/>
<a NAME="E25:35_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="c10">k</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E72">A169</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E75">A189</font></span></a>, <a class="txt" href="#E7:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E78">A192</font></span></a>, <a class="txt" href="#E8:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E79">A193</font></span></a></span>;<br/>
<a NAME="E26:35_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="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="txt" href="#E12:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E81">A195</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E27:35_2_1_1_1_2_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A215</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#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#K4" title="INTEGRA1:func.4">upper_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="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_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="c10">k</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E72">A169</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E75">A189</font></span></a>, <a class="txt" href="#E7:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E78">A192</font></span></a>, <a class="txt" href="#E8:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E79">A193</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_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="c10">k</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E80">A194</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E74">A188</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#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><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="c10">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="c10">k</font>)</span>)</span>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E90">A215</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E29:35_2_1_1_1_2_1_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_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#K4" title="INTEGRA1:func.4">upper_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="#E24:35_2_1_1_1_2_1_5"><span class="lab"><font color="Green" title="E89">A214</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="E40:35_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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E7:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E51">A43</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E41:35_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="E42:35_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A216</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#K4" title="INTEGRA1:func.4">upper_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#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/>
<a NAME="E43:35_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="#E7:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E51">A43</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E44:35_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A217</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#K4" title="INTEGRA1:func.4">upper_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#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/>
<a NAME="E45:35_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="#E12:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E55">A47</font></span></a>, <a class="txt" href="#E36:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E70">A167</font></span></a>, <a class="txt" href="#E34:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E69">A166</font></span></a>, <a class="ref" href="integra3.html#T6" target="_self" title="INTEGRA3:th.6">Th6</a></span>;<br/>
<a NAME="E46:35_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="#E26:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E65">A133</font></span></a>, <a class="txt" href="#E24:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E63">A131</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>;<br/>
<a NAME="E47:35_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#K4" title="INTEGRA1:func.4">upper_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="#E44:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E75">A217</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>;<br/>
<a NAME="E48:35_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#K4" title="INTEGRA1:func.4">upper_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#K4" title="INTEGRA1:func.4">upper_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="#E42:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E74">A216</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>;<br/>
<a NAME="E49:35_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A218</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_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#K4" title="INTEGRA1:func.4">upper_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="#E39:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E73">A187</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</a></span>;<br/>
<a NAME="E50:35_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E77">A219</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="#E5:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A38</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="E51:35_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A220</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="E52:35_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="#E21:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E60">A128</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/>
<a NAME="E53:35_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A221</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="E54:35_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="#E21:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E60">A128</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E55:35_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A222</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#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/>
<a NAME="E56:35_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="#E50:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E77">A219</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/>
<a NAME="E57:35_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">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#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="E58:35_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="#E50:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E77">A219</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E59:35_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A224</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#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/>
<a NAME="E60:35_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="#E3:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E49">A41</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="E61:35_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#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/>
<a NAME="E62:35_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="E63:35_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="#E29:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E67">A146</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="#E29:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E67">A146</font></span></a>, <a class="txt" href="#E59:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E82">A224</font></span></a>, <a class="txt" href="#E19:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E59">A127</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="#E51:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E78">A220</font></span></a>, <a class="ref" href="finseq_6.html#T116" title="FINSEQ_6:th.116">FINSEQ_6:116</a></span>
;<br/>
<a NAME="E65:35_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A225</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#K4" title="INTEGRA1:func.4">upper_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="#E57:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E81">A223</font></span></a>, <a class="ref" href="integra1.html#D20" title="INTEGRA1:def.20">INTEGRA1:def 20</a></span>;<br/>
<a NAME="E66:35_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="#E8:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E52">A44</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="E67:35_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#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/>
<a NAME="E68:35_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="E69:35_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#K4" title="INTEGRA1:func.4">upper_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="#E10:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E54">A46</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="#E10:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E54">A46</font></span></a>, <a class="txt" href="#E15:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E56">A48</font></span></a>, <a class="txt" href="#E55:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E80">A222</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="#E22:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E61">A129</font></span></a>, <a class="ref" href="finseq_6.html#T116" title="FINSEQ_6:th.116">FINSEQ_6:116</a></span>
;<br/>
<a NAME="E71:35_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A226</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#K4" title="INTEGRA1:func.4">upper_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="#E53:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E79">A221</font></span></a>, <a class="ref" href="integra1.html#D20" title="INTEGRA1:def.20">INTEGRA1:def 20</a></span>;<br/>
<a NAME="E72:35_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="#E8:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E52">A44</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="E73:35_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#K4" title="INTEGRA1:func.4">upper_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#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/>
<a NAME="E74:35_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#K4" title="INTEGRA1:func.4">upper_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#K4" title="INTEGRA1:func.4">upper_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="#E49:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E76">A218</font></span></a>, <a class="txt" href="#E6:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E50">A42</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="E76:35_2_1_1_1_2_1"/>
<font color="Maroon">H<sub>2</sub></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="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="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="#E17:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E58">A50</font></span></a>, <a class="txt" href="#E71:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E84">A226</font></span></a>, <a class="txt" href="#E65:35_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E83">A225</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="E8:35_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="#E5:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A38</font></span></a>, <a class="txt" href="#E6:35_2_1_1_1_2"><span class="lab"><font color="Green" title="E47">A39</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="#E2:35_2_1_1_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E6:35_2_1_1_1"/><span class="lab"><font color="Green" title="E45">A227</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="35_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="E1:35_2_1_1_1_3"/><span class="lab"><font color="Green" title="E46">A228</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="E2:35_2_1_1_1_3"/><span class="lab"><font color="Green" title="E47">A229</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="E3:35_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="35_2_1_1_1_3_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:35_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E48">A230</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="E2:35_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E49">A231</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="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="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="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="E4:35_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E50">A232</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="E5:35_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E51">A233</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="E6:35_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E52">A234</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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E7:35_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="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="#E2:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E49">A231</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="E8:35_2_1_1_1_3_1"/><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>  <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="E9:35_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="#E1:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E48">A230</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E10:35_2_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A235</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="#E1:35_2_1_1_1_3"><span class="lab"><font color="Green" title="E46">A228</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E11:35_2_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A236</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/>
<a NAME="E12:35_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E55">A237</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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E13:35_2_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A238</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="E14:35_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E57">A239</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="#E12:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E55">A237</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E15:35_2_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A240</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#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</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="E17:35_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E59">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="E18:35_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E60">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="E19:35_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E61">A243</font></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#K9" title="REAL_1:func.9">-</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="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="#E2:35_2_1_1_1_3"><span class="lab"><font color="Green" title="E47">A229</font></span></a>, <a class="txt" href="#E10:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E53">A235</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="E20:35_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E62">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="E21:35_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E63">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="35_2_1_1_1_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:35_2_1_1_1_3_1_1"/><span class="lab"><font color="Green" title="E64">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="35_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="35_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="E1:35_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="#E1:35_2_1_1_1_3_1_1"><span class="lab"><font color="Green" title="E64">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="35_2_1_1_1_3_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_1_1_3_1_1_1_1_1"/><span class="lab"><font color="Green" title="E65">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="E2:35_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="#E11:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E54">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="E3:35_2_1_1_1_3_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">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="#E18:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E60">A242</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E65">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="E4:35_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="#E2:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E49">A231</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="E5:35_2_1_1_1_3_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">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="#E5:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E51">A233</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="E6:35_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="E7:35_2_1_1_1_3_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_2"><span class="lab"><font color="Green" title="E35">A8</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E49">A231</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</font></span></a>, <a class="txt" href="#E11:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E54">A236</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E66">A248</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E67">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="35_2_1_1_1_3_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_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="E2:35_2_1_1_1_3_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">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="E3:35_2_1_1_1_3_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">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="E4:35_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="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E5:35_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="E6:35_2_1_1_1_3_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">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="#E2:35_2_1_1_1_3_1_1_1_1_2"><span class="lab"><font color="Green" title="E65">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="E7:35_2_1_1_1_3_1_1_1_1_2"/><span class="kw">then </span>
<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="#E17:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E59">A241</font></span></a>, <a class="ref" href="integra1.html#T7" title="INTEGRA1:th.7">INTEGRA1:7</a></span>;<br/><a NAME="E8:35_2_1_1_1_3_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A253</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> <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="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_3_1_1_1_1_2"><span class="lab"><font color="Green" title="E66">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="E9:35_2_1_1_1_3_1_1_1_1_2"/><span class="lab"><font color="Green" title="E69">A254</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="35_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="35_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="E1:35_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="35_2_1_1_1_3_1_1_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_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="E2:35_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="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</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="35_2_1_1_1_3_1_1_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_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="E2:35_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="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</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="E10:35_2_1_1_1_3_1_1_1_1_2"/><span class="lab"><font color="Green" title="E70">A255</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">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="#E18:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E60">A242</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/><a NAME="E11:35_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="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="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="#E17:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E59">A241</font></span></a>, <a class="txt" href="#E6:35_2_1_1_1_3_1_1_1_1_2"><span class="lab"><font color="Green" title="E67">A252</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E12:35_2_1_1_1_3_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A256</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="#E10:35_2_1_1_1_3_1_1_1_1_2"><span class="lab"><font color="Green" title="E70">A255</font></span></a>, <a class="txt" href="#E8:35_2_1_1_1_3_1_1_1_1_2"><span class="lab"><font color="Green" title="E68">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="E13:35_2_1_1_1_3_1_1_1_1_2"/><span class="lab"><font color="Green" title="E72">A257</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="E14:35_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="#E5:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E51">A233</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/><a NAME="E15:35_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="#E9:35_2_1_1_1_3_1_1_1_1_2"><span class="lab"><font color="Green" title="E69">A254</font></span></a>, <a class="txt" href="#E12:35_2_1_1_1_3_1_1_1_1_2"><span class="lab"><font color="Green" title="E71">A256</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="E16:35_2_1_1_1_3_1_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E49">A231</font></span></a>, <a class="txt" href="#E11:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E54">A236</font></span></a>, <a class="txt" href="#E13:35_2_1_1_1_3_1_1_1_1_2"><span class="lab"><font color="Green" title="E72">A257</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="E3:35_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="E22:35_2_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A258</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="E23:35_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E65">A259</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="#E17:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E59">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="E24:35_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E66">A260</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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E17:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E59">A241</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E25:35_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E67">A261</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="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E26:35_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E68">A262</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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E17:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E59">A241</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E27:35_2_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A263</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="E28:35_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E70">A264</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="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E29:35_2_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A265</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="#E22:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E64">A258</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E30:35_2_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A266</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="#E20:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">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="E31:35_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="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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E32:35_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E74">A268</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="#E28:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E70">A264</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/>
<a NAME="E33:35_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="#E25:35_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#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E34:35_2_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A269</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="E35:35_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E76">A270</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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E30:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E36:35_2_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A271</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="E37:35_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="#E17:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E59">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="E38:35_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#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/>
<a NAME="E39:35_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="#E23:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E65">A259</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="#E21:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A245</font></span></a>, <a class="txt" href="#E23:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E65">A259</font></span></a>, <a class="txt" href="#E32:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E74">A268</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="#E25:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E67">A261</font></span></a>, <a class="ref" href="finseq_6.html#T116" title="FINSEQ_6:th.116">FINSEQ_6:116</a></span>
;<br/>
<a NAME="E42:35_2_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A272</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="#E34:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E75">A269</font></span></a>, <a class="ref" href="integra1.html#D20" title="INTEGRA1:def.20">INTEGRA1:def 20</a></span>;<br/>
<a NAME="E43:35_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="#E12:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E55">A237</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="E44:35_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#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/>
<a NAME="E45:35_2_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A273</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="E46:35_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E80">A274</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="#E35:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E76">A270</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E47:35_2_1_1_1_3_1"/>
<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> <font color="Maroon" title="c10">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</font></span></a>, <a class="txt" href="#E22:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E64">A258</font></span></a>, <a class="txt" href="#E30:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/>
<a NAME="E48:35_2_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A275</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="#E35:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E76">A270</font></span></a>, <a class="txt" href="#E31:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E73">A267</font></span></a>, <a class="txt" href="#E12:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E55">A237</font></span></a>, <a class="txt" href="#E6:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E52">A234</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="E49:35_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="E50:35_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="E82">A276</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="#E48:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E81">A275</font></span></a>, <a class="txt" href="#E36:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E77">A271</font></span></a>, <a class="txt" href="#E14:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E57">A239</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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E30:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>
;<br/>
<a NAME="E52:35_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E83">A277</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="#E17:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E59">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="E53:35_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E84">A278</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="#E22:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E64">A258</font></span></a>, <a class="ref" href="xreal_1.html#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/>
<a NAME="E54:35_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="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="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="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="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="35_2_1_1_1_3_1_5"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_1_1_3_1_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="E1:35_2_1_1_1_3_1_5_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="#E22:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E64">A258</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="35_2_1_1_1_3_1_5_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_1_1_3_1_5_1_1"/><span class="lab"><font color="Green" title="E85">A279</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="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="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="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="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="E2:35_2_1_1_1_3_1_5_1_1"/><span class="lab"><font color="Green" title="E86">A280</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="35_2_1_1_1_3_1_5_1_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:35_2_1_1_1_3_1_5_1_1_1"/><span class="lab"><font color="Green" title="E87">A281</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="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</font></span></a>, <a class="txt" href="#E21:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A245</font></span></a>, <a class="txt" href="#E52:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E83">A277</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/>
<a NAME="E2:35_2_1_1_1_3_1_5_1_1_1"/><span class="lab"><font color="Green" title="E88">A282</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="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</font></span></a>, <a class="txt" href="#E21:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A245</font></span></a>, <a class="txt" href="#E52:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E83">A277</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/>
<a NAME="E3:35_2_1_1_1_3_1_5_1_1_1"/><span class="lab"><font color="Green" title="E89">A283</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="#E27:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E69">A263</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="E4:35_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="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="E5:35_2_1_1_1_3_1_5_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A284</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="E6:35_2_1_1_1_3_1_5_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A285</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="#E14:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E57">A239</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E7:35_2_1_1_1_3_1_5_1_1_1"/><span class="lab"><font color="Green" title="E92">A286</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="E8:35_2_1_1_1_3_1_5_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A287</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="E9:35_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"><a href="integra1.html#K13" 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="#E27:35_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="E10:35_2_1_1_1_3_1_5_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="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="#E6:35_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E91">A285</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E11:35_2_1_1_1_3_1_5_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A289</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="#E12:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E55">A237</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E90">A284</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/>
<a NAME="E12:35_2_1_1_1_3_1_5_1_1_1"/><span class="lab"><font color="Green" title="E96">A290</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="35_2_1_1_1_3_1_5_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:35_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="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="E3:35_2_1_1_1_3_1_5_1_1_1_1"/><span class="lab"><font color="Green" title="E97">A291</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="E4:35_2_1_1_1_3_1_5_1_1_1_1"/><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"><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="E5:35_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#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="#E12:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E55">A237</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E90">A284</font></span></a>, <a class="txt" href="#E10:35_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E94">A288</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="E6:35_2_1_1_1_3_1_5_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A293</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="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</font></span></a>, <a class="txt" href="#E6:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E52">A234</font></span></a>, <a class="txt" href="#E3:35_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="txt" href="#E4:35_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="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/>
<a NAME="E7:35_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"><a href="integra1.html#K13" 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="#E26:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E68">A262</font></span></a>, <a class="txt" href="#E8:35_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E93">A287</font></span></a>, <a class="txt" href="#E10:35_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E94">A288</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="E8:35_2_1_1_1_3_1_5_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="#E17:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E59">A241</font></span></a>, <a class="txt" href="#E24:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E66">A260</font></span></a>, <a class="txt" href="#E3:35_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="txt" href="#E4:35_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="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:35_2_1_1_1_3_1_5_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_1"><span class="lab"><font color="Green" title="E85">A279</font></span></a>, <a class="txt" href="#E6:35_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="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="E13:35_2_1_1_1_3_1_5_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="#E10:35_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E94">A288</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="E14:35_2_1_1_1_3_1_5_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A294</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="#E4:35_2_1"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="txt" href="#E12:35_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E96">A290</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="E15:35_2_1_1_1_3_1_5_1_1_1"/><span class="lab"><font color="Green" title="E98">A295</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="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</font></span></a>, <a class="txt" href="#E21:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A245</font></span></a>, <a class="txt" href="#E52:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E83">A277</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/>
<a NAME="E16:35_2_1_1_1_3_1_5_1_1_1"/><span class="lab"><font color="Green" title="E99">A296</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="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</font></span></a>, <a class="txt" href="#E21:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A245</font></span></a>, <a class="txt" href="#E52:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E83">A277</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/>
<a NAME="E17:35_2_1_1_1_3_1_5_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="#E26:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E68">A262</font></span></a>, <a class="txt" href="#E8:35_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E93">A287</font></span></a>, <a class="txt" href="#E10:35_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E94">A288</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/>
<a NAME="E18:35_2_1_1_1_3_1_5_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="#E24:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E66">A260</font></span></a>, <a class="txt" href="#E6:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E52">A234</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_1"><span class="lab"><font color="Green" title="E85">A279</font></span></a>, <a class="txt" href="#E15:35_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E98">A295</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E87">A281</font></span></a>, <a class="txt" href="#E11:35_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E95">A289</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/>
<a NAME="E19:35_2_1_1_1_3_1_5_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> <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="#E14:35_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E97">A294</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="E20:35_2_1_1_1_3_1_5_1_1_1"/><span class="lab"><font color="Green" title="E101">A298</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="#E5:35_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E90">A284</font></span></a>, <a class="txt" href="#E7:35_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E92">A286</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E21:35_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="#E14:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E57">A239</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E22:35_2_1_1_1_3_1_5_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A299</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="#E3:35_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="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E23:35_2_1_1_1_3_1_5_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"><a href="integra1.html#K13" 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="#E12:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E55">A237</font></span></a>, <a class="txt" href="#E20:35_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E101">A298</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/>
<a NAME="E24:35_2_1_1_1_3_1_5_1_1_1"/><span class="lab"><font color="Green" title="E104">A301</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="E25:35_2_1_1_1_3_1_5_1_1_1"/><span class="lab"><font color="Green" title="E105">A302</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="35_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="E1:35_2_1_1_1_3_1_5_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="E3:35_2_1_1_1_3_1_5_1_1_1_2"/><span class="lab"><font color="Green" title="E106">A303</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="E4:35_2_1_1_1_3_1_5_1_1_1_2"/><span class="lab"><font color="Green" title="E107">A304</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="E5:35_2_1_1_1_3_1_5_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="#E12:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E55">A237</font></span></a>, <a class="txt" href="#E20:35_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E101">A298</font></span></a>, <a class="txt" href="#E22:35_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E102">A299</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="E6:35_2_1_1_1_3_1_5_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A305</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="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</font></span></a>, <a class="txt" href="#E6:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E52">A234</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_3_1_5_1_1_1_2"><span class="lab"><font color="Green" title="E106">A303</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_3_1_5_1_1_1_2"><span class="lab"><font color="Green" title="E107">A304</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/>
<a NAME="E7:35_2_1_1_1_3_1_5_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="#E26:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E68">A262</font></span></a>, <a class="txt" href="#E24:35_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E104">A301</font></span></a>, <a class="txt" href="#E22:35_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E102">A299</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="E8:35_2_1_1_1_3_1_5_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="#E17:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E59">A241</font></span></a>, <a class="txt" href="#E24:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E66">A260</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_3_1_5_1_1_1_2"><span class="lab"><font color="Green" title="E106">A303</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_3_1_5_1_1_1_2"><span class="lab"><font color="Green" title="E107">A304</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="E9:35_2_1_1_1_3_1_5_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_1"><span class="lab"><font color="Green" title="E85">A279</font></span></a>, <a class="txt" href="#E6:35_2_1_1_1_3_1_5_1_1_1_2"><span class="lab"><font color="Green" title="E108">A305</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="E26:35_2_1_1_1_3_1_5_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="#E22:35_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E102">A299</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="E27:35_2_1_1_1_3_1_5_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A306</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="#E4:35_2_1"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="txt" href="#E25:35_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E105">A302</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="E28:35_2_1_1_1_3_1_5_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="#E26:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E68">A262</font></span></a>, <a class="txt" href="#E24:35_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E104">A301</font></span></a>, <a class="txt" href="#E22:35_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E102">A299</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/>
<a NAME="E29:35_2_1_1_1_3_1_5_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="#E24:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E66">A260</font></span></a>, <a class="txt" href="#E6:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E52">A234</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_1"><span class="lab"><font color="Green" title="E85">A279</font></span></a>, <a class="txt" href="#E2:35_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="#E16:35_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E99">A296</font></span></a>, <a class="txt" href="#E23:35_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E103">A300</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/>
<a NAME="E30:35_2_1_1_1_3_1_5_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="#E27:35_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E106">A306</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="E31:35_2_1_1_1_3_1_5_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="#E1:35_2"><span class="lab"><font color="Green" title="E35">A8</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</font></span></a>, <a class="txt" href="#E19:35_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E100">A297</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="E32:35_2_1_1_1_3_1_5_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E92">A286</font></span></a>, <a class="txt" href="#E22:35_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E102">A299</font></span></a>, <a class="txt" href="#E10:35_2_1_1_1_3_1_5_1_1_1"><span class="lab"><font color="Green" title="E94">A288</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="E3:35_2_1_1_1_3_1_5_1_1"/><span class="lab"><font color="Green" title="E87">A307</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="35_2_1_1_1_3_1_5_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:35_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#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="E2:35_2_1_1_1_3_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A308</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="E3:35_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> 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="#E2:35_2_1_1_1_3_1_5_1_1"><span class="lab"><font color="Green" title="E86">A280</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="E4:35_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> 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="#E2:35_2_1_1_1_3_1_5_1_1_2"><span class="lab"><font color="Green" title="E88">A308</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="E4:35_2_1_1_1_3_1_5_1_1"/><span class="lab"><font color="Green" title="E88">A309</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="E5:35_2_1_1_1_3_1_5_1_1"/><span class="lab"><font color="Green" title="E89">A310</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="#E14:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E57">A239</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/><a NAME="E6:35_2_1_1_1_3_1_5_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="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</font></span></a>, <a class="txt" href="#E17:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E59">A241</font></span></a>, <a class="txt" href="#E21:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">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="E7:35_2_1_1_1_3_1_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A311</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="#E26:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E68">A262</font></span></a>, <a class="txt" href="#E24:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E66">A260</font></span></a>, <a class="txt" href="#E12:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E55">A237</font></span></a>, <a class="txt" href="#E6:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E52">A234</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/><a NAME="E8:35_2_1_1_1_3_1_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A312</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="E9:35_2_1_1_1_3_1_5_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="#E14:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E57">A239</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E10:35_2_1_1_1_3_1_5_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#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A313</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="#E13:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E56">A238</font></span></a>, <a class="txt" href="#E8:35_2_1_1_1_3_1_5_1_1"><span class="lab"><font color="Green" title="E91">A312</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_3_1_5_1_1"><span class="lab"><font color="Green" title="E88">A309</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1_3_1_5_1_1"><span class="lab"><font color="Green" title="E89">A310</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="#E8:35_2_1_1_1_3_1_5_1_1"><span class="lab"><font color="Green" title="E91">A312</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="E12:35_2_1_1_1_3_1_5_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="#E7:35_2_1_1_1_3_1_5_1_1"><span class="lab"><font color="Green" title="E90">A311</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E13:35_2_1_1_1_3_1_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A314</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="E14:35_2_1_1_1_3_1_5_1_1"/><span class="lab"><font color="Green" title="E94">A315</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#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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1_1_1_3_1_5_1_1_4"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1_1_1_3_1_5_1_1_4_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_1_1_3_1_5_1_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="E1:35_2_1_1_1_3_1_5_1_1_4_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="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="#E13:35_2_1_1_1_3_1_5_1_1"><span class="lab"><font color="Green" title="E93">A314</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_3_1_5_1_1"><span class="lab"><font color="Green" title="E87">A307</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_1_1_3_1_5_1_1_4_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_1_1_3_1_5_1_1_4_1_1_1"/><span class="lab"><font color="Green" title="E95">A316</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#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><br/><div class="add"><a NAME="E2:35_2_1_1_1_3_1_5_1_1_4_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#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="#E27:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E69">A263</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E3:35_2_1_1_1_3_1_5_1_1_4_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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><a NAME="E4:35_2_1_1_1_3_1_5_1_1_4_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="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="#E12:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E55">A237</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_1_4_1_1_1"><span class="lab"><font color="Green" title="E95">A316</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E5:35_2_1_1_1_3_1_5_1_1_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A318</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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_1_4_1_1_1"><span class="lab"><font color="Green" title="E95">A316</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/><a NAME="E6:35_2_1_1_1_3_1_5_1_1_4_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="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="#E12:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E55">A237</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_1_4_1_1_1"><span class="lab"><font color="Green" title="E95">A316</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_3_1_5_1_1_4_1_1_1"><span class="lab"><font color="Green" title="E96">A317</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E7:35_2_1_1_1_3_1_5_1_1_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A319</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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E17:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E59">A241</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/><a NAME="E8:35_2_1_1_1_3_1_5_1_1_4_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="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="#E21:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A245</font></span></a>, <a class="txt" href="#E52:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E83">A277</font></span></a>, <a class="txt" href="#E30:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_1"><span class="lab"><font color="Green" title="E85">A279</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E9:35_2_1_1_1_3_1_5_1_1_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A320</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="#E21:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A245</font></span></a>, <a class="txt" href="#E52:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E83">A277</font></span></a>, <a class="txt" href="#E30:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_1"><span class="lab"><font color="Green" title="E85">A279</font></span></a>, <a class="txt" href="#E7:35_2_1_1_1_3_1_5_1_1_4_1_1_1"><span class="lab"><font color="Green" title="E98">A319</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1_3_1_5_1_1_4_1_1_1"><span class="lab"><font color="Green" title="E97">A318</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E10:35_2_1_1_1_3_1_5_1_1_4_1_1_1"/><span class="lab"><font color="Green" title="E100">A321</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="E11:35_2_1_1_1_3_1_5_1_1_4_1_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="#E1:35_2_1_1_1_3_1_5_1_1_4_1_1_1"><span class="lab"><font color="Green" title="E95">A316</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="#E4:35_2_1_1_1_3_1_5_1_1"><span class="lab"><font color="Green" title="E88">A309</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1_3_1_5_1_1"><span class="lab"><font color="Green" title="E89">A310</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="E13:35_2_1_1_1_3_1_5_1_1_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A322</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="#E11:35_2_1_1_1_3_1_5_1_1"><span class="lab"><font color="Green" title="E92">A313</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_1_4_1_1_1"><span class="lab"><font color="Green" title="E95">A316</font></span></a>, <a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>;<br/><a NAME="E14:35_2_1_1_1_3_1_5_1_1_4_1_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#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><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="#E12:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E55">A237</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_1_4_1_1_1"><span class="lab"><font color="Green" title="E95">A316</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/><a NAME="E15:35_2_1_1_1_3_1_5_1_1_4_1_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#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="#E1:35"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E12:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E55">A237</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_1_4_1_1_1"><span class="lab"><font color="Green" title="E95">A316</font></span></a>, <a class="txt" href="#E9:35_2_1_1_1_3_1_5_1_1_4_1_1_1"><span class="lab"><font color="Green" title="E99">A320</font></span></a>, <a class="txt" href="#E10:35_2_1_1_1_3_1_5_1_1_4_1_1_1"><span class="lab"><font color="Green" title="E100">A321</font></span></a>, <a class="ref" href="integra3.html#T19" target="_self" title="INTEGRA3:th.19">Th19</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="E16:35_2_1_1_1_3_1_5_1_1_4_1_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#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="#E13:35_2_1_1_1_3_1_5_1_1_4_1_1_1"><span class="lab"><font color="Green" title="E101">A322</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="35_2_1_1_1_3_1_5_1_1_4_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_1_1_3_1_5_1_1_4_1_1_2"/><span class="lab"><font color="Green" title="E95">A323</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#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><br/><div class="add"><a NAME="E2:35_2_1_1_1_3_1_5_1_1_4_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> 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="#E27:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E69">A263</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E3:35_2_1_1_1_3_1_5_1_1_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A324</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="E4:35_2_1_1_1_3_1_5_1_1_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A325</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="#E12:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E55">A237</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E95">A323</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E5:35_2_1_1_1_3_1_5_1_1_4_1_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="E6:35_2_1_1_1_3_1_5_1_1_4_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="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="#E12:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E55">A237</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E95">A323</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_3_1_5_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E96">A324</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E7:35_2_1_1_1_3_1_5_1_1_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A326</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="#E6:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E52">A234</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_3_1_5_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E97">A325</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><a NAME="E8:35_2_1_1_1_3_1_5_1_1_4_1_1_2"/><span class="lab"><font color="Green" title="E99">A327</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="#E21:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A245</font></span></a>, <a class="txt" href="#E52:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E83">A277</font></span></a>, <a class="txt" href="#E30:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_1"><span class="lab"><font color="Green" title="E85">A279</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E9:35_2_1_1_1_3_1_5_1_1_4_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="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="#E21:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A245</font></span></a>, <a class="txt" href="#E52:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E83">A277</font></span></a>, <a class="txt" href="#E30:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_1"><span class="lab"><font color="Green" title="E85">A279</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E10:35_2_1_1_1_3_1_5_1_1_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A328</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="#E8:35_2_1_1_1_3_1_5_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E99">A327</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><a NAME="E11:35_2_1_1_1_3_1_5_1_1_4_1_1_2"/><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="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="E12:35_2_1_1_1_3_1_5_1_1_4_1_1_2"/><span class="lab"><font color="Green" title="E102">A330</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="#E14:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E57">A239</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/><a NAME="E13:35_2_1_1_1_3_1_5_1_1_4_1_1_2"/><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="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="E14:35_2_1_1_1_3_1_5_1_1_4_1_1_2"/><span class="lab"><font color="Green" title="E104">A332</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="E15:35_2_1_1_1_3_1_5_1_1_4_1_1_2"/><span class="lab"><font color="Green" title="E105">A333</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="E16:35_2_1_1_1_3_1_5_1_1_4_1_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="#E14:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E57">A239</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E95">A323</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E17:35_2_1_1_1_3_1_5_1_1_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A334</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="#E14:35_2_1_1_1_3_1_5_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E104">A332</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E18:35_2_1_1_1_3_1_5_1_1_4_1_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#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><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#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/><a NAME="E19:35_2_1_1_1_3_1_5_1_1_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A335</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#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="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="#E1:35"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E17:35_2_1_1_1_3_1_5_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E106">A334</font></span></a>, <a class="txt" href="#E13:35_2_1_1_1_3_1_5_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E103">A331</font></span></a>, <a class="ref" href="integra3.html#T19" target="_self" title="INTEGRA3:th.19">Th19</a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/><a NAME="E20:35_2_1_1_1_3_1_5_1_1_4_1_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="#E1:35_2_1_1_1_3_1_5_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E95">A323</font></span></a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A336</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="#E14:35_2_1_1_1_3_1_5_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E104">A332</font></span></a>, <a class="txt" href="#E15:35_2_1_1_1_3_1_5_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E105">A333</font></span></a>, <a class="txt" href="#E12:35_2_1_1_1_3_1_5_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E102">A330</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="E22:35_2_1_1_1_3_1_5_1_1_4_1_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="#E1:35_2_1_1_1_3_1_5_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E95">A323</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="#E1:35_2_1_1_1_3_1_5_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E95">A323</font></span></a>, <a class="txt" href="#E14:35_2_1_1_1_3_1_5_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E104">A332</font></span></a>, <a class="txt" href="#E15:35_2_1_1_1_3_1_5_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E105">A333</font></span></a>, <a class="txt" href="#E12:35_2_1_1_1_3_1_5_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E102">A330</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="E24:35_2_1_1_1_3_1_5_1_1_4_1_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="#E11:35_2_1_1_1_3_1_5_1_1"><span class="lab"><font color="Green" title="E92">A313</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E95">A323</font></span></a>, <a class="txt" href="#E21:35_2_1_1_1_3_1_5_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E108">A336</font></span></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/><a NAME="E25:35_2_1_1_1_3_1_5_1_1_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A337</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="E26:35_2_1_1_1_3_1_5_1_1_4_1_1_2"/><span class="lab"><font color="Green" title="E110">A338</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="#E27:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E69">A263</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E27:35_2_1_1_1_3_1_5_1_1_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">A339</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="#E17:35_2_1_1_1_3_1_5_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E106">A334</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E28:35_2_1_1_1_3_1_5_1_1_4_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="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="#E17:35_2_1_1_1_3_1_5_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E106">A334</font></span></a>, <a class="txt" href="#E26:35_2_1_1_1_3_1_5_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E110">A338</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E29:35_2_1_1_1_3_1_5_1_1_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E112">A340</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="#E24:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E66">A260</font></span></a>, <a class="txt" href="#E27:35_2_1_1_1_3_1_5_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E111">A339</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><a NAME="E30:35_2_1_1_1_3_1_5_1_1_4_1_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#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><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="#E12:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E55">A237</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E95">A323</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/><a NAME="E31:35_2_1_1_1_3_1_5_1_1_4_1_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#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="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="#E1:35"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E12:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E55">A237</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E95">A323</font></span></a>, <a class="txt" href="#E11:35_2_1_1_1_3_1_5_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E101">A329</font></span></a>, <a class="ref" href="integra3.html#T19" target="_self" title="INTEGRA3:th.19">Th19</a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/><a NAME="E32:35_2_1_1_1_3_1_5_1_1_4_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="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#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="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#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="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="#E25:35_2_1_1_1_3_1_5_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E109">A337</font></span></a>, <a class="txt" href="#E19:35_2_1_1_1_3_1_5_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E107">A335</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="E33:35_2_1_1_1_3_1_5_1_1_4_1_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#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="#E1:35_2_1_1_1_3_1_5_1_1"><span class="lab"><font color="Green" title="E85">A279</font></span></a>, <a class="txt" href="#E29:35_2_1_1_1_3_1_5_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E112">A340</font></span></a>, <a class="txt" href="#E7:35_2_1_1_1_3_1_5_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E98">A326</font></span></a>, <a class="txt" href="#E10:35_2_1_1_1_3_1_5_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E100">A328</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>
<span class="kw">hence </span><a NAME="E2:35_2_1_1_1_3_1_5_1_1_4"/>
 <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#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>
 ; <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="E15:35_2_1_1_1_3_1_5_1_1"/><span class="lab"><font color="Green" title="E95">A341</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="#E29:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E71">A265</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/><a NAME="E16:35_2_1_1_1_3_1_5_1_1"/>
<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> <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="#E1:35_2_1_1_1_3_1_5_1_1"><span class="lab"><font color="Green" title="E85">A279</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/><a NAME="E17:35_2_1_1_1_3_1_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A342</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A244</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_1"><span class="lab"><font color="Green" title="E85">A279</font></span></a>, <a class="txt" href="#E15:35_2_1_1_1_3_1_5_1_1"><span class="lab"><font color="Green" title="E95">A341</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>;<br/><a NAME="E18:35_2_1_1_1_3_1_5_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="#E22:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E64">A258</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E19:35_2_1_1_1_3_1_5_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="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="#E20:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A244</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_1"><span class="lab"><font color="Green" title="E85">A279</font></span></a>, <a class="txt" href="#E15:35_2_1_1_1_3_1_5_1_1"><span class="lab"><font color="Green" title="E95">A341</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#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><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="#E30:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>
;<br/><a NAME="E21:35_2_1_1_1_3_1_5_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#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> <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="#E17:35_2_1_1_1_3_1_5_1_1"><span class="lab"><font color="Green" title="E96">A342</font></span></a>, <a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>;<br/><a NAME="E22:35_2_1_1_1_3_1_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A343</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#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><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="E23:35_2_1_1_1_3_1_5_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="#E30:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="ref" href="integra1.html#T8" title="INTEGRA1:th.8">INTEGRA1:8</a></span>;<br/><a NAME="E24:35_2_1_1_1_3_1_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A344</font></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"><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#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="#E1:35"><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="E25:35_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#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="#E30:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</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="E26:35_2_1_1_1_3_1_5_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="E27:35_2_1_1_1_3_1_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A345</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="E28:35_2_1_1_1_3_1_5_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="#E30:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="ref" href="integra1.html#T20" title="INTEGRA1:th.20">INTEGRA1:20</a></span>;<br/><a NAME="E29:35_2_1_1_1_3_1_5_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="#E27:35_2_1_1_1_3_1_5_1_1"><span class="lab"><font color="Green" title="E99">A345</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="E30:35_2_1_1_1_3_1_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A346</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="E31:35_2_1_1_1_3_1_5_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="#E1:35"><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="E32:35_2_1_1_1_3_1_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A347</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="#E30:35_2_1_1_1_3_1_5_1_1"><span class="lab"><font color="Green" title="E100">A346</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/><a NAME="E33:35_2_1_1_1_3_1_5_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="E34:35_2_1_1_1_3_1_5_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#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="#E22:35_2_1_1_1_3_1_5_1_1"><span class="lab"><font color="Green" title="E97">A343</font></span></a>, <a class="txt" href="#E24:35_2_1_1_1_3_1_5_1_1"><span class="lab"><font color="Green" title="E98">A344</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/><a NAME="E35:35_2_1_1_1_3_1_5_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="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="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="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="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="#E14:35_2_1_1_1_3_1_5_1_1"><span class="lab"><font color="Green" title="E94">A315</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="E36:35_2_1_1_1_3_1_5_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="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="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="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="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="#E32:35_2_1_1_1_3_1_5_1_1"><span class="lab"><font color="Green" title="E101">A347</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="35_2_1_1_1_3_1_5_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_1_1_3_1_5_1_2"/><span class="lab"><font color="Green" title="E85">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="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="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="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="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="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="E2:35_2_1_1_1_3_1_5_1_2"/><span class="lab"><font color="Green" title="E86">A349</font></span>: 
<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="#E22:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E64">A258</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/><a NAME="E3:35_2_1_1_1_3_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A350</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="E4:35_2_1_1_1_3_1_5_1_2"/><span class="lab"><font color="Green" title="E88">A351</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="E5:35_2_1_1_1_3_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A352</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="#E17:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E59">A241</font></span></a>, <a class="txt" href="#E30:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</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="E7:35_2_1_1_1_3_1_5_1_2"/><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="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="E8:35_2_1_1_1_3_1_5_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="E9:35_2_1_1_1_3_1_5_1_2"/><span class="lab"><font color="Green" title="E91">A354</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="E10:35_2_1_1_1_3_1_5_1_2"/><span class="lab"><font color="Green" title="E92">A355</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="E11:35_2_1_1_1_3_1_5_1_2"/><span class="lab"><font color="Green" title="E93">A356</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="E12:35_2_1_1_1_3_1_5_1_2"/><span class="lab"><font color="Green" title="E94">A357</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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</font></span></a>, <a class="txt" href="#E22:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E64">A258</font></span></a>, <a class="txt" href="#E30:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="txt" href="#E51:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E82">A276</font></span></a>, <a class="ref" href="integra3.html#T15" target="_self" title="INTEGRA3:th.15">Th15</a></span>;<br/><a NAME="E13:35_2_1_1_1_3_1_5_1_2"/><span class="lab"><font color="Green" title="E95">A358</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="#E22:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E64">A258</font></span></a>, <a class="txt" href="#E25:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E67">A261</font></span></a>, <a class="txt" href="#E28:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E70">A264</font></span></a>, <a class="txt" href="#E20:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A244</font></span></a>, <a class="txt" href="#E29:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E71">A265</font></span></a>, <a class="txt" href="#E11:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E93">A356</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>;<br/><a NAME="E14:35_2_1_1_1_3_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A359</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="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="#E22:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E64">A258</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/><a NAME="E15:35_2_1_1_1_3_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A360</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 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j</font>
 ;<br/><a NAME="E16:35_2_1_1_1_3_1_5_1_2"/><span class="lab"><font color="Green" title="E98">A361</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="E17:35_2_1_1_1_3_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A362</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="E18:35_2_1_1_1_3_1_5_1_2"/><span class="lab"><font color="Green" title="E100">A363</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="35_2_1_1_1_3_1_5_1_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_1_1_3_1_5_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="E1:35_2_1_1_1_3_1_5_1_2_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 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="35_2_1_1_1_3_1_5_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_1_1_3_1_5_1_2_1_1_1"/><span class="lab"><font color="Green" title="E101">A364</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="E2:35_2_1_1_1_3_1_5_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A365</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="#E16:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E98">A361</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E3:35_2_1_1_1_3_1_5_1_2_1_1_1"/><span class="lab"><font color="Green" title="E103">A366</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="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</font></span></a>, <a class="txt" href="#E21:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A245</font></span></a>, <a class="txt" href="#E52:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E83">A277</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E4:35_2_1_1_1_3_1_5_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="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="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</font></span></a>, <a class="txt" href="#E21:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A245</font></span></a>, <a class="txt" href="#E52:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E83">A277</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="E5:35_2_1_1_1_3_1_5_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="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="#E25:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E67">A261</font></span></a>, <a class="txt" href="#E28:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E70">A264</font></span></a>, <a class="txt" href="#E7:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E90">A353</font></span></a>, <a class="txt" href="#E11:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E93">A356</font></span></a>, <a class="txt" href="#E14:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E96">A359</font></span></a>, <a class="txt" href="#E16:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E98">A361</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_2_1_1_1"><span class="lab"><font color="Green" title="E101">A364</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_3_1_5_1_2_1_1_1"><span class="lab"><font color="Green" title="E102">A365</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_3_1_5_1_2_1_1_1"><span class="lab"><font color="Green" title="E103">A366</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="35_2_1_1_1_3_1_5_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_1_1_3_1_5_1_2_1_1_2"/><span class="lab"><font color="Green" title="E101">A367</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="E2:35_2_1_1_1_3_1_5_1_2_1_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="#E16:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E98">A361</font></span></a>, <a class="ref" href="integra1.html#T7" title="INTEGRA1:th.7">INTEGRA1:7</a></span>;<br/><a NAME="E3:35_2_1_1_1_3_1_5_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A368</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="E4:35_2_1_1_1_3_1_5_1_2_1_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="E5:35_2_1_1_1_3_1_5_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A369</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="E6:35_2_1_1_1_3_1_5_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="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="#E16:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E98">A361</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E101">A367</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E7:35_2_1_1_1_3_1_5_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A370</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="#E22:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E64">A258</font></span></a>, <a class="txt" href="#E28:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E70">A264</font></span></a>, <a class="txt" href="#E20:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A244</font></span></a>, <a class="txt" href="#E11:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E93">A356</font></span></a>, <a class="txt" href="#E13:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E95">A358</font></span></a>, <a class="txt" href="#E15:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E97">A360</font></span></a>, <a class="txt" href="#E17:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E99">A362</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>;<br/><a NAME="E8:35_2_1_1_1_3_1_5_1_2_1_1_2"/><span class="lab"><font color="Green" title="E105">A371</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="#E13:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E95">A358</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E86">A349</font></span></a></span>;<br/><a NAME="E9:35_2_1_1_1_3_1_5_1_2_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>,<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="#E16:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E98">A361</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E101">A367</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E10:35_2_1_1_1_3_1_5_1_2_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>,<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="#E22:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E64">A258</font></span></a>, <a class="txt" href="#E28:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E70">A264</font></span></a>, <a class="txt" href="#E20:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A244</font></span></a>, <a class="txt" href="#E11:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E93">A356</font></span></a>, <a class="txt" href="#E13:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E95">A358</font></span></a>, <a class="txt" href="#E8:35_2_1_1_1_3_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E105">A371</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_3_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E102">A368</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1_3_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E103">A369</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="E11:35_2_1_1_1_3_1_5_1_2_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>,<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="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</font></span></a>, <a class="txt" href="#E21:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A245</font></span></a>, <a class="txt" href="#E52:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E83">A277</font></span></a>, <a class="txt" href="#E7:35_2_1_1_1_3_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E104">A370</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="E19:35_2_1_1_1_3_1_5_1_2"/><span class="lab"><font color="Green" title="E101">A372</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="35_2_1_1_1_3_1_5_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="E1:35_2_1_1_1_3_1_5_1_2_2"/><span class="lab"><font color="Green" title="E102">A373</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="E2:35_2_1_1_1_3_1_5_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="#E17:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E59">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="E3:35_2_1_1_1_3_1_5_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A374</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="#E1:35_2_1_1_1_3_1_5_1_2_2"><span class="lab"><font color="Green" title="E102">A373</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="E4:35_2_1_1_1_3_1_5_1_2_2"/><span class="lab"><font color="Green" title="E104">A375</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="E6:35_2_1_1_1_3_1_5_1_2_2"/><span class="lab"><font color="Green" title="E105">A376</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="#E9:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E91">A354</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_3_1_5_1_2_2"><span class="lab"><font color="Green" title="E104">A375</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/>
<a NAME="E7:35_2_1_1_1_3_1_5_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="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</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="E8:35_2_1_1_1_3_1_5_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A377</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="#E1:35_2_1_1_1_3_1_5_1_2_2"><span class="lab"><font color="Green" title="E102">A373</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/>
<a NAME="E9:35_2_1_1_1_3_1_5_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="#E7:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E90">A353</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_3_1_5_1_2_2"><span class="lab"><font color="Green" title="E104">A375</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/>
<a NAME="E10:35_2_1_1_1_3_1_5_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A378</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="#E3:35_2_1_1_1_3_1_5_1_2_2"><span class="lab"><font color="Green" title="E103">A374</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="#E22:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E64">A258</font></span></a>, <a class="txt" href="#E53:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E84">A278</font></span></a>, <a class="txt" href="#E28:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E70">A264</font></span></a>, <a class="txt" href="#E20:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A244</font></span></a>, <a class="txt" href="#E11:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E93">A356</font></span></a>, <a class="txt" href="#E13:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E95">A358</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E86">A349</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="E12:35_2_1_1_1_3_1_5_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="#E6:35_2_1_1_1_3_1_5_1_2_2"><span class="lab"><font color="Green" title="E105">A376</font></span></a>, <a class="txt" href="#E8:35_2_1_1_1_3_1_5_1_2_2"><span class="lab"><font color="Green" title="E106">A377</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="E13:35_2_1_1_1_3_1_5_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="#E10:35_2_1_1_1_3_1_5_1_2_2"><span class="lab"><font color="Green" title="E107">A378</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/><a NAME="E21:35_2_1_1_1_3_1_5_1_2"/><span class="lab"><font color="Green" title="E102">A379</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="E22:35_2_1_1_1_3_1_5_1_2"/><span class="lab"><font color="Green" title="E103">A380</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="35_2_1_1_1_3_1_5_1_2_3"><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="E2:35_2_1_1_1_3_1_5_1_2_3"/><span class="lab"><font color="Green" title="E104">A381</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="#E1:35"><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="E3:35_2_1_1_1_3_1_5_1_2_3"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="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="35_2_1_1_1_3_1_5_1_2_3_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="E1:35_2_1_1_1_3_1_5_1_2_3_1"/><span class="lab"><font color="Green" title="E105">A382</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="#E19:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E101">A372</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="E2:35_2_1_1_1_3_1_5_1_2_3_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="E3:35_2_1_1_1_3_1_5_1_2_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A383</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="E4:35_2_1_1_1_3_1_5_1_2_3_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="E5:35_2_1_1_1_3_1_5_1_2_3_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="#E2:35_2_1_1_1_3_1_5_1_2_3"><span class="lab"><font color="Green" title="E104">A381</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_2_3_1"><span class="lab"><font color="Green" title="E105">A382</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:35_2_1_1_1_3_1_5_1_2_3_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="#E3:35_2_1_1_1_3_1_5_1_2_3_1"><span class="lab"><font color="Green" title="E106">A383</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="E4:35_2_1_1_1_3_1_5_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A384</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="E6:35_2_1_1_1_3_1_5_1_2_3"/><span class="lab"><font color="Green" title="E106">A385</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="#E1:35"><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="E7:35_2_1_1_1_3_1_5_1_2_3"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="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="35_2_1_1_1_3_1_5_1_2_3_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="E1:35_2_1_1_1_3_1_5_1_2_3_2"/><span class="lab"><font color="Green" title="E107">A386</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="#E19:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E101">A372</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="E2:35_2_1_1_1_3_1_5_1_2_3_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="E3:35_2_1_1_1_3_1_5_1_2_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A387</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="E4:35_2_1_1_1_3_1_5_1_2_3_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="E5:35_2_1_1_1_3_1_5_1_2_3_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="#E6:35_2_1_1_1_3_1_5_1_2_3"><span class="lab"><font color="Green" title="E106">A385</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_2_3_2"><span class="lab"><font color="Green" title="E107">A386</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:35_2_1_1_1_3_1_5_1_2_3_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="#E3:35_2_1_1_1_3_1_5_1_2_3_2"><span class="lab"><font color="Green" title="E108">A387</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="E8:35_2_1_1_1_3_1_5_1_2_3"/><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="E9:35_2_1_1_1_3_1_5_1_2_3"/>
<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="#E4:35_2_1_1_1_3_1_5_1_2_3"><span class="lab"><font color="Green" title="E105">A384</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="E23:35_2_1_1_1_3_1_5_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="#E5:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E51">A233</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/><a NAME="E24:35_2_1_1_1_3_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A388</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="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</font></span></a>, <a class="txt" href="#E21:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A245</font></span></a>, <a class="txt" href="#E52:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E83">A277</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E25:35_2_1_1_1_3_1_5_1_2"/><span class="lab"><font color="Green" title="E105">A389</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> <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="#E22:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E64">A258</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/><span class="lab"><font color="Green" title="E106">A390</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<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#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</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="#E22:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E64">A258</font></span></a>, <a class="txt" href="#E25:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E67">A261</font></span></a>, <a class="txt" href="#E28:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E70">A264</font></span></a>, <a class="txt" href="#E20:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A244</font></span></a>, <a class="txt" href="#E29:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E71">A265</font></span></a>, <a class="txt" href="#E11:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E93">A356</font></span></a>, <a class="txt" href="#E25:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E105">A389</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>
;<br/><a NAME="E27:35_2_1_1_1_3_1_5_1_2"/><span class="lab"><font color="Green" title="E107">A391</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="#E28:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E70">A264</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/><a NAME="E28:35_2_1_1_1_3_1_5_1_2"/><span class="lab"><font color="Green" title="E108">A392</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#K4" title="INTEGRA1:func.4">upper_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#K4" title="INTEGRA1:func.4">upper_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="35_2_1_1_1_3_1_5_1_2_5"><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#K4" title="INTEGRA1:func.4">upper_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#K4" title="INTEGRA1:func.4">upper_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="E1:35_2_1_1_1_3_1_5_1_2_5"/><span class="lab"><font color="Green" title="E109">A393</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="E2:35_2_1_1_1_3_1_5_1_2_5"/><span class="lab"><font color="Green" title="E110">A394</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#K4" title="INTEGRA1:func.4">upper_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#K4" title="INTEGRA1:func.4">upper_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="E3:35_2_1_1_1_3_1_5_1_2_5"/><span class="lab"><font color="Green" title="E111">A395</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> (<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="#E1:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E109">A393</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E110">A394</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E4:35_2_1_1_1_3_1_5_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E112">A396</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="E5:35_2_1_1_1_3_1_5_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">A397</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="E6:35_2_1_1_1_3_1_5_1_2_5"/>
<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="#E4:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E112">A396</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="E7:35_2_1_1_1_3_1_5_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">A398</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_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#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><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#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</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="E9:35_2_1_1_1_3_1_5_1_2_5"/><span class="lab"><font color="Green" title="E115">A399</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="E10:35_2_1_1_1_3_1_5_1_2_5"/><span class="lab"><font color="Green" title="E116">A400</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="#E1:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E109">A393</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E11:35_2_1_1_1_3_1_5_1_2_5"/>
<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="#E26:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E106">A390</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E110">A394</font></span></a></span>;<br/>
<a NAME="E12:35_2_1_1_1_3_1_5_1_2_5"/><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="E13:35_2_1_1_1_3_1_5_1_2_5"/><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="#E28:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E70">A264</font></span></a>, <a class="txt" href="#E9:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E115">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="E14:35_2_1_1_1_3_1_5_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">A401</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="#E10:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E116">A400</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E15:35_2_1_1_1_3_1_5_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E118">A402</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="E16:35_2_1_1_1_3_1_5_1_2_5"/>
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="#E23:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E65">A259</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E109">A393</font></span></a>, <a class="txt" href="#E9:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E115">A399</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<a NAME="E17:35_2_1_1_1_3_1_5_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">A403</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="E18:35_2_1_1_1_3_1_5_1_2_5"/><span class="lab"><font color="Green" title="E120">A404</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="#E9:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E115">A399</font></span></a></span>;<br/>
<a NAME="E19:35_2_1_1_1_3_1_5_1_2_5"/><span class="lab"><font color="Green" title="E121">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>,<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="35_2_1_1_1_3_1_5_1_2_5_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_1_1_3_1_5_1_2_5_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_2_1_1_1_3_1_5_1_2_5_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="35_2_1_1_1_3_1_5_1_2_5_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_1_1_3_1_5_1_2_5_1_1_1"/><span class="lab"><font color="Green" title="E122">A406</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="E2:35_2_1_1_1_3_1_5_1_2_5_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="#E5:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E113">A397</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E3:35_2_1_1_1_3_1_5_1_2_5_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E123">A407</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="#E22:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E64">A258</font></span></a>, <a class="txt" href="#E28:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E70">A264</font></span></a>, <a class="txt" href="#E20:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A244</font></span></a>, <a class="txt" href="#E11:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E93">A356</font></span></a>, <a class="txt" href="#E26:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E106">A390</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E109">A393</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E110">A394</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E111">A395</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>;<br/><a NAME="E4:35_2_1_1_1_3_1_5_1_2_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="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="#E7:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E90">A353</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E113">A397</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_2_5_1_1_1"><span class="lab"><font color="Green" title="E122">A406</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="E5:35_2_1_1_1_3_1_5_1_2_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>,<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="#E18:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E120">A404</font></span></a>, <a class="txt" href="#E17:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E119">A403</font></span></a>, <a class="txt" href="#E15:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E118">A402</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_2_5_1_1_1"><span class="lab"><font color="Green" title="E122">A406</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_3_1_5_1_2_5_1_1_1"><span class="lab"><font color="Green" title="E123">A407</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="35_2_1_1_1_3_1_5_1_2_5_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_1_1_3_1_5_1_2_5_1_1_2"/><span class="lab"><font color="Green" title="E122">A408</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="E2:35_2_1_1_1_3_1_5_1_2_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="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="#E5:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E113">A397</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E3:35_2_1_1_1_3_1_5_1_2_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E123">A409</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="#E22:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E64">A258</font></span></a>, <a class="txt" href="#E28:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E70">A264</font></span></a>, <a class="txt" href="#E20:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A244</font></span></a>, <a class="txt" href="#E11:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E93">A356</font></span></a>, <a class="txt" href="#E26:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E106">A390</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E109">A393</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E110">A394</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E111">A395</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>;<br/><a NAME="E4:35_2_1_1_1_3_1_5_1_2_5_1_1_2"/><span class="lab"><font color="Green" title="E124">A410</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="#E26:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E106">A390</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E110">A394</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="E5:35_2_1_1_1_3_1_5_1_2_5_1_1_2"/><span class="lab"><font color="Green" title="E125">A411</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="#E5:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E113">A397</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_2_5_1_1_2"><span class="lab"><font color="Green" title="E122">A408</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E6:35_2_1_1_1_3_1_5_1_2_5_1_1_2"/><span class="lab"><font color="Green" title="E126">A412</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="#E5:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E113">A397</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_2_5_1_1_2"><span class="lab"><font color="Green" title="E122">A408</font></span></a>, <a class="ref" href="integra1.html#T7" title="INTEGRA1:th.7">INTEGRA1:7</a></span>;<br/><a NAME="E7:35_2_1_1_1_3_1_5_1_2_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="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="E8:35_2_1_1_1_3_1_5_1_2_5_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="#E22:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E64">A258</font></span></a>, <a class="txt" href="#E28:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E70">A264</font></span></a>, <a class="txt" href="#E20:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A244</font></span></a>, <a class="txt" href="#E11:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E93">A356</font></span></a>, <a class="txt" href="#E6:35_2_1_1_1_3_1_5_1_2_5_1_1_2"><span class="lab"><font color="Green" title="E126">A412</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_3_1_5_1_2_5_1_1_2"><span class="lab"><font color="Green" title="E124">A410</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1_3_1_5_1_2_5_1_1_2"><span class="lab"><font color="Green" title="E125">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="kw">hence </span><a NAME="E9:35_2_1_1_1_3_1_5_1_2_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>,<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="#E9:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E115">A399</font></span></a>, <a class="txt" href="#E17:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E119">A403</font></span></a>, <a class="txt" href="#E15:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E118">A402</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_3_1_5_1_2_5_1_1_2"><span class="lab"><font color="Green" title="E123">A409</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="E20:35_2_1_1_1_3_1_5_1_2_5"/>
 <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="E21:35_2_1_1_1_3_1_5_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E122">A413</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="#E19:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E121">A405</font></span></a>, <a class="ref" href="integra1.html#T4" title="INTEGRA1:th.4">INTEGRA1:4</a></span>;<br/>
<a NAME="E22:35_2_1_1_1_3_1_5_1_2_5"/><span class="lab"><font color="Green" title="E123">A414</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="#E14:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E117">A401</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="E124">A415</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="#E22:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E64">A258</font></span></a>, <a class="txt" href="#E20:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A244</font></span></a>, <a class="txt" href="#E26:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E106">A390</font></span></a>, <a class="txt" href="#E27:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E107">A391</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E109">A393</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E110">A394</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E111">A395</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#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><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="#E9:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E115">A399</font></span></a>, <a class="txt" href="#E22:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E123">A414</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>
;<br/>
<a NAME="E24:35_2_1_1_1_3_1_5_1_2_5"/>
<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="#E4:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E112">A396</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="E25:35_2_1_1_1_3_1_5_1_2_5"/><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="#E21:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E122">A413</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="E26:35_2_1_1_1_3_1_5_1_2_5"/>
<span class="p1">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_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="#E7:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E114">A398</font></span></a>, <a class="txt" href="#E23:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E124">A415</font></span></a>, <a class="txt" href="#E21:35_2_1_1_1_3_1_5_1_2_5"><span class="lab"><font color="Green" title="E122">A413</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="E29:35_2_1_1_1_3_1_5_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="#E29:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E71">A265</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/><a NAME="E30:35_2_1_1_1_3_1_5_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#K4" title="INTEGRA1:func.4">upper_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="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="kw">by</span> <span class="lab"><a class="txt" href="#E22:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E64">A258</font></span></a>, <a class="txt" href="#E25:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E67">A261</font></span></a>, <a class="txt" href="#E20:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A244</font></span></a>, <a class="txt" href="#E25:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E105">A389</font></span></a>, <a class="txt" href="#E26:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E106">A390</font></span></a>, <a class="txt" href="#E27:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E107">A391</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>;<br/><a NAME="E31:35_2_1_1_1_3_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A416</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#K4" title="INTEGRA1:func.4">upper_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="#E28:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E108">A392</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</a></span>;<br/><a NAME="E32:35_2_1_1_1_3_1_5_1_2"/><span class="lab"><font color="Green" title="E110">A417</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="#E1:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E85">A348</font></span></a>, <a class="ref" href="xreal_1.html#T20" title="XREAL_1:th.20">XREAL_1:20</a></span>;<br/><a NAME="E33:35_2_1_1_1_3_1_5_1_2"/><span class="lab"><font color="Green" title="E111">A418</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="#E27:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E69">A263</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E34:35_2_1_1_1_3_1_5_1_2"/><span class="lab"><font color="Green" title="E112">A419</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="E35:35_2_1_1_1_3_1_5_1_2"/><span class="lab"><font color="Green" title="E113">A420</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="35_2_1_1_1_3_1_5_1_2_6"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_1_1_3_1_5_1_2_6_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="E1:35_2_1_1_1_3_1_5_1_2_6_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="35_2_1_1_1_3_1_5_1_2_6_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_1_1_3_1_5_1_2_6_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="E2:35_2_1_1_1_3_1_5_1_2_6_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="#E34:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E112">A419</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="35_2_1_1_1_3_1_5_1_2_6_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_1_1_3_1_5_1_2_6_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="E2:35_2_1_1_1_3_1_5_1_2_6_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="#E34:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E112">A419</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="E36:35_2_1_1_1_3_1_5_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="#E7:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E90">A353</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><a NAME="E37:35_2_1_1_1_3_1_5_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="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</font></span></a>, <a class="txt" href="#E21:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A245</font></span></a>, <a class="txt" href="#E52:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E83">A277</font></span></a>, <a class="txt" href="#E9:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E91">A354</font></span></a>, <a class="txt" href="#E18:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E100">A363</font></span></a>, <a class="txt" href="#E35:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E113">A420</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E38:35_2_1_1_1_3_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">A421</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="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</font></span></a>, <a class="txt" href="#E17:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E59">A241</font></span></a>, <a class="txt" href="#E21:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">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="E39:35_2_1_1_1_3_1_5_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="#E1:35"><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="E40:35_2_1_1_1_3_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">A422</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="E41:35_2_1_1_1_3_1_5_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="#E1:35"><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="E42:35_2_1_1_1_3_1_5_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="E43:35_2_1_1_1_3_1_5_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="#E40:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E115">A422</font></span></a>, <a class="ref" href="xreal_1.html#T13" title="XREAL_1:th.13">XREAL_1:13</a></span>;<br/><a NAME="E44:35_2_1_1_1_3_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A423</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="#E21:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E102">A379</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/><a NAME="E45:35_2_1_1_1_3_1_5_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="#E17:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E59">A241</font></span></a>, <a class="txt" href="#E30:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E88">A351</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="E46:35_2_1_1_1_3_1_5_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="#E26:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E68">A262</font></span></a>, <a class="txt" href="#E24:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E66">A260</font></span></a>, <a class="txt" href="#E35:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E76">A270</font></span></a>, <a class="txt" href="#E31:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E73">A267</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/><a NAME="E47:35_2_1_1_1_3_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">A424</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="E48:35_2_1_1_1_3_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E118">A425</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="#E46:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E80">A274</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E49:35_2_1_1_1_3_1_5_1_2"/><span class="lab"><font color="Green" title="E119">A426</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="E50:35_2_1_1_1_3_1_5_1_2"/><span class="lab"><font color="Green" title="E120">A427</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="35_2_1_1_1_3_1_5_1_2_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:35_2_1_1_1_3_1_5_1_2_7"/>
 <a 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="E2:35_2_1_1_1_3_1_5_1_2_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E121">A428</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="#E47:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E117">A424</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E3:35_2_1_1_1_3_1_5_1_2_7"/><span class="lab"><font color="Green" title="E122">A429</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="#E33:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E111">A418</font></span></a>, <a class="txt" href="#E48:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E118">A425</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E4:35_2_1_1_1_3_1_5_1_2_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E123">A430</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="35_2_1_1_1_3_1_5_1_2_7_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_1_1_3_1_5_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="E1:35_2_1_1_1_3_1_5_1_2_7_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="#E4:35_2_1"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_3_1_5_1_2_7"><span class="lab"><font color="Green" title="E123">A430</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="35_2_1_1_1_3_1_5_1_2_7_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_1_1_3_1_5_1_2_7_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="E3:35_2_1_1_1_3_1_5_1_2_7_1_1_1"/><span class="lab"><font color="Green" title="E124">A431</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="E4:35_2_1_1_1_3_1_5_1_2_7_1_1_1"/><span class="lab"><font color="Green" title="E125">A432</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="E5:35_2_1_1_1_3_1_5_1_2_7_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="#E26:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E68">A262</font></span></a>, <a class="txt" href="#E49:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E119">A426</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_3_1_5_1_2_7"><span class="lab"><font color="Green" title="E122">A429</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="E6:35_2_1_1_1_3_1_5_1_2_7_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="#E17:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E59">A241</font></span></a>, <a class="txt" href="#E24:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E66">A260</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_3_1_5_1_2_7_1_1_1"><span class="lab"><font color="Green" title="E124">A431</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_3_1_5_1_2_7_1_1_1"><span class="lab"><font color="Green" title="E125">A432</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/><a NAME="E7:35_2_1_1_1_3_1_5_1_2_7_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E126">A433</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="E8:35_2_1_1_1_3_1_5_1_2_7_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="#E35:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E76">A270</font></span></a>, <a class="txt" href="#E31:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E73">A267</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_3_1_5_1_2_7"><span class="lab"><font color="Green" title="E121">A428</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_3_1_5_1_2_7"><span class="lab"><font color="Green" title="E122">A429</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_3_1_5_1_2_7_1_1_1"><span class="lab"><font color="Green" title="E125">A432</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="E9:35_2_1_1_1_3_1_5_1_2_7_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_3_1_5_1_2_7_1_1_1"><span class="lab"><font color="Green" title="E124">A431</font></span></a>, <a class="txt" href="#E7:35_2_1_1_1_3_1_5_1_2_7_1_1_1"><span class="lab"><font color="Green" title="E126">A433</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="35_2_1_1_1_3_1_5_1_2_7_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_1_1_3_1_5_1_2_7_1_1_2"/><span class="lab"><font color="Green" title="E124">A434</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="E2:35_2_1_1_1_3_1_5_1_2_7_1_1_2"/><span class="lab"><font color="Green" title="E125">A435</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="#E18:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E60">A242</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/><a NAME="E3:35_2_1_1_1_3_1_5_1_2_7_1_1_2"/><span class="lab"><font color="Green" title="E126">A436</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="35_2_1_1_1_3_1_5_1_2_7_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_1_1_3_1_5_1_2_7_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="E1:35_2_1_1_1_3_1_5_1_2_7_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="35_2_1_1_1_3_1_5_1_2_7_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_1_1_3_1_5_1_2_7_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="E2:35_2_1_1_1_3_1_5_1_2_7_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="#E17:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E59">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="35_2_1_1_1_3_1_5_1_2_7_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_1_1_3_1_5_1_2_7_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="E2:35_2_1_1_1_3_1_5_1_2_7_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="#E17:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E59">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="E5:35_2_1_1_1_3_1_5_1_2_7_1_1_2"/><span class="lab"><font color="Green" title="E127">A437</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="E6:35_2_1_1_1_3_1_5_1_2_7_1_1_2"/><span class="lab"><font color="Green" title="E128">A438</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="#E1:35_2_1_1_1_3_1_5_1_2_7_1_1_2"><span class="lab"><font color="Green" title="E124">A434</font></span></a>, <a class="ref" href="partfun1.html#T3" title="PARTFUN1:th.3">PARTFUN1:3</a></span>;<br/><a NAME="E7:35_2_1_1_1_3_1_5_1_2_7_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="#E26:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E68">A262</font></span></a>, <a class="txt" href="#E24:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E66">A260</font></span></a>, <a class="txt" href="#E49:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E119">A426</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_3_1_5_1_2_7"><span class="lab"><font color="Green" title="E122">A429</font></span></a>, <a class="txt" href="#E6:35_2_1_1_1_3_1_5_1_2_7_1_1_2"><span class="lab"><font color="Green" title="E128">A438</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="E8:35_2_1_1_1_3_1_5_1_2_7_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="#E2:35_2_1_1_1_3_1_5_1_2_7_1_1_2"><span class="lab"><font color="Green" title="E125">A435</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_3_1_5_1_2_7_1_1_2"><span class="lab"><font color="Green" title="E126">A436</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E9:35_2_1_1_1_3_1_5_1_2_7_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="#E11:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E54">A236</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1_3_1_5_1_2_7_1_1_2"><span class="lab"><font color="Green" title="E127">A437</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/><a NAME="E10:35_2_1_1_1_3_1_5_1_2_7_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E129">A439</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="E11:35_2_1_1_1_3_1_5_1_2_7_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="#E1:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E85">A348</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E12:35_2_1_1_1_3_1_5_1_2_7_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E130">A440</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="E13:35_2_1_1_1_3_1_5_1_2_7_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="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</font></span></a>, <a class="txt" href="#E21:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A245</font></span></a>, <a class="txt" href="#E52:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E83">A277</font></span></a>, <a class="ref" href="integra1.html#T7" title="INTEGRA1:th.7">INTEGRA1:7</a></span>;<br/><a NAME="E14:35_2_1_1_1_3_1_5_1_2_7_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E131">A441</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="#E30:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="txt" href="#E12:35_2_1_1_1_3_1_5_1_2_7_1_1_2"><span class="lab"><font color="Green" title="E130">A440</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/><a NAME="E15:35_2_1_1_1_3_1_5_1_2_7_1_1_2"/><span class="lab"><font color="Green" title="E132">A442</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="#E5:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E51">A233</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/><a NAME="E16:35_2_1_1_1_3_1_5_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>,<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="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</font></span></a>, <a class="txt" href="#E21:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A245</font></span></a>, <a class="txt" href="#E52:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E83">A277</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E17:35_2_1_1_1_3_1_5_1_2_7_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E133">A443</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="#E14:35_2_1_1_1_3_1_5_1_2_7_1_1_2"><span class="lab"><font color="Green" title="E131">A441</font></span></a>, <a class="txt" href="#E15:35_2_1_1_1_3_1_5_1_2_7_1_1_2"><span class="lab"><font color="Green" title="E132">A442</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E18:35_2_1_1_1_3_1_5_1_2_7_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="#E35:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E76">A270</font></span></a>, <a class="txt" href="#E31:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E73">A267</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_3_1_5_1_2_7"><span class="lab"><font color="Green" title="E121">A428</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_3_1_5_1_2_7"><span class="lab"><font color="Green" title="E122">A429</font></span></a>, <a class="txt" href="#E6:35_2_1_1_1_3_1_5_1_2_7_1_1_2"><span class="lab"><font color="Green" title="E128">A438</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="E19:35_2_1_1_1_3_1_5_1_2_7_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="#E17:35_2_1_1_1_3_1_5_1_2_7_1_1_2"><span class="lab"><font color="Green" title="E133">A443</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="E20:35_2_1_1_1_3_1_5_1_2_7_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E49">A231</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1_3_1_5_1_2_7_1_1_2"><span class="lab"><font color="Green" title="E127">A437</font></span></a>, <a class="txt" href="#E10:35_2_1_1_1_3_1_5_1_2_7_1_1_2"><span class="lab"><font color="Green" title="E129">A439</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="E6:35_2_1_1_1_3_1_5_1_2_7"/>
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="E51:35_2_1_1_1_3_1_5_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="#E48:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E81">A275</font></span></a>, <a class="txt" href="#E36:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E77">A271</font></span></a>, <a class="txt" href="#E46:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E80">A274</font></span></a>, <a class="txt" href="#E13:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E56">A238</font></span></a>, <a class="txt" href="#E14:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E57">A239</font></span></a>, <a class="txt" href="#E12:35_2_1_1_1_3_1_5_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="E52:35_2_1_1_1_3_1_5_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="#E48:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E81">A275</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/><a NAME="E53:35_2_1_1_1_3_1_5_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#K4" title="INTEGRA1:func.4">upper_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="#E50:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E120">A427</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/><a NAME="E54:35_2_1_1_1_3_1_5_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="35_2_1_1_1_3_1_5_1_2_8"><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="E1:35_2_1_1_1_3_1_5_1_2_8"/><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="35_2_1_1_1_3_1_5_1_2_8_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_1_1_3_1_5_1_2_8_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="E1:35_2_1_1_1_3_1_5_1_2_8_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="#E1:35_2_1_1_1_3_1_5_1_2_8"><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="35_2_1_1_1_3_1_5_1_2_8_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_1_1_3_1_5_1_2_8_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="E2:35_2_1_1_1_3_1_5_1_2_8_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="E3:35_2_1_1_1_3_1_5_1_2_8_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="E4:35_2_1_1_1_3_1_5_1_2_8_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="E5:35_2_1_1_1_3_1_5_1_2_8_1_1_1"/><span class="lab"><font color="Green" title="E127">A450</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="#E22:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E64">A258</font></span></a>, <a class="txt" href="#E25:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E67">A261</font></span></a>, <a class="txt" href="#E28:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E70">A264</font></span></a>, <a class="txt" href="#E20:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A244</font></span></a>, <a class="txt" href="#E29:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E71">A265</font></span></a>, <a class="txt" href="#E11:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E93">A356</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="#E22:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E64">A258</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="E7:35_2_1_1_1_3_1_5_1_2_8_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E128">A451</font></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="#E22:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E64">A258</font></span></a>, <a class="txt" href="#E28:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E70">A264</font></span></a>, <a class="txt" href="#E20:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A244</font></span></a>, <a class="txt" href="#E11:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E93">A356</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_3_1_5_1_2_8_1_1_1"><span class="lab"><font color="Green" title="E126">A449</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1_3_1_5_1_2_8_1_1_1"><span class="lab"><font color="Green" title="E127">A450</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>;<br/><a NAME="E8:35_2_1_1_1_3_1_5_1_2_8_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="#E11:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E93">A356</font></span></a>, <a class="ref" href="finseq_6.html#T119" title="FINSEQ_6:th.119">FINSEQ_6:119</a></span>;<br/><a NAME="E9:35_2_1_1_1_3_1_5_1_2_8_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E129">A452</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="#E1:35_2_1_1_1_3_1_5_1_2_8_1_1_1"><span class="lab"><font color="Green" title="E125">A448</font></span></a></span>;<br/><a NAME="E10:35_2_1_1_1_3_1_5_1_2_8_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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</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="E12:35_2_1_1_1_3_1_5_1_2_8_1_1_1"/><span class="lab"><font color="Green" title="E130">A453</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="E13:35_2_1_1_1_3_1_5_1_2_8_1_1_1"/><span class="lab"><font color="Green" title="E131">A454</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="#E9:35_2_1_1_1_3_1_5_1_2_8_1_1_1"><span class="lab"><font color="Green" title="E129">A452</font></span></a>, <a class="ref" href="partfun1.html#T3" title="PARTFUN1:th.3">PARTFUN1:3</a></span>;<br/><a NAME="E14:35_2_1_1_1_3_1_5_1_2_8_1_1_1"/>
<font color="Maroon" title="c17">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="#E1:35_2_1_1_1_3_1_5_1_2_8_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="E15:35_2_1_1_1_3_1_5_1_2_8_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="#E12:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E55">A237</font></span></a>, <a class="txt" href="#E6:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E52">A234</font></span></a>, <a class="txt" href="#E7:35_2_1_1_1_3_1_5_1_2_8_1_1_1"><span class="lab"><font color="Green" title="E128">A451</font></span></a>, <a class="txt" href="#E12:35_2_1_1_1_3_1_5_1_2_8_1_1_1"><span class="lab"><font color="Green" title="E130">A453</font></span></a>, <a class="txt" href="#E13:35_2_1_1_1_3_1_5_1_2_8_1_1_1"><span class="lab"><font color="Green" title="E131">A454</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="E16:35_2_1_1_1_3_1_5_1_2_8_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="E17:35_2_1_1_1_3_1_5_1_2_8_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E132">A455</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="E18:35_2_1_1_1_3_1_5_1_2_8_1_1_1"/><span class="lab"><font color="Green" title="E133">A456</font></span>: 
<font color="Maroon" title="c13">MD1</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="c17">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_2_8_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="E19:35_2_1_1_1_3_1_5_1_2_8_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="#E25:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E67">A261</font></span></a>, <a class="txt" href="#E28:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E70">A264</font></span></a>, <a class="txt" href="#E20:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A244</font></span></a>, <a class="txt" href="#E29:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E71">A265</font></span></a>, <a class="txt" href="#E11:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E93">A356</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>;<br/><a NAME="E20:35_2_1_1_1_3_1_5_1_2_8_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E134">A457</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="#E35:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E76">A270</font></span></a>, <a class="txt" href="#E31:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E73">A267</font></span></a>, <a class="txt" href="#E18:35_2_1_1_1_3_1_5_1_2_8_1_1_1"><span class="lab"><font color="Green" title="E133">A456</font></span></a>, <a class="txt" href="#E12:35_2_1_1_1_3_1_5_1_2_8_1_1_1"><span class="lab"><font color="Green" title="E130">A453</font></span></a>, <a class="txt" href="#E13:35_2_1_1_1_3_1_5_1_2_8_1_1_1"><span class="lab"><font color="Green" title="E131">A454</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="E22:35_2_1_1_1_3_1_5_1_2_8_1_1_1"/><span class="lab"><font color="Green" title="E135">A458</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="E23:35_2_1_1_1_3_1_5_1_2_8_1_1_1"/><span class="lab"><font color="Green" title="E136">A459</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">n</font> <a href="nat_1.html#K1" title="NAT_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>,<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#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:35_2_1_1_1_3_1_5_1_2_8_1_1_1"><span class="lab"><font color="Green" title="E135">A458</font></span></a></span>;<br/><a NAME="E24:35_2_1_1_1_3_1_5_1_2_8_1_1_1"/><span class="lab"><font color="Green" title="E137">A460</font></span>: 
<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/><a NAME="E25:35_2_1_1_1_3_1_5_1_2_8_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="#E20:35_2_1_1_1_3_1_5_1_2_8_1_1_1"><span class="lab"><font color="Green" title="E134">A457</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E26:35_2_1_1_1_3_1_5_1_2_8_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E138">A461</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="E27:35_2_1_1_1_3_1_5_1_2_8_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="c14">MD2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E122">A445</font></span></a>, <a class="txt" href="#E17:35_2_1_1_1_3_1_5_1_2_8_1_1_1"><span class="lab"><font color="Green" title="E132">A455</font></span></a>, <a class="txt" href="#E22:35_2_1_1_1_3_1_5_1_2_8_1_1_1"><span class="lab"><font color="Green" title="E135">A458</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E28:35_2_1_1_1_3_1_5_1_2_8_1_1_1"/><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> <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="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="E29:35_2_1_1_1_3_1_5_1_2_8_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="#E48:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E81">A275</font></span></a>, <a class="txt" href="#E36:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E77">A271</font></span></a>, <a class="txt" href="#E14:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E57">A239</font></span></a>, <a class="txt" href="#E12:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E94">A357</font></span></a>, <a class="txt" href="#E13:35_2_1_1_1_3_1_5_1_2_8_1_1_1"><span class="lab"><font color="Green" title="E131">A454</font></span></a>, <a class="txt" href="#E26:35_2_1_1_1_3_1_5_1_2_8_1_1_1"><span class="lab"><font color="Green" title="E138">A461</font></span></a>, <a class="txt" href="#E17:35_2_1_1_1_3_1_5_1_2_8_1_1_1"><span class="lab"><font color="Green" title="E132">A455</font></span></a>, <a class="txt" href="#E24:35_2_1_1_1_3_1_5_1_2_8_1_1_1"><span class="lab"><font color="Green" title="E137">A460</font></span></a>, <a class="txt" href="#E23:35_2_1_1_1_3_1_5_1_2_8_1_1_1"><span class="lab"><font color="Green" title="E136">A459</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</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="35_2_1_1_1_3_1_5_1_2_8_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_1_1_3_1_5_1_2_8_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="E2:35_2_1_1_1_3_1_5_1_2_8_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E125">A462</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="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</font></span></a>, <a class="txt" href="#E21:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A245</font></span></a>, <a class="txt" href="#E52:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E83">A277</font></span></a>, <a class="ref" href="integra1.html#T7" title="INTEGRA1:th.7">INTEGRA1:7</a></span>;<br/><a NAME="E4:35_2_1_1_1_3_1_5_1_2_8_1_1_2"/><span class="lab"><font color="Green" title="E126">A463</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="#E2:35_2_1"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="ref" href="integra1.html#D18" title="INTEGRA1:def.18">INTEGRA1:def 18</a></span>;<br/><a NAME="E5:35_2_1_1_1_3_1_5_1_2_8_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="#E2:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E49">A231</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="E7:35_2_1_1_1_3_1_5_1_2_8_1_1_2"/><span class="lab"><font color="Green" title="E127">A464</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="E8:35_2_1_1_1_3_1_5_1_2_8_1_1_2"/><span class="lab"><font color="Green" title="E128">A465</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="#E2:35_2_1_1_1_3_1_5_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E125">A462</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_3_1_5_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E126">A463</font></span></a>, <a class="ref" href="partfun1.html#T3" title="PARTFUN1:th.3">PARTFUN1:3</a></span>;<br/><a NAME="E9:35_2_1_1_1_3_1_5_1_2_8_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="#E5:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E51">A233</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/><a NAME="E10:35_2_1_1_1_3_1_5_1_2_8_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="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</font></span></a>, <a class="txt" href="#E21:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A245</font></span></a>, <a class="txt" href="#E52:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E83">A277</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_3_1_5_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E125">A462</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E11:35_2_1_1_1_3_1_5_1_2_8_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E129">A466</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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</font></span></a>, <a class="txt" href="#E8:35_2_1_1_1_3_1_5_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E128">A465</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/><a NAME="E12:35_2_1_1_1_3_1_5_1_2_8_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="#E1:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E85">A348</font></span></a>, <a class="ref" href="xreal_1.html#T20" title="XREAL_1:th.20">XREAL_1:20</a></span>;<br/><a NAME="E13:35_2_1_1_1_3_1_5_1_2_8_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E130">A467</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="E14:35_2_1_1_1_3_1_5_1_2_8_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="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</font></span></a>, <a class="txt" href="#E21:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A245</font></span></a>, <a class="txt" href="#E52:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E83">A277</font></span></a>, <a class="ref" href="integra1.html#T7" title="INTEGRA1:th.7">INTEGRA1:7</a></span>;<br/><a NAME="E15:35_2_1_1_1_3_1_5_1_2_8_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E131">A468</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="#E30:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="txt" href="#E13:35_2_1_1_1_3_1_5_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E130">A467</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/><a NAME="E16:35_2_1_1_1_3_1_5_1_2_8_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="#E5:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E51">A233</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/><a NAME="E17:35_2_1_1_1_3_1_5_1_2_8_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="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</font></span></a>, <a class="txt" href="#E21:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A245</font></span></a>, <a class="txt" href="#E52:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E83">A277</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_3_1_5_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E125">A462</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E18:35_2_1_1_1_3_1_5_1_2_8_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="#E31:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E73">A267</font></span></a>, <a class="txt" href="#E8:35_2_1_1_1_3_1_5_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E128">A465</font></span></a>, <a class="txt" href="#E15:35_2_1_1_1_3_1_5_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E131">A468</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="E19:35_2_1_1_1_3_1_5_1_2_8_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="#E35:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E76">A270</font></span></a>, <a class="txt" href="#E12:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E55">A237</font></span></a>, <a class="txt" href="#E12:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E94">A357</font></span></a>, <a class="txt" href="#E7:35_2_1_1_1_3_1_5_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E127">A464</font></span></a>, <a class="txt" href="#E8:35_2_1_1_1_3_1_5_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E128">A465</font></span></a>, <a class="txt" href="#E11:35_2_1_1_1_3_1_5_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E129">A466</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="E4:35_2_1_1_1_3_1_5_1_2_8"/>
<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="E55:35_2_1_1_1_3_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A469</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="E56:35_2_1_1_1_3_1_5_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="E57:35_2_1_1_1_3_1_5_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="E58:35_2_1_1_1_3_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E125">A470</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="E59:35_2_1_1_1_3_1_5_1_2"/><span class="lab"><font color="Green" title="E126">A471</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="#E52:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E122">A445</font></span></a></span>;<br/><a NAME="E60:35_2_1_1_1_3_1_5_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="35_2_1_1_1_3_1_5_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>  <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="E1:35_2_1_1_1_3_1_5_1_2_9"/><span class="lab"><font color="Green" title="E127">A472</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="E3:35_2_1_1_1_3_1_5_1_2_9"/><span class="lab"><font color="Green" title="E128">A473</font></span>: 
<font color="Maroon" title="c14">MD2</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="c17">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_2_9"><span class="lab"><font color="Green" title="E127">A472</font></span></a>, <a class="ref" href="integra3.html#T16" target="_self" title="INTEGRA3:th.16">Th16</a></span>;<br/>
<a NAME="E4:35_2_1_1_1_3_1_5_1_2_9"/><span class="lab"><font color="Green" title="E129">A474</font></span>: 
<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="#E48:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E81">A275</font></span></a>, <a class="txt" href="#E36:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E77">A271</font></span></a>, <a class="txt" href="#E14:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E57">A239</font></span></a>, <a class="txt" href="#E12:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E94">A357</font></span></a>, <a class="txt" href="#E58:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E125">A470</font></span></a>, <a class="txt" href="#E51:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E121">A444</font></span></a>, <a class="txt" href="#E59:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E126">A471</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>;<br/>
<a NAME="E5:35_2_1_1_1_3_1_5_1_2_9"/><span class="lab"><font color="Green" title="E130">A475</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="#E12:35_2_1_1_1_3_1_5_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="E6:35_2_1_1_1_3_1_5_1_2_9"/><span class="lab"><font color="Green" title="E131">A476</font></span>: 
<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="#E36:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E77">A271</font></span></a>, <a class="txt" href="#E46:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E80">A274</font></span></a>, <a class="txt" href="#E13:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E56">A238</font></span></a>, <a class="txt" href="#E14:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E57">A239</font></span></a>, <a class="txt" href="#E12:35_2_1_1_1_3_1_5_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="E7:35_2_1_1_1_3_1_5_1_2_9"/><span class="lab"><font color="Green" title="E132">A477</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="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">by</span> <span class="lab"><a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_2_9"><span class="lab"><font color="Green" title="E127">A472</font></span></a>, <a class="ref" href="integra3.html#T16" target="_self" title="INTEGRA3:th.16">Th16</a></span>;<br/>
<a NAME="E8:35_2_1_1_1_3_1_5_1_2_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E133">A478</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="#E6:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E52">A234</font></span></a>, <a class="txt" href="#E48:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E81">A275</font></span></a>, <a class="txt" href="#E36:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E77">A271</font></span></a>, <a class="txt" href="#E14:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E57">A239</font></span></a>, <a class="txt" href="#E12:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E94">A357</font></span></a>, <a class="txt" href="#E58:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E125">A470</font></span></a>, <a class="txt" href="#E51:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E121">A444</font></span></a>, <a class="txt" href="#E59:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E126">A471</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1_1_1_3_1_5_1_2_9_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_1_1_3_1_5_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="E1:35_2_1_1_1_3_1_5_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="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="#E4:35_2_1"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_2_9"><span class="lab"><font color="Green" title="E127">A472</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1_3_1_5_1_2_9"><span class="lab"><font color="Green" title="E130">A475</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="35_2_1_1_1_3_1_5_1_2_9_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_1_1_3_1_5_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="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="E3:35_2_1_1_1_3_1_5_1_2_9_1_1_1"/><span class="lab"><font color="Green" title="E134">A479</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="E4:35_2_1_1_1_3_1_5_1_2_9_1_1_1"/><span class="lab"><font color="Green" title="E135">A480</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="E5:35_2_1_1_1_3_1_5_1_2_9_1_1_1"/><span class="lab"><font color="Green" title="E136">A481</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="#E30:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="txt" href="#E31:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E73">A267</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_3_1_5_1_2_9"><span class="lab"><font color="Green" title="E128">A473</font></span></a>, <a class="txt" href="#E6:35_2_1_1_1_3_1_5_1_2_9"><span class="lab"><font color="Green" title="E131">A476</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_3_1_5_1_2_9_1_1_1"><span class="lab"><font color="Green" title="E134">A479</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_3_1_5_1_2_9_1_1_1"><span class="lab"><font color="Green" title="E135">A480</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="E6:35_2_1_1_1_3_1_5_1_2_9_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E137">A482</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="E7:35_2_1_1_1_3_1_5_1_2_9_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="E9:35_2_1_1_1_3_1_5_1_2_9_1_1_1"/><span class="lab"><font color="Green" title="E138">A483</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="#E5:35_2_1_1_1_3_1_5_1_2_9_1_1_1"><span class="lab"><font color="Green" title="E136">A481</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="E10:35_2_1_1_1_3_1_5_1_2_9_1_1_1"/><span class="lab"><font color="Green" title="E139">A484</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="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</font></span></a>, <a class="txt" href="#E6:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E52">A234</font></span></a>, <a class="txt" href="#E7:35_2_1_1_1_3_1_5_1_2_9"><span class="lab"><font color="Green" title="E132">A477</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_3_1_5_1_2_9"><span class="lab"><font color="Green" title="E129">A474</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_3_1_5_1_2_9_1_1_1"><span class="lab"><font color="Green" title="E134">A479</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_3_1_5_1_2_9_1_1_1"><span class="lab"><font color="Green" title="E135">A480</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="E11:35_2_1_1_1_3_1_5_1_2_9_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E140">A485</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> <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="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E12:35_2_1_1_1_3_1_5_1_2_9_1_1_1"/><span class="lab"><font color="Green" title="E141">A486</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="txt" href="#E5:35_2_1_1_1_3_1_5_1_2_9_1_1_1"><span class="lab"><font color="Green" title="E136">A481</font></span></a>, <a class="ref" href="xreal_1.html#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/><a NAME="E13:35_2_1_1_1_3_1_5_1_2_9_1_1_1"/><span class="lab"><font color="Green" title="E142">A487</font></span>: 
<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="E14:35_2_1_1_1_3_1_5_1_2_9_1_1_1"/>
<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="#E13:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E95">A358</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E87">A350</font></span></a>, <a class="txt" href="#E10:35_2_1_1_1_3_1_5_1_2_9_1_1_1"><span class="lab"><font color="Green" title="E139">A484</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E15:35_2_1_1_1_3_1_5_1_2_9_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="#E12:35_2_1_1_1_3_1_5_1_2_9_1_1_1"><span class="lab"><font color="Green" title="E141">A486</font></span></a>, <a class="txt" href="#E9:35_2_1_1_1_3_1_5_1_2_9_1_1_1"><span class="lab"><font color="Green" title="E138">A483</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E16:35_2_1_1_1_3_1_5_1_2_9_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E143">A488</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="E17:35_2_1_1_1_3_1_5_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="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="#E22:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E64">A258</font></span></a>, <a class="txt" href="#E28:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E70">A264</font></span></a>, <a class="txt" href="#E20:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A244</font></span></a>, <a class="txt" href="#E11:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E93">A356</font></span></a>, <a class="txt" href="#E6:35_2_1_1_1_3_1_5_1_2_9_1_1_1"><span class="lab"><font color="Green" title="E137">A482</font></span></a>, <a class="txt" href="#E11:35_2_1_1_1_3_1_5_1_2_9_1_1_1"><span class="lab"><font color="Green" title="E140">A485</font></span></a>, <a class="txt" href="#E13:35_2_1_1_1_3_1_5_1_2_9_1_1_1"><span class="lab"><font color="Green" title="E142">A487</font></span></a>, <a class="txt" href="#E9:35_2_1_1_1_3_1_5_1_2_9_1_1_1"><span class="lab"><font color="Green" title="E138">A483</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="E19:35_2_1_1_1_3_1_5_1_2_9_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="#E4:35_2_1_1_1_3_1_5_1_2_9_1_1_1"><span class="lab"><font color="Green" title="E135">A480</font></span></a>, <a class="txt" href="#E16:35_2_1_1_1_3_1_5_1_2_9_1_1_1"><span class="lab"><font color="Green" title="E143">A488</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="35_2_1_1_1_3_1_5_1_2_9_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_1_1_3_1_5_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="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="E3:35_2_1_1_1_3_1_5_1_2_9_1_1_2"/><span class="lab"><font color="Green" title="E134">A489</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="E4:35_2_1_1_1_3_1_5_1_2_9_1_1_2"/><span class="lab"><font color="Green" title="E135">A490</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="E5:35_2_1_1_1_3_1_5_1_2_9_1_1_2"/><span class="lab"><font color="Green" title="E136">A491</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="35_2_1_1_1_3_1_5_1_2_9_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:35_2_1_1_1_3_1_5_1_2_9_1_1_2_1"/><span class="lab"><font color="Green" title="E137">A492</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="35_2_1_1_1_3_1_5_1_2_9_1_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_1_1_3_1_5_1_2_9_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="E1:35_2_1_1_1_3_1_5_1_2_9_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="35_2_1_1_1_3_1_5_1_2_9_1_1_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_1_1_3_1_5_1_2_9_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="E2:35_2_1_1_1_3_1_5_1_2_9_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="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</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="35_2_1_1_1_3_1_5_1_2_9_1_1_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_1_1_3_1_5_1_2_9_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="E2:35_2_1_1_1_3_1_5_1_2_9_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="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</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="E3:35_2_1_1_1_3_1_5_1_2_9_1_1_2_1"/><span class="lab"><font color="Green" title="E138">A493</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="E4:35_2_1_1_1_3_1_5_1_2_9_1_1_2_1"/><span class="lab"><font color="Green" title="E139">A494</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="#E3:35_2_1_1_1_3_1_5_1_2_9_1_1_2"><span class="lab"><font color="Green" title="E134">A489</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="E5:35_2_1_1_1_3_1_5_1_2_9_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="E6:35_2_1_1_1_3_1_5_1_2_9_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E140">A495</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="#E2:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E49">A231</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_3_1_5_1_2_9_1_1_2"><span class="lab"><font color="Green" title="E134">A489</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="E7:35_2_1_1_1_3_1_5_1_2_9_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="#E5:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E51">A233</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/>
<a NAME="E8:35_2_1_1_1_3_1_5_1_2_9_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="#E6:35_2_1_1_1_3_1_5_1_2_9_1_1_2_1"><span class="lab"><font color="Green" title="E140">A495</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E9:35_2_1_1_1_3_1_5_1_2_9_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="#E8:35_2_1_1_1_3_1_5_1_2_9"><span class="lab"><font color="Green" title="E133">A478</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_3_1_5_1_2_9_1_1_2"><span class="lab"><font color="Green" title="E135">A490</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_2_9_1_1_2_1"><span class="lab"><font color="Green" title="E137">A492</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/>
<a NAME="E10:35_2_1_1_1_3_1_5_1_2_9_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E141">A496</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="#E4:35_2_1_1_1_3_1_5_1_2_9_1_1_2"><span class="lab"><font color="Green" title="E135">A490</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_3_1_5_1_2_9_1_1_2_1"><span class="lab"><font color="Green" title="E139">A494</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="E11:35_2_1_1_1_3_1_5_1_2_9_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="#E2:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E49">A231</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="E12:35_2_1_1_1_3_1_5_1_2_9_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="#E5:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E51">A233</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_3_1_5_1_2_9_1_1_2_1"><span class="lab"><font color="Green" title="E138">A493</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="E13:35_2_1_1_1_3_1_5_1_2_9_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_2"><span class="lab"><font color="Green" title="E35">A8</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_3_1_5_1_2_9_1_1_2"><span class="lab"><font color="Green" title="E135">A490</font></span></a>, <a class="txt" href="#E6:35_2_1_1_1_3_1_5_1_2_9_1_1_2_1"><span class="lab"><font color="Green" title="E140">A495</font></span></a>, <a class="txt" href="#E10:35_2_1_1_1_3_1_5_1_2_9_1_1_2_1"><span class="lab"><font color="Green" title="E141">A496</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_3_1_5_1_2_9_1_1_2_1"><span class="lab"><font color="Green" title="E138">A493</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="E6:35_2_1_1_1_3_1_5_1_2_9_1_1_2"/><span class="lab"><font color="Green" title="E137">A497</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="35_2_1_1_1_3_1_5_1_2_9_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_1_1_3_1_5_1_2_9_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="E1:35_2_1_1_1_3_1_5_1_2_9_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="35_2_1_1_1_3_1_5_1_2_9_1_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_1_1_3_1_5_1_2_9_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="E2:35_2_1_1_1_3_1_5_1_2_9_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="#E17:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E59">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="35_2_1_1_1_3_1_5_1_2_9_1_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_1_1_3_1_5_1_2_9_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="E2:35_2_1_1_1_3_1_5_1_2_9_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="#E17:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E59">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="E7:35_2_1_1_1_3_1_5_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> <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="#E18:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E60">A242</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/><a NAME="E8:35_2_1_1_1_3_1_5_1_2_9_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="#E5:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E89">A352</font></span></a>, <a class="txt" href="#E6:35_2_1_1_1_3_1_5_1_2_9_1_1_2"><span class="lab"><font color="Green" title="E137">A497</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E9:35_2_1_1_1_3_1_5_1_2_9_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="#E31:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E73">A267</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_3_1_5_1_2_9"><span class="lab"><font color="Green" title="E128">A473</font></span></a>, <a class="txt" href="#E6:35_2_1_1_1_3_1_5_1_2_9"><span class="lab"><font color="Green" title="E131">A476</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_3_1_5_1_2_9_1_1_2"><span class="lab"><font color="Green" title="E135">A490</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E10:35_2_1_1_1_3_1_5_1_2_9_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="#E11:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E54">A236</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_3_1_5_1_2_9_1_1_2"><span class="lab"><font color="Green" title="E134">A489</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/><a NAME="E11:35_2_1_1_1_3_1_5_1_2_9_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="E12:35_2_1_1_1_3_1_5_1_2_9_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="E13:35_2_1_1_1_3_1_5_1_2_9_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="#E4:35_2_1_1_1_3_1_5_1_2_9_1_1_2"><span class="lab"><font color="Green" title="E135">A490</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1_3_1_5_1_2_9_1_1_2"><span class="lab"><font color="Green" title="E136">A491</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="E14:35_2_1_1_1_3_1_5_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">(<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="E10:35_2_1_1_1_3_1_5_1_2_9"/>
<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="E61:35_2_1_1_1_3_1_5_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="E62:35_2_1_1_1_3_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E127">A498</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="#E55:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E124">A469</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="E63:35_2_1_1_1_3_1_5_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="E65:35_2_1_1_1_3_1_5_1_2"/><span class="lab"><font color="Green" title="E128">A499</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="E66:35_2_1_1_1_3_1_5_1_2"/><span class="lab"><font color="Green" title="E129">A500</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="E67:35_2_1_1_1_3_1_5_1_2"/><span class="lab"><font color="Green" title="E130">A501</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="#E65:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E128">A499</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="E68:35_2_1_1_1_3_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E131">A502</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="E69:35_2_1_1_1_3_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E132">A503</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="E70:35_2_1_1_1_3_1_5_1_2"/><span class="lab"><font color="Green" title="E133">A504</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="#E68:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E131">A502</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E71:35_2_1_1_1_3_1_5_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="#E13:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E95">A358</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E87">A350</font></span></a>, <a class="ref" href="xreal_1.html#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/><a NAME="E72:35_2_1_1_1_3_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E134">A505</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="#E28:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E70">A264</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E73:35_2_1_1_1_3_1_5_1_2"/><span class="lab"><font color="Green" title="E135">A506</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="#E67:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E130">A501</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E74:35_2_1_1_1_3_1_5_1_2"/><span class="lab"><font color="Green" title="E136">A507</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="#E52:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E83">A277</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E75:35_2_1_1_1_3_1_5_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="E76:35_2_1_1_1_3_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E137">A508</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="E77:35_2_1_1_1_3_1_5_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="#E73:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E135">A506</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E78:35_2_1_1_1_3_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E138">A509</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="#E72:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E134">A505</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E79:35_2_1_1_1_3_1_5_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="35_2_1_1_1_3_1_5_1_2_10"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_1_1_3_1_5_1_2_10_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="E1:35_2_1_1_1_3_1_5_1_2_10_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="35_2_1_1_1_3_1_5_1_2_10_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_1_1_3_1_5_1_2_10_1_1"/><span class="lab"><font color="Green" title="E139">A510</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="E2:35_2_1_1_1_3_1_5_1_2_10_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="#E69:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E132">A503</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E3:35_2_1_1_1_3_1_5_1_2_10_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E140">A511</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="#E22:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E64">A258</font></span></a>, <a class="txt" href="#E28:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E70">A264</font></span></a>, <a class="txt" href="#E20:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A244</font></span></a>, <a class="txt" href="#E11:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E93">A356</font></span></a>, <a class="txt" href="#E13:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E95">A358</font></span></a>, <a class="txt" href="#E73:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E135">A506</font></span></a>, <a class="txt" href="#E70:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E133">A504</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>;<br/><a NAME="E4:35_2_1_1_1_3_1_5_1_2_10_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="#E73:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E135">A506</font></span></a>, <a class="txt" href="#E76:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E137">A508</font></span></a>, <a class="txt" href="#E78:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E138">A509</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="E5:35_2_1_1_1_3_1_5_1_2_10_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="#E7:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E90">A353</font></span></a>, <a class="txt" href="#E74:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E136">A507</font></span></a>, <a class="txt" href="#E69:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E132">A503</font></span></a>, <a class="txt" href="#E78:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E138">A509</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_2_10_1_1"><span class="lab"><font color="Green" title="E139">A510</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_3_1_5_1_2_10_1_1"><span class="lab"><font color="Green" title="E140">A511</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="35_2_1_1_1_3_1_5_1_2_10_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_1_1_3_1_5_1_2_10_1_2"/><span class="lab"><font color="Green" title="E139">A512</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="E2:35_2_1_1_1_3_1_5_1_2_10_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="#E69:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E132">A503</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E3:35_2_1_1_1_3_1_5_1_2_10_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E140">A513</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="#E22:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E64">A258</font></span></a>, <a class="txt" href="#E28:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E70">A264</font></span></a>, <a class="txt" href="#E20:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A244</font></span></a>, <a class="txt" href="#E11:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E93">A356</font></span></a>, <a class="txt" href="#E13:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E95">A358</font></span></a>, <a class="txt" href="#E73:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E135">A506</font></span></a>, <a class="txt" href="#E70:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E133">A504</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>;<br/><a NAME="E4:35_2_1_1_1_3_1_5_1_2_10_1_2"/><span class="lab"><font color="Green" title="E141">A514</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="#E69:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E132">A503</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_2_10_1_2"><span class="lab"><font color="Green" title="E139">A512</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E5:35_2_1_1_1_3_1_5_1_2_10_1_2"/><span class="lab"><font color="Green" title="E142">A515</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="#E69:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E132">A503</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_2_10_1_2"><span class="lab"><font color="Green" title="E139">A512</font></span></a>, <a class="ref" href="integra1.html#T7" title="INTEGRA1:th.7">INTEGRA1:7</a></span>;<br/><a NAME="E6:35_2_1_1_1_3_1_5_1_2_10_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E143">A516</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="E7:35_2_1_1_1_3_1_5_1_2_10_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="#E5:35_2_1_1_1_3_1_5_1_2_10_1_2"><span class="lab"><font color="Green" title="E142">A515</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E8:35_2_1_1_1_3_1_5_1_2_10_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="#E22:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E64">A258</font></span></a>, <a class="txt" href="#E28:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E70">A264</font></span></a>, <a class="txt" href="#E20:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A244</font></span></a>, <a class="txt" href="#E11:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E93">A356</font></span></a>, <a class="txt" href="#E13:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E95">A358</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1_3_1_5_1_2_10_1_2"><span class="lab"><font color="Green" title="E142">A515</font></span></a>, <a class="txt" href="#E6:35_2_1_1_1_3_1_5_1_2_10_1_2"><span class="lab"><font color="Green" title="E143">A516</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_3_1_5_1_2_10_1_2"><span class="lab"><font color="Green" title="E141">A514</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="E9:35_2_1_1_1_3_1_5_1_2_10_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="#E73:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E135">A506</font></span></a>, <a class="txt" href="#E76:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E137">A508</font></span></a>, <a class="txt" href="#E78:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E138">A509</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_3_1_5_1_2_10_1_2"><span class="lab"><font color="Green" title="E140">A513</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="E80:35_2_1_1_1_3_1_5_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="E81:35_2_1_1_1_3_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E139">A517</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="E82:35_2_1_1_1_3_1_5_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="#E78:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E138">A509</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="E83:35_2_1_1_1_3_1_5_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="E84:35_2_1_1_1_3_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E140">A518</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="E85:35_2_1_1_1_3_1_5_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="#E68:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E131">A502</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="E86:35_2_1_1_1_3_1_5_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="#E66:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E129">A500</font></span></a>, <a class="ref" href="integra1.html#T20" title="INTEGRA1:th.20">INTEGRA1:20</a></span>;<br/><a NAME="E87:35_2_1_1_1_3_1_5_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="#E78:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E138">A509</font></span></a>, <a class="txt" href="#E81:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E139">A517</font></span></a>, <a class="ref" href="integra1.html#T20" title="INTEGRA1:th.20">INTEGRA1:20</a></span>;<br/><a NAME="E88:35_2_1_1_1_3_1_5_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="#E84:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E140">A518</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="E89:35_2_1_1_1_3_1_5_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="E90:35_2_1_1_1_3_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E141">A519</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="E91:35_2_1_1_1_3_1_5_1_2"/><span class="lab"><font color="Green" title="E142">A520</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="#E5:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E51">A233</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/><a NAME="E92:35_2_1_1_1_3_1_5_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="#E5:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E51">A233</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/><a NAME="E93:35_2_1_1_1_3_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E143">A521</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="#E18:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E100">A363</font></span></a>, <a class="txt" href="#E91:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E142">A520</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/><a NAME="E94:35_2_1_1_1_3_1_5_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="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</font></span></a>, <a class="txt" href="#E21:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A245</font></span></a>, <a class="txt" href="#E52:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E83">A277</font></span></a>, <a class="ref" href="integra1.html#T7" title="INTEGRA1:th.7">INTEGRA1:7</a></span>;<br/><a NAME="E95:35_2_1_1_1_3_1_5_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="#E17:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E59">A241</font></span></a>, <a class="txt" href="#E32:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E110">A417</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="E96:35_2_1_1_1_3_1_5_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="#E7:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E90">A353</font></span></a>, <a class="txt" href="#E24:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E104">A388</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E97:35_2_1_1_1_3_1_5_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#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c13">MD1</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#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c14">MD2</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="#E10:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E92">A355</font></span></a>, <a class="txt" href="#E22:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E103">A380</font></span></a>, <a class="txt" href="#E62:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E127">A498</font></span></a>, <a class="txt" href="#E93:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E143">A521</font></span></a>, <a class="txt" href="#E38:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E114">A421</font></span></a>, <a class="ref" href="integra3.html#T14" target="_self" title="INTEGRA3:th.14">Th14</a></span>;<br/><a NAME="E98:35_2_1_1_1_3_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E144">A522</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#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c13">MD1</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#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c14">MD2</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="#E44:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E116">A423</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E99:35_2_1_1_1_3_1_5_1_2"/><span class="lab"><font color="Green" title="E145">A523</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="#E14:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E57">A239</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/><a NAME="E100:35_2_1_1_1_3_1_5_1_2"/><span class="lab"><font color="Green" title="E146">A524</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="#E48:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E81">A275</font></span></a>, <a class="txt" href="#E47:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E117">A424</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E101:35_2_1_1_1_3_1_5_1_2"/><span class="lab"><font color="Green" title="E147">A525</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#K4" title="INTEGRA1:func.4">upper_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#K4" title="INTEGRA1:func.4">upper_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="35_2_1_1_1_3_1_5_1_2_11"><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#K4" title="INTEGRA1:func.4">upper_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#K4" title="INTEGRA1:func.4">upper_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="E1:35_2_1_1_1_3_1_5_1_2_11"/><span class="lab"><font color="Green" title="E148">A526</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="E2:35_2_1_1_1_3_1_5_1_2_11"/><span class="lab"><font color="Green" title="E149">A527</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#K4" title="INTEGRA1:func.4">upper_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#K4" title="INTEGRA1:func.4">upper_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="E3:35_2_1_1_1_3_1_5_1_2_11"/><span class="lab"><font color="Green" title="E150">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#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> (<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="#E1:35_2_1_1_1_3_1_5_1_2_11"><span class="lab"><font color="Green" title="E148">A526</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_3_1_5_1_2_11"><span class="lab"><font color="Green" title="E149">A527</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E4:35_2_1_1_1_3_1_5_1_2_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E151">A529</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="#E33:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E111">A418</font></span></a>, <a class="txt" href="#E53:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E123">A446</font></span></a>, <a class="txt" href="#E99:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E145">A523</font></span></a>, <a class="txt" href="#E100:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E146">A524</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_2_11"><span class="lab"><font color="Green" title="E148">A526</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_3_1_5_1_2_11"><span class="lab"><font color="Green" title="E149">A527</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>;<br/>
<a NAME="E5:35_2_1_1_1_3_1_5_1_2_11"/><span class="lab"><font color="Green" title="E152">A530</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="#E3:35_2_1_1_1_3_1_5_1_2_11"><span class="lab"><font color="Green" title="E150">A528</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/>
<a NAME="E6:35_2_1_1_1_3_1_5_1_2_11"/><span class="kw">then </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="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E7:35_2_1_1_1_3_1_5_1_2_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E153">A531</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><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="c14">MD2</font>,<font color="Maroon" title="c17">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="c14">MD2</font>,<font color="Maroon" title="c17">k</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/>
<a NAME="E8:35_2_1_1_1_3_1_5_1_2_11"/>
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="E9:35_2_1_1_1_3_1_5_1_2_11"/><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="#E1:35_2_1_1_1_3_1_5_1_2_11"><span class="lab"><font color="Green" title="E148">A526</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<a NAME="E10:35_2_1_1_1_3_1_5_1_2_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E154">A532</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="E12:35_2_1_1_1_3_1_5_1_2_11"/><span class="lab"><font color="Green" title="E155">A533</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="E13:35_2_1_1_1_3_1_5_1_2_11"/>
<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="#E52:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E122">A445</font></span></a>, <a class="txt" href="#E50:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E120">A427</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_3_1_5_1_2_11"><span class="lab"><font color="Green" title="E149">A527</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/>
<a NAME="E14:35_2_1_1_1_3_1_5_1_2_11"/><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="E15:35_2_1_1_1_3_1_5_1_2_11"/><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="#E99:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E145">A523</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E16:35_2_1_1_1_3_1_5_1_2_11"/><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="#E10:35_2_1_1_1_3_1_5_1_2_11"><span class="lab"><font color="Green" title="E154">A532</font></span></a>, <a class="txt" href="#E12:35_2_1_1_1_3_1_5_1_2_11"><span class="lab"><font color="Green" title="E155">A533</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E17:35_2_1_1_1_3_1_5_1_2_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E156">A534</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#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/>
<a NAME="E18:35_2_1_1_1_3_1_5_1_2_11"/><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="E19:35_2_1_1_1_3_1_5_1_2_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E157">A535</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#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><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="#E4:35_2_1_1_1_3_1_5_1_2_11"><span class="lab"><font color="Green" title="E151">A529</font></span></a>, <a class="txt" href="#E12:35_2_1_1_1_3_1_5_1_2_11"><span class="lab"><font color="Green" title="E155">A533</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/>
<a NAME="E20:35_2_1_1_1_3_1_5_1_2_11"/><span class="lab"><font color="Green" title="E158">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>  <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="35_2_1_1_1_3_1_5_1_2_11_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:35_2_1_1_1_3_1_5_1_2_11_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="#E27:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E69">A263</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_2_11"><span class="lab"><font color="Green" title="E148">A526</font></span></a>, <a class="txt" href="#E12:35_2_1_1_1_3_1_5_1_2_11"><span class="lab"><font color="Green" title="E155">A533</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<a NAME="E2:35_2_1_1_1_3_1_5_1_2_11_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E159">A537</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="E3:35_2_1_1_1_3_1_5_1_2_11_1"/><span class="lab"><font color="Green" title="E160">A538</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="#E5:35_2_1_1_1_3_1_5_1_2_11"><span class="lab"><font color="Green" title="E152">A530</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="E4:35_2_1_1_1_3_1_5_1_2_11_1"/><span class="lab"><font color="Green" title="E161">A539</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="#E17:35_2_1_1_1_3_1_5_1_2_11"><span class="lab"><font color="Green" title="E156">A534</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="35_2_1_1_1_3_1_5_1_2_11_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="E1:35_2_1_1_1_3_1_5_1_2_11_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="35_2_1_1_1_3_1_5_1_2_11_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_1_1_3_1_5_1_2_11_1_1_1"/><span class="lab"><font color="Green" title="E162">A540</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="E2:35_2_1_1_1_3_1_5_1_2_11_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E163">A541</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"><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">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">k2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_2_1_1_1_3_1_5_1_2_11_1"><span class="lab"><font color="Green" title="E159">A537</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_3_1_5_1_2_11_1"><span class="lab"><font color="Green" title="E161">A539</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E3:35_2_1_1_1_3_1_5_1_2_11_1_1_1"/><span class="lab"><font color="Green" title="E164">A542</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> <font color="Maroon" title="c12">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_2_1_1_1_3_1_5_1_2_11_1"><span class="lab"><font color="Green" title="E160">A538</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_2_11_1_1_1"><span class="lab"><font color="Green" title="E162">A540</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E4:35_2_1_1_1_3_1_5_1_2_11_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="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="#E3:35_2_1_1_1_3_1_5_1_2_11_1"><span class="lab"><font color="Green" title="E160">A538</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_2_11_1_1_1"><span class="lab"><font color="Green" title="E162">A540</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><span class="lab"><font color="Green" title="E165">A543</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> = 
<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"><a 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="#E48:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E81">A275</font></span></a>, <a class="txt" href="#E14:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E57">A239</font></span></a>, <a class="txt" href="#E12:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E94">A357</font></span></a>, <a class="txt" href="#E33:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E111">A418</font></span></a>, <a class="txt" href="#E50:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E120">A427</font></span></a>, <a class="txt" href="#E53:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E123">A446</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_3_1_5_1_2_11"><span class="lab"><font color="Green" title="E149">A527</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_2_11_1_1_1"><span class="lab"><font color="Green" title="E162">A540</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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E30:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>
;<br/><a NAME="E6:35_2_1_1_1_3_1_5_1_2_11_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="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">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="#E2:35_2_1_1_1_3_1_5_1_2_11_1"><span class="lab"><font color="Green" title="E159">A537</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_3_1_5_1_2_11_1"><span class="lab"><font color="Green" title="E161">A539</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_2_11_1_1_1"><span class="lab"><font color="Green" title="E162">A540</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="E7:35_2_1_1_1_3_1_5_1_2_11_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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E17:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E59">A241</font></span></a>, <a class="txt" href="#E30:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</font></span></a>, <a class="txt" href="#E7:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E90">A353</font></span></a>, <a class="txt" href="#E50:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E120">A427</font></span></a>, <a class="txt" href="#E12:35_2_1_1_1_3_1_5_1_2_11"><span class="lab"><font color="Green" title="E155">A533</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_3_1_5_1_2_11_1_1_1"><span class="lab"><font color="Green" title="E164">A542</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1_3_1_5_1_2_11_1_1_1"><span class="lab"><font color="Green" title="E165">A543</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_3_1_5_1_2_11_1_1_1"><span class="lab"><font color="Green" title="E163">A541</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_1_1_3_1_5_1_2_11_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_1_1_3_1_5_1_2_11_1_1_2"/><span class="lab"><font color="Green" title="E162">A544</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="E2:35_2_1_1_1_3_1_5_1_2_11_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="#E3:35_2_1_1_1_3_1_5_1_2_11_1"><span class="lab"><font color="Green" title="E160">A538</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E3:35_2_1_1_1_3_1_5_1_2_11_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E163">A545</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="#E48:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E81">A275</font></span></a>, <a class="txt" href="#E14:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E57">A239</font></span></a>, <a class="txt" href="#E12:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E94">A357</font></span></a>, <a class="txt" href="#E33:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E111">A418</font></span></a>, <a class="txt" href="#E50:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E120">A427</font></span></a>, <a class="txt" href="#E53:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E123">A446</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_2_11"><span class="lab"><font color="Green" title="E148">A526</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_3_1_5_1_2_11"><span class="lab"><font color="Green" title="E149">A527</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_3_1_5_1_2_11"><span class="lab"><font color="Green" title="E150">A528</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>;<br/><a NAME="E4:35_2_1_1_1_3_1_5_1_2_11_1_1_2"/><span class="lab"><font color="Green" title="E164">A546</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="#E53:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E123">A446</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_3_1_5_1_2_11"><span class="lab"><font color="Green" title="E149">A527</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="E5:35_2_1_1_1_3_1_5_1_2_11_1_1_2"/><span class="lab"><font color="Green" title="E165">A547</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="#E3:35_2_1_1_1_3_1_5_1_2_11_1"><span class="lab"><font color="Green" title="E160">A538</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_2_11_1_1_2"><span class="lab"><font color="Green" title="E162">A544</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E6:35_2_1_1_1_3_1_5_1_2_11_1_1_2"/><span class="lab"><font color="Green" title="E166">A548</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="#E3:35_2_1_1_1_3_1_5_1_2_11_1"><span class="lab"><font color="Green" title="E160">A538</font></span></a>, <a class="txt" href="#E1:35_2_1_1_1_3_1_5_1_2_11_1_1_2"><span class="lab"><font color="Green" title="E162">A544</font></span></a>, <a class="ref" href="integra1.html#T7" title="INTEGRA1:th.7">INTEGRA1:7</a></span>;<br/><a NAME="E7:35_2_1_1_1_3_1_5_1_2_11_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="E8:35_2_1_1_1_3_1_5_1_2_11_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="#E48:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E81">A275</font></span></a>, <a class="txt" href="#E14:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E57">A239</font></span></a>, <a class="txt" href="#E12:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E94">A357</font></span></a>, <a class="txt" href="#E33:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E111">A418</font></span></a>, <a class="txt" href="#E50:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E120">A427</font></span></a>, <a class="txt" href="#E6:35_2_1_1_1_3_1_5_1_2_11_1_1_2"><span class="lab"><font color="Green" title="E166">A548</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_3_1_5_1_2_11_1_1_2"><span class="lab"><font color="Green" title="E164">A546</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1_3_1_5_1_2_11_1_1_2"><span class="lab"><font color="Green" title="E165">A547</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="E9:35_2_1_1_1_3_1_5_1_2_11_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="#E12:35_2_1_1_1_3_1_5_1_2_11"><span class="lab"><font color="Green" title="E155">A533</font></span></a>, <a class="txt" href="#E2:35_2_1_1_1_3_1_5_1_2_11_1"><span class="lab"><font color="Green" title="E159">A537</font></span></a>, <a class="txt" href="#E4:35_2_1_1_1_3_1_5_1_2_11_1"><span class="lab"><font color="Green" title="E161">A539</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1_3_1_5_1_2_11_1_1_2"><span class="lab"><font color="Green" title="E163">A545</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="E21:35_2_1_1_1_3_1_5_1_2_11"/>
 <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="E22:35_2_1_1_1_3_1_5_1_2_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E159">A549</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="#E20:35_2_1_1_1_3_1_5_1_2_11"><span class="lab"><font color="Green" title="E158">A536</font></span></a>, <a class="ref" href="integra1.html#T4" title="INTEGRA1:th.4">INTEGRA1:4</a></span>;<br/>
<a NAME="E23:35_2_1_1_1_3_1_5_1_2_11"/>
<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="#E5:35_2_1_1_1_3_1_5_1_2_11"><span class="lab"><font color="Green" title="E152">A530</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="E24:35_2_1_1_1_3_1_5_1_2_11"/><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="#E22:35_2_1_1_1_3_1_5_1_2_11"><span class="lab"><font color="Green" title="E159">A549</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="E25:35_2_1_1_1_3_1_5_1_2_11"/>
<span class="p1">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_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="#E7:35_2_1_1_1_3_1_5_1_2_11"><span class="lab"><font color="Green" title="E153">A531</font></span></a>, <a class="txt" href="#E19:35_2_1_1_1_3_1_5_1_2_11"><span class="lab"><font color="Green" title="E157">A535</font></span></a>, <a class="txt" href="#E22:35_2_1_1_1_3_1_5_1_2_11"><span class="lab"><font color="Green" title="E159">A549</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="E102:35_2_1_1_1_3_1_5_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="#E48:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E118">A425</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/><a NAME="E103:35_2_1_1_1_3_1_5_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="#E13:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E56">A238</font></span></a>, <a class="txt" href="#E33:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E111">A418</font></span></a>, <a class="txt" href="#E99:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E145">A523</font></span></a>, <a class="txt" href="#E100:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E146">A524</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>;<br/><a NAME="E104:35_2_1_1_1_3_1_5_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#K4" title="INTEGRA1:func.4">upper_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="#E48:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E81">A275</font></span></a>, <a class="txt" href="#E47:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E117">A424</font></span></a>, <a class="txt" href="#E53:35_2_1_1_1_3_1_5_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="E105:35_2_1_1_1_3_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E148">A550</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#K4" title="INTEGRA1:func.4">upper_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="#E101:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E147">A525</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</a></span>;<br/><a NAME="E106:35_2_1_1_1_3_1_5_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="#E1:35"><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="E107:35_2_1_1_1_3_1_5_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="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="#E90:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E141">A519</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="E108:35_2_1_1_1_3_1_5_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="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="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="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="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="#E98:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E144">A522</font></span></a>, <a class="txt" href="#E105:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E148">A550</font></span></a>, <a class="txt" href="#E31:35_2_1_1_1_3_1_5_1_2"><span class="lab"><font color="Green" title="E109">A416</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>
<a NAME="E55:35_2_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A551</font></span>: 
<span class="p1">(<span class="default"><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#K9" title="REAL_1:func.9">-</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>))</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="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> <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="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></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="#E19:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E61">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="E56:35_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="E57:35_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="#E17:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E59">A241</font></span></a>, <a class="txt" href="#E30:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A266</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="E58:35_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="#E26:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E68">A262</font></span></a>, <a class="txt" href="#E24:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E66">A260</font></span></a>, <a class="txt" href="#E35:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E76">A270</font></span></a>, <a class="txt" href="#E31:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E73">A267</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/>
<a NAME="E59:35_2_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A552</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="#E48:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E81">A275</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E60:35_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="#E26:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E68">A262</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="E61:35_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#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/>
<a NAME="E62:35_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="#E27:35_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#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="#E27:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E69">A263</font></span></a>, <a class="txt" href="#E59:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E86">A552</font></span></a>, <a class="txt" href="#E15:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E58">A240</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="#E13:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E56">A238</font></span></a>, <a class="ref" href="finseq_6.html#T116" title="FINSEQ_6:th.116">FINSEQ_6:116</a></span>
;<br/>
<a NAME="E65:35_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="#E45:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E79">A273</font></span></a>, <a class="ref" href="integra1.html#D20" title="INTEGRA1:def.20">INTEGRA1:def 20</a></span>;<br/>
<a NAME="E66:35_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="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="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>))</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="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> <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="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></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="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="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>))
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E78">A272</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E67:35_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="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="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="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="#E4:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">A232</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E51">A233</font></span></a>, <a class="txt" href="#E55:35_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E85">A551</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="E4:35_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="E7:35_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="#E4:35_2_1_1_1"><span class="lab"><font color="Green" title="E44">A36</font></span></a>, <a class="txt" href="#E6:35_2_1_1_1"><span class="lab"><font color="Green" title="E45">A227</font></span></a>);<br/></span>
<a NAME="E8:35_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="E9:35_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="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="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="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="#E1:35_2_1_1_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>
<a NAME="E3:35_2_1_1"/><span class="lab"><font color="Green" title="E41">A553</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="E4:35_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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E5:35_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="E6:35_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A554</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="E7:35_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="E9:35_2_1_1"/><span class="lab"><font color="Green" title="E43">A555</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="E10:35_2_1_1"/><span class="lab"><font color="Green" title="E44">A556</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="E11:35_2_1_1"/><span class="lab"><font color="Green" title="E45">A557</font></span>: 
<font color="Maroon">H<sub>2</sub></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="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="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="#E2:35_2_1_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a></span>;<br/>
<a NAME="E12:35_2_1_1"/><span class="lab"><font color="Green" title="E46">A558</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="35_2_1_1_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:35_2_1_1_2"/>
<font color="Maroon" title="c7">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="#E9:35_2_1_1"><span class="lab"><font color="Green" title="E43">A555</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="E2:35_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A559</font></span>: 
<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="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:35_2_1_1_2"/>
<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="E4:35_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="#E2:35_2_1_1_2"><span class="lab"><font color="Green" title="E47">A559</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E5:35_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="#E9:35_2_1_1"><span class="lab"><font color="Green" title="E43">A555</font></span></a>, <a class="txt" href="#E3:35_2_1_1"><span class="lab"><font color="Green" title="E41">A553</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="E6:35_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A560</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="E7:35_2_1_1_2"/><span class="lab"><font color="Green" title="E49">A561</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="35_2_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_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="E1:35_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="35_2_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_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="E2:35_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="#E9:35_2_1_1"><span class="lab"><font color="Green" title="E43">A555</font></span></a>, <a class="txt" href="#E6:35_2_1_1_2"><span class="lab"><font color="Green" title="E48">A560</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="35_2_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_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="E2:35_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="#E9:35_2_1_1"><span class="lab"><font color="Green" title="E43">A555</font></span></a>, <a class="txt" href="#E6:35_2_1_1_2"><span class="lab"><font color="Green" title="E48">A560</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="E8:35_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="#E10:35_2_1_1"><span class="lab"><font color="Green" title="E44">A556</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="E9:35_2_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_2_1_1_2"><span class="lab"><font color="Green" title="E49">A561</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="E13:35_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="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E3:35_2_1_1"><span class="lab"><font color="Green" title="E41">A553</font></span></a>, <a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></span>;<br/>
<a NAME="E14:35_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="#E1:35_2_1_1"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="txt" href="#E6:35_2_1_1"><span class="lab"><font color="Green" title="E42">A554</font></span></a>, <a class="ref" href="seq_4.html#T138" title="SEQ_4:th.138">SEQ_4:138</a></span>;<br/>
<a NAME="E15:35_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <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="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="#E11:35_2_1_1"><span class="lab"><font color="Green" title="E45">A557</font></span></a>, <a class="txt" href="#E12:35_2_1_1"><span class="lab"><font color="Green" title="E46">A558</font></span></a>, <a class="ref" href="integra1.html#T42" title="INTEGRA1:th.42">INTEGRA1:42</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:35_2_1_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</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#T42" title="INTEGRA1:th.42">INTEGRA1:42</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:35_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#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">D2</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="#E2:35_2_1"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="txt" href="#E3:35_2_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E4:35_2_1"><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>
<span class="kw">hence </span><a NAME="E3:35_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#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">D2</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>
<span class="kw">assume </span><a NAME="E4:35"/><span class="lab"><font color="Green" title="E35">A562</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not  <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>  or (  <a href="integra2.html#K3" title="INTEGRA2:func.3">upper_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#K3" title="INTEGRA2:func.3">upper_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#K10" title="INTEGRA1:func.10">upper_integral</a> <font color="Maroon" title="c2">f</font> ) )</span><br/>

<a NAME="E5:35"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A563</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="ref" href="fdiff_1.html#D1" title="FDIFF_1:def.1">FDIFF_1:def 1</a></span>;<br/>
<a NAME="E6:35"/><span class="lab"><font color="Green" title="E37">A564</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="#E4:35"><span class="lab"><font color="Green" title="E35">A562</font></span></a>, <a class="ref" href="fdiff_1.html#D1" title="FDIFF_1:def.1">FDIFF_1:def 1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E7:35"/><span class="lab"><font color="Green" title="E38">A565</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#K3" title="INTEGRA2:func.3">upper_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#K3" title="INTEGRA2:func.3">upper_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#K10" title="INTEGRA1:func.10">upper_integral</a> <font color="Maroon" title="c2">f</font> )</span><br/>

<a NAME="E8:35"/><span class="lab"><font color="Green" title="E39">A566</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="#E4:35"><span class="lab"><font color="Green" title="E35">A562</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="E9:35"/><span class="lab"><font color="Green" title="E40">A567</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="35_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="E1:35_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="E3:35_3"/><span class="lab"><font color="Green" title="E41">A568</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="#E5:35"><span class="lab"><font color="Green" title="E36">A563</font></span></a>, <a class="txt" href="#E6:35"><span class="lab"><font color="Green" title="E37">A564</font></span></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">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="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> )</span><br/>

<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/>

<span class="kw">assume </span><a NAME="E4:35_3"/>
<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="E5:35_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"><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="#E3:35_3"><span class="lab"><font color="Green" title="E41">A568</font></span></a></span>;<br/>
<a NAME="E6:35_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A569</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/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">D</font> = <font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c6">m</font> as    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font> ;<br/>
<a NAME="E8:35_3"/><span class="lab"><font color="Green" title="E43">A570</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="E9:35_3"/>
 <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="c8">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="E11:35_3"/><span class="lab"><font color="Green" title="E44">A571</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>,<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="E12:35_3"/><span class="lab"><font color="Green" title="E45">A572</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="c8">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/>
<a NAME="E13:35_3"/>
<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>,<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="#E11:35_3"><span class="lab"><font color="Green" title="E44">A571</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="E14:35_3"/><span class="kw">then </span>
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">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="E15:35_3"/><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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">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="E16:35_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A573</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="c8">i</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:35_3"><span class="lab"><font color="Green" title="E45">A572</font></span></a>, <a class="ref" href="integra1.html#T20" title="INTEGRA1:th.20">INTEGRA1:20</a></span>;<br/>
<a NAME="E17:35_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="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="#E8:35"><span class="lab"><font color="Green" title="E39">A566</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="E18:35_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="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="#E6:35_3"><span class="lab"><font color="Green" title="E42">A569</font></span></a>, <a class="txt" href="#E8:35_3"><span class="lab"><font color="Green" title="E43">A570</font></span></a>, <a class="txt" href="#E16:35_3"><span class="lab"><font color="Green" title="E46">A573</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>
<a NAME="E10:35"/><span class="lab"><font color="Green" title="E41">A574</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#K3" title="INTEGRA2:func.3">upper_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#K10" title="INTEGRA1:func.10">upper_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="35_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">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="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="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#K3" title="INTEGRA2:func.3">upper_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#K10" title="INTEGRA1:func.10">upper_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="c4">e</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_4"/><span class="lab"><font color="Green" title="E42">A576</font></span>: 
<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="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#K3" title="INTEGRA2:func.3">upper_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#K10" title="INTEGRA1:func.10">upper_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="c4">e</font></span><br/>

<a NAME="E2:35_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A577</font></span>: 
<font color="Maroon" title="c4">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="c5">e</font> = <font color="Maroon" title="c4">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="E4:35_4"/><span class="lab"><font color="Green" title="E44">A578</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#K8" title="INTEGRA1:func.8">upper_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span> 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="#E1:35"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="integra2.html#T35" title="INTEGRA2:th.35">INTEGRA2:35</a></span>;<br/>
<a NAME="E5:35_4"/>
 <a href="integra1.html#K10" title="INTEGRA1:func.10">upper_integral</a> <font color="Maroon" title="c2">f</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> <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#K8" title="INTEGRA1:func.8">upper_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#D14" title="INTEGRA1:def.14">INTEGRA1:def 14</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">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="E7:35_4"/><span class="lab"><font color="Green" title="E45">A579</font></span>: 
<font color="Maroon" title="c6">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#K8" title="INTEGRA1:func.8">upper_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:35_4"/><span class="lab"><font color="Green" title="E46">A580</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">upper_integral</a> <font color="Maroon" title="c2">f</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="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_4"><span class="lab"><font color="Green" title="E43">A577</font></span></a>, <a class="txt" href="#E4:35_4"><span class="lab"><font color="Green" title="E44">A578</font></span></a>, <a class="ref" href="seq_4.html#D2" title="SEQ_4:def.2">SEQ_4:def 2</a></span>;<br/>
<a NAME="E9:35_4"/>
 ex <font color="Olive" title="b1">D</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b1">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#K8" title="INTEGRA1:func.8">upper_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K8" title="INTEGRA1:func.8">upper_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="Olive" title="b1">D</font> &amp; <font color="Olive" title="b1">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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c7">D3</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="integra1.html#K1" title="INTEGRA1:func.1">divs</a> <font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E2:35_4_1"/><span class="lab"><font color="Green" title="E47">A581</font></span>: 
<font color="Maroon" title="c7">D3</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#K8" title="INTEGRA1:func.8">upper_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:35_4_1"/><span class="lab"><font color="Green" title="E48">A582</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K8" title="INTEGRA1:func.8">upper_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="c7">D3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_4"><span class="lab"><font color="Green" title="E45">A579</font></span></a>, <a class="ref" href="partfun1.html#T3" title="PARTFUN1:th.3">PARTFUN1:3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">D3</font> = <font color="Maroon" title="c7">D3</font> as    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <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="E5:35_4_1"/><span class="lab"><font color="Green" title="E49">A583</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">D3</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">D3</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="E6:35_4_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="c8">D3</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="E7:35_4_1"/><span class="kw">then </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="c8">D3</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="E8:35_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A584</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="c8">D3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_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="E1:35_4_1_1"/>
( <font color="Maroon" title="c8">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font> or <font color="Maroon" title="c8">D3</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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_4_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_4_1_1_1"/><span class="lab"><font color="Green" title="E51">A585</font></span>: 
<font color="Maroon" title="c8">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <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">D</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="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#K8" title="INTEGRA1:func.8">upper_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K8" title="INTEGRA1:func.8">upper_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="Olive" title="b1">D</font> &amp; <font color="Olive" title="b1">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><br/><div class="add"><a NAME="E2:35_4_1_1_1"/>
<font color="Maroon" title="c8">D3</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> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_4_1"><span class="lab"><font color="Green" title="E50">A584</font></span></a>, <a class="ref" href="integra1.html#T6" title="INTEGRA1:th.6">INTEGRA1:6</a></span>;<br/><a NAME="E3:35_4_1_1_1"/><span class="kw">then </span>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
 <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="E4:35_4_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">D3</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="#E1:35_4_1_1_1"><span class="lab"><font color="Green" title="E51">A585</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="E5:35_4_1_1_1"/>
 ex <font color="Olive" title="b1">D</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b1">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#K8" title="INTEGRA1:func.8">upper_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K8" title="INTEGRA1:func.8">upper_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="Olive" title="b1">D</font> &amp; <font color="Olive" title="b1">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="#E2:35_4_1"><span class="lab"><font color="Green" title="E47">A581</font></span></a>, <a class="txt" href="#E3:35_4_1"><span class="lab"><font color="Green" title="E48">A582</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="35_4_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_4_1_1_2"/><span class="lab"><font color="Green" title="E51">A586</font></span>: 
<font color="Maroon" title="c8">D3</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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <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">D</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="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#K8" title="INTEGRA1:func.8">upper_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K8" title="INTEGRA1:func.8">upper_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="Olive" title="b1">D</font> &amp; <font color="Olive" title="b1">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><br/><div class="add"><a NAME="E2:35_4_1_1_2"/>
 ex <font color="Olive" title="b1">D</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b1">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#K8" title="INTEGRA1:func.8">upper_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K8" title="INTEGRA1:func.8">upper_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="Olive" title="b1">D</font> &amp; <font color="Olive" title="b1">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> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_4_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:35_4_1_1_2_1"/><span class="lab"><font color="Green" title="E52">A587</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><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="c8">D3</font>,1)</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="c8">D3</font>,1)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_4_1"><span class="lab"><font color="Green" title="E50">A584</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/>
<a NAME="E2:35_4_1_1_2_1"/>
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</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="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a></span>;<br/>
<a NAME="E3:35_4_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A588</font></span>: 
<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="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#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="#E7:35"><span class="lab"><font color="Green" title="E38">A565</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/>
<span class="lab"><font color="Green" title="E54">A589</font></span>: <font color="Maroon" title="c6">y</font> = 
 <a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_4_1"><span class="lab"><font color="Green" title="E48">A582</font></span></a>, <a class="ref" href="integra1.html#D10" title="INTEGRA1:def.10">INTEGRA1:def 10</a></span>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></span>
<br/>.= 
 <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"><span class="p3">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq.html#T8" title="RFINSEQ:th.8">RFINSEQ:8</a></span>
;<br/>
<a NAME="E5:35_4_1_1_2_1"/><span class="lab"><font color="Green" title="E55">A590</font></span>: 
<font color="Maroon" title="c8">D3</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="c8">D3</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> <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="E6:35_4_1_1_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">D3</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">D3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_4_1"><span class="lab"><font color="Green" title="E49">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="E7:35_4_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A591</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">D3</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="#E8:35_4_1"><span class="lab"><font color="Green" title="E50">A584</font></span></a>, <a class="txt" href="#E1:35_4_1_1_2"><span class="lab"><font color="Green" title="E51">A586</font></span></a>, <a class="txt" href="#E5:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E55">A590</font></span></a>, <a class="txt" href="#E3:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E53">A588</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a>, <a class="ref" href="xreal_1.html#T47" title="XREAL_1:th.47">XREAL_1:47</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">D</font> = <font color="Maroon" title="c8">D3</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1 as   <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</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">by</span> <span class="lab"><a class="ref" href="integra1.html#T34" title="INTEGRA1:th.34">INTEGRA1:34</a></span>;<br/>
<a NAME="E9:35_4_1_1_2_1"/><span class="lab"><font color="Green" title="E57">A592</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">D3</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E56">A591</font></span></a>, <a class="ref" href="rfinseq.html#D1" title="RFINSEQ:def.1">RFINSEQ:def 1</a></span>;<br/>
<a NAME="E10:35_4_1_1_2_1"/>
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font> <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="#E3:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E53">A588</font></span></a>, <a class="ref" href="xreal_1.html#T47" title="XREAL_1:th.47">XREAL_1:47</a></span>;<br/>
<a NAME="E11:35_4_1_1_2_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">D</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="#E1:35_4_1_1_2"><span class="lab"><font color="Green" title="E51">A586</font></span></a>, <a class="txt" href="#E9:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E57">A592</font></span></a>, <a class="ref" href="integra1.html#D2" title="INTEGRA1:def.2">INTEGRA1:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">D</font> = <font color="Maroon" title="c9">D</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</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>  ;<br/>
<a NAME="E13:35_4_1_1_2_1"/><span class="lab"><font color="Green" title="E58">A593</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">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="c10">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/>
<a NAME="E14:35_4_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">D</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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">D3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E57">A592</font></span></a></span>;<br/>
<a NAME="E15:35_4_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A594</font></span>: 
<font color="Maroon" title="c10">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="c10">D</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> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E55">A590</font></span></a>, <a class="txt" href="#E7:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E56">A591</font></span></a>, <a class="txt" href="#E13:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E58">A593</font></span></a>, <a class="ref" href="rfinseq.html#D1" title="RFINSEQ:def.1">RFINSEQ:def 1</a></span>;<br/>
<a NAME="E16:35_4_1_1_2_1"/><span class="lab"><font color="Green" title="E60">A595</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">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="c10">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="E17:35_4_1_1_2_1"/>
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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">D3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E56">A591</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E18:35_4_1_1_2_1"/><span class="kw">then </span>
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="c8">D3</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="E19:35_4_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A596</font></span>: 
<font color="Maroon" title="c8">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_4_1"><span class="lab"><font color="Green" title="E50">A584</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="E20:35_4_1_1_2_1"/><span class="lab"><font color="Green" title="E62">A597</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c8">D3</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="E21:35_4_1_1_2_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c10">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="c8">D3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T33" title="FINSEQ_5:th.33">FINSEQ_5:33</a></span>;<br/>
<a NAME="E22:35_4_1_1_2_1"/><span class="kw">then </span>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c10">D</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="#E20:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E62">A597</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">D</font> = <font color="Maroon" title="c10">D</font> as    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E59">A594</font></span></a>, <a class="ref" href="integra1.html#D2" title="INTEGRA1:def.2">INTEGRA1:def 2</a></span>;<br/>
<a NAME="E24:35_4_1_1_2_1"/><span class="lab"><font color="Green" title="E63">A598</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> 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="E25:35_4_1_1_2_1"/><span class="lab"><font color="Green" title="E64">A599</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">D3</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="#E7:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E56">A591</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E26:35_4_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A600</font></span>: 
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> <span class="p1">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</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="E27:35_4_1_1_2_1"/>
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#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E56">A591</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E66">A601</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#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</span>)</span>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</span>)</span></span>)</span>)</span>)</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> <span class="p3">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2</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="#E26:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E65">A600</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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">D3</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2</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="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">D3</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2</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="#E25:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E64">A599</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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">D3</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1

;<br/>
<a NAME="E29:35_4_1_1_2_1"/><span class="lab"><font color="Green" title="E67">A602</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <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#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</span>)</span>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</span>)</span></span>)</span>)</span>)</span> holds <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#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</span>)</span>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</span>)</span></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c11">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_4_1_1_2_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c12">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> <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#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</span>)</span>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</span>)</span></span>)</span>)</span>)</span> implies <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#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</span>)</span>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</span>)</span></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</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> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c11">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35_4_1_1_2_1_3"/><span class="lab"><font color="Green" title="E68">A603</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font>
 <span class="kw">and </span><br/><a NAME="E2:35_4_1_1_2_1_3"/><span class="lab"><font color="Green" title="E69">A604</font></span>: 
<font color="Maroon" title="c12">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> <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#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</span>)</span>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</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"> <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#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</span>)</span>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</span>)</span></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</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> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c11">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span><br/>

<a NAME="E3:35_4_1_1_2_1_3"/><span class="lab"><font color="Green" title="E70">A605</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">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#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E4:35_4_1_1_2_1_3"/>
<font color="Maroon" title="c12">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="c8">D3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E66">A601</font></span></a>, <a class="txt" href="#E2:35_4_1_1_2_1_3"><span class="lab"><font color="Green" title="E69">A604</font></span></a>, <a class="ref" href="xreal_1.html#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/>
<a NAME="E5:35_4_1_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A606</font></span>: 
<font color="Maroon" title="c12">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#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">D3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_4_1_1_2_1_3"><span class="lab"><font color="Green" title="E70">A605</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E6:35_4_1_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A607</font></span>: 
<font color="Maroon" title="c12">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="c8">D3</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="E7:35_4_1_1_2_1_3"/><span class="lab"><font color="Green" title="E73">A608</font></span>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c8">D3</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c11">D</font>,<font color="Maroon" title="c12">i</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_4_1_1_2_1_3_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:35_4_1_1_2_1_3_1"/><span class="lab"><font color="Green" title="E74">A609</font></span>: 
<font color="Maroon" title="c12">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="c8">D3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_4_1_1_2_1_3"><span class="lab"><font color="Green" title="E71">A606</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="E2:35_4_1_1_2_1_3_1"/><span class="lab"><font color="Green" title="E75">A610</font></span>: 
1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">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="txt" href="#E1:35_4_1_1_2_1_3"><span class="lab"><font color="Green" title="E68">A603</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E3:35_4_1_1_2_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A611</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="c8">D3</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">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="#E1:35_4_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E74">A609</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/>
<a NAME="E4:35_4_1_1_2_1_3_1"/><span class="lab"><font color="Green" title="E77">A612</font></span>: 
<font color="Maroon" title="c12">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="c11">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E57">A592</font></span></a>, <a class="txt" href="#E28:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E66">A601</font></span></a>, <a class="txt" href="#E1:35_4_1_1_2_1_3"><span class="lab"><font color="Green" title="E68">A603</font></span></a>, <a class="txt" href="#E2:35_4_1_1_2_1_3"><span class="lab"><font color="Green" title="E69">A604</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E5:35_4_1_1_2_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A613</font></span>: 
<font color="Maroon" title="c11">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">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="#E7:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E56">A591</font></span></a>, <a class="ref" href="rfinseq.html#D1" title="RFINSEQ:def.1">RFINSEQ:def 1</a></span>;<br/>
<a NAME="E6:35_4_1_1_2_1_3_1"/><span class="lab"><font color="Green" title="E79">A614</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="c8">D3</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">D3</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="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="#E2:35_4_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E75">A610</font></span></a>, <a class="txt" href="#E1:35_4_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E74">A609</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_4_1_1_2_1_3_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="E1:35_4_1_1_2_1_3_1_1"/>
( <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c12">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="35_4_1_1_2_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_4_1_1_2_1_3_1_1_1"/><span class="lab"><font color="Green" title="E80">A615</font></span>: 
<font color="Maroon" title="c12">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="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c8">D3</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c11">D</font>,<font color="Maroon" title="c12">i</font>)</span><br/><div class="add"><a NAME="E2:35_4_1_1_2_1_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A616</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="c11">D</font>,<font color="Maroon" title="c12">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_4_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E77">A612</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E3:35_4_1_1_2_1_3_1_1_1"/><span class="lab"><font color="Green" title="E82">A617</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="c11">D</font>,<font color="Maroon" title="c12">i</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="#E4:35_4_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E77">A612</font></span></a>, <a class="txt" href="#E1:35_4_1_1_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E80">A615</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E4:35_4_1_1_2_1_3_1_1_1"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c8">D3</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font></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="#E1:35_4_1_1_2"><span class="lab"><font color="Green" title="E51">A586</font></span></a>, <a class="txt" href="#E3:35_4_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E76">A611</font></span></a>, <a class="txt" href="#E6:35_4_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E79">A614</font></span></a>, <a class="txt" href="#E5:35_4_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E78">A613</font></span></a>, <a class="txt" href="#E1:35_4_1_1_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E80">A615</font></span></a>, <a class="ref" href="integra1.html#T4" title="INTEGRA1:th.4">INTEGRA1:4</a></span>;<br/><span class="kw">hence </span><a NAME="E5:35_4_1_1_2_1_3_1_1_1"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c8">D3</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c11">D</font>,<font color="Maroon" title="c12">i</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_4_1_1_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E82">A617</font></span></a>, <a class="txt" href="#E2:35_4_1_1_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E81">A616</font></span></a>, <a class="ref" href="integra1.html#T4" title="INTEGRA1:th.4">INTEGRA1:4</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="35_4_1_1_2_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_4_1_1_2_1_3_1_1_2"/><span class="lab"><font color="Green" title="E80">A618</font></span>: 
<font color="Maroon" title="c12">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="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c8">D3</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c11">D</font>,<font color="Maroon" title="c12">i</font>)</span><br/><div class="add"><a NAME="E2:35_4_1_1_2_1_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">i</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="c11">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_4_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E77">A612</font></span></a>, <a class="ref" href="integra1.html#T7" title="INTEGRA1:th.7">INTEGRA1:7</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A619</font></span>: <font color="Maroon" title="c11">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> = 
<font color="Maroon" title="c8">D3</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="c12">i</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> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E56">A591</font></span></a>, <a class="ref" href="rfinseq.html#D1" title="RFINSEQ:def.1">RFINSEQ:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c8">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font>

;<br/><a NAME="E4:35_4_1_1_2_1_3_1_1_2"/><span class="lab"><font color="Green" title="E82">A620</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="c11">D</font>,<font color="Maroon" title="c12">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_4_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E77">A612</font></span></a>, <a class="txt" href="#E1:35_4_1_1_2_1_3_1_1_2"><span class="lab"><font color="Green" title="E80">A618</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E5:35_4_1_1_2_1_3_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="c11">D</font>,<font color="Maroon" title="c12">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_4_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E77">A612</font></span></a>, <a class="txt" href="#E1:35_4_1_1_2_1_3_1_1_2"><span class="lab"><font color="Green" title="E80">A618</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E6:35_4_1_1_2_1_3_1_1_2"/><span class="kw">then </span>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c8">D3</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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="c11">D</font>,<font color="Maroon" title="c12">i</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="c11">D</font>,<font color="Maroon" title="c12">i</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="#E3:35_4_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E76">A611</font></span></a>, <a class="txt" href="#E6:35_4_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E79">A614</font></span></a>, <a class="txt" href="#E5:35_4_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E78">A613</font></span></a>, <a class="txt" href="#E4:35_4_1_1_2_1_3_1_1_2"><span class="lab"><font color="Green" title="E82">A620</font></span></a>, <a class="txt" href="#E3:35_4_1_1_2_1_3_1_1_2"><span class="lab"><font color="Green" title="E81">A619</font></span></a>, <a class="ref" href="integra1.html#T4" title="INTEGRA1:th.4">INTEGRA1:4</a></span>;<br/><span class="kw">hence </span><a NAME="E7:35_4_1_1_2_1_3_1_1_2"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c8">D3</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c11">D</font>,<font color="Maroon" title="c12">i</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T4" title="INTEGRA1:th.4">INTEGRA1:4</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="E8:35_4_1_1_2_1_3"/>
<font color="Maroon" title="c12">i</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> <span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</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="#E28:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E66">A601</font></span></a>, <a class="txt" href="#E2:35_4_1_1_2_1_3"><span class="lab"><font color="Green" title="E69">A604</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/>
<a NAME="E9:35_4_1_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A621</font></span>: 
<font color="Maroon" title="c12">i</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> <span class="p3">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
;<br/>
<a NAME="E10:35_4_1_1_2_1_3"/>
<font color="Maroon" title="c12">i</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><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#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</span>)</span>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</span>)</span></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font> = 
<span class="p1">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</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"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</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="#E26:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E65">A600</font></span></a>, <a class="txt" href="#E1:35_4_1_1_2_1_3"><span class="lab"><font color="Green" title="E68">A603</font></span></a>, <a class="txt" href="#E9:35_4_1_1_2_1_3"><span class="lab"><font color="Green" title="E74">A621</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="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</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="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>

;<br/>
<a NAME="E12:35_4_1_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A622</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#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</span>)</span>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</span>)</span></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><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="c8">D3</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c8">D3</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_4_1_1_2_1_3"><span class="lab"><font color="Green" title="E72">A607</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/>
<a NAME="E13:35_4_1_1_2_1_3"/>
<font color="Maroon" title="c12">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="c11">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E57">A592</font></span></a>, <a class="txt" href="#E28:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E66">A601</font></span></a>, <a class="txt" href="#E1:35_4_1_1_2_1_3"><span class="lab"><font color="Green" title="E68">A603</font></span></a>, <a class="txt" href="#E2:35_4_1_1_2_1_3"><span class="lab"><font color="Green" title="E69">A604</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E14:35_4_1_1_2_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c12">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="c11">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">hence </span><a NAME="E15:35_4_1_1_2_1_3"/>
<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#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</span>)</span>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</span>)</span></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</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> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c11">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:35_4_1_1_2_1_3"><span class="lab"><font color="Green" title="E75">A622</font></span></a>, <a class="txt" href="#E7:35_4_1_1_2_1_3"><span class="lab"><font color="Green" title="E73">A608</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1: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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E30:35_4_1_1_2_1"/><span class="lab"><font color="Green" title="E68">A623</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="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E56">A591</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/>
<a NAME="E31:35_4_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A624</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<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="E32:35_4_1_1_2_1"/>
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> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E68">A623</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E33:35_4_1_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1</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#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</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="#E24:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E63">A598</font></span></a>, <a class="ref" href="rfinseq.html#T6" title="RFINSEQ:th.6">RFINSEQ:6</a></span>;<br/>
<a NAME="E34:35_4_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A625</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1 <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#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</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="#E31:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E69">A624</font></span></a>, <a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>;<br/>
<span class="lab"><font color="Green" title="E71">A626</font></span>: 2 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 = 
2 <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#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>
<br/>.= 
1

;<br/>
<a NAME="E36:35_4_1_1_2_1"/>
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="c11">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E60">A595</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E37:35_4_1_1_2_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="c11">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="lab"><font color="Green" title="E72">A627</font></span>: <font color="Maroon" title="c11">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 = 
<font color="Maroon" title="c8">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default">1 <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="#E7:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E56">A591</font></span></a>, <a class="ref" href="rfinseq.html#D1" title="RFINSEQ:def.1">RFINSEQ:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c8">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2

;<br/>
<a NAME="E39:35_4_1_1_2_1"/>
<font color="Maroon" title="c11">D</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="E40:35_4_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A628</font></span>: 
<font color="Maroon" title="c11">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#K8" title="INTEGRA1:func.8">upper_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="E41:35_4_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="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E64">A599</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/>
<a NAME="E42:35_4_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A629</font></span>: 
 <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p1">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</span>)</span>,2,<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> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</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> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E71">A626</font></span></a>, <a class="ref" href="finseq_6.html#T117" title="FINSEQ_6:th.117">FINSEQ_6:117</a></span>;<br/>
<a NAME="E43:35_4_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#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</span>)</span>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</span>)</span></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"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c11">D</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E57">A592</font></span></a>, <a class="txt" href="#E28:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E66">A601</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/>
<a NAME="E44:35_4_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A630</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">D3</font>)</span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c11">D</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E74">A629</font></span></a>, <a class="txt" href="#E29:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E67">A602</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</a></span>;<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="c8">D3</font>,1)</span>)</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="c8">D3</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="c8">D3</font>,1)</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/>.= 
<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="c8">D3</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> <font color="Maroon" title="c1">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_4_1"><span class="lab"><font color="Green" title="E50">A584</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">D3</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="#E8:35_4_1"><span class="lab"><font color="Green" title="E50">A584</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_4_1_1_2"><span class="lab"><font color="Green" title="E51">A586</font></span></a></span>
;<br/>
<span class="kw">then </span><font color="Maroon" title="c6">y</font> = 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <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="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c11">D</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E54">A589</font></span></a>, <a class="txt" href="#E34:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E70">A625</font></span></a>, <a class="txt" href="#E1:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E52">A587</font></span></a>, <a class="txt" href="#E44:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E75">A630</font></span></a>, <a class="ref" href="rvsum_1.html#T76" title="RVSUM_1:th.76">RVSUM_1:76</a></span>
<br/>.= 
 <a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c11">D</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></span>
;<br/>
<a NAME="E47:35_4_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K8" title="INTEGRA1:func.8">upper_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="c11">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D10" title="INTEGRA1:def.10">INTEGRA1:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E48:35_4_1_1_2_1"/>
 ex <font color="Olive" title="b1">D</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b1">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#K8" title="INTEGRA1:func.8">upper_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K8" title="INTEGRA1:func.8">upper_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="Olive" title="b1">D</font> &amp; <font color="Olive" title="b1">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="#E1:35_4_1_1_2"><span class="lab"><font color="Green" title="E51">A586</font></span></a>, <a class="txt" href="#E40:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E73">A628</font></span></a>, <a class="txt" href="#E38:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E72">A627</font></span></a>, <a class="txt" href="#E19:35_4_1_1_2_1"><span class="lab"><font color="Green" title="E61">A596</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="E3:35_4_1_1_2"/>
 ex <font color="Olive" title="b1">D</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b1">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#K8" title="INTEGRA1:func.8">upper_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K8" title="INTEGRA1:func.8">upper_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="Olive" title="b1">D</font> &amp; <font color="Olive" title="b1">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> )
 ; <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="c7">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="E11:35_4"/>
<font color="Maroon" title="c7">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#K8" title="INTEGRA1:func.8">upper_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E12:35_4"/><span class="lab"><font color="Green" title="E47">A632</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K8" title="INTEGRA1:func.8">upper_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="c7">D</font>
 <span class="kw">and </span><br/><a NAME="E13:35_4"/><span class="lab"><font color="Green" title="E48">A633</font></span>: 
<font color="Maroon" title="c7">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>
 ;<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="c7">D</font>,$1)</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">p</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">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">set </span><font color="Maroon" title="c10">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">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="E15:35_4"/><span class="lab"><font color="Green" title="E49">A634</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="c7">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>
<a NAME="E16:35_4"/><span class="lab"><font color="Green" title="E50">A635</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="c7">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/>
<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="E18:35_4"/><span class="lab"><font color="Green" title="E51">A636</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="E19:35_4"/><span class="lab"><font color="Green" title="E52">A637</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="c7">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_4"><span class="lab"><font color="Green" title="E49">A634</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="E20:35_4"/><span class="lab"><font color="Green" title="E53">A638</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="35_4_2"><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="E2:35_4_2"/>
 <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="E3:35_4_2"/>
 <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="E4:35_4_2"/>
<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="E5:35_4_2"/><span class="lab"><font color="Green" title="E54">A639</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="#E18:35_4"><span class="lab"><font color="Green" title="E51">A636</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="E7:35_4_2"/><span class="lab"><font color="Green" title="E55">A640</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="c7">D</font>
 <span class="kw">and </span><br/><a NAME="E8:35_4_2"/><span class="lab"><font color="Green" title="E56">A641</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="c7">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="#E7:35"><span class="lab"><font color="Green" title="E38">A565</font></span></a>, <a class="ref" href="integra3.html#T2" target="_self" title="INTEGRA3:th.2">Th2</a></span>;<br/>
<a NAME="E9:35_4_2"/><span class="lab"><font color="Green" title="E57">A642</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">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="c7">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="E10:35_4_2"/><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="#E19:35_4"><span class="lab"><font color="Green" title="E52">A637</font></span></a>, <a class="txt" href="#E5:35_4_2"><span class="lab"><font color="Green" title="E54">A639</font></span></a>, <a class="txt" href="#E7:35_4_2"><span class="lab"><font color="Green" title="E55">A640</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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E12:35_4_2"/>
<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="#E15:35_4"><span class="lab"><font color="Green" title="E49">A634</font></span></a>, <a class="txt" href="#E19:35_4"><span class="lab"><font color="Green" title="E52">A637</font></span></a>, <a class="txt" href="#E7:35_4_2"><span class="lab"><font color="Green" title="E55">A640</font></span></a>, <a class="txt" href="#E8:35_4_2"><span class="lab"><font color="Green" title="E56">A641</font></span></a>, <a class="txt" href="#E9:35_4_2"><span class="lab"><font color="Green" title="E57">A642</font></span></a></span>;<br/>
<a NAME="E13:35_4_2"/><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="#E19:35_4"><span class="lab"><font color="Green" title="E52">A637</font></span></a>, <a class="txt" href="#E5:35_4_2"><span class="lab"><font color="Green" title="E54">A639</font></span></a>, <a class="txt" href="#E7:35_4_2"><span class="lab"><font color="Green" title="E55">A640</font></span></a>, <a class="txt" href="#E9:35_4_2"><span class="lab"><font color="Green" title="E57">A642</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="E14:35_4_2"/>
 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="E22:35_4"/><span class="lab"><font color="Green" title="E54">A643</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="#E20:35_4"><span class="lab"><font color="Green" title="E53">A638</font></span></a>);<br/></span>
<a NAME="E23:35_4"/><span class="lab"><font color="Green" title="E55">A644</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="#E1:35"><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="E24:35_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A645</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="c7">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="#E16:35_4"><span class="lab"><font color="Green" title="E50">A635</font></span></a>, <a class="ref" href="xreal_1.html#T129" title="XREAL_1:th.129">XREAL_1:129</a></span>;<br/>
<a NAME="E25:35_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="c5">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="c7">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="35_4_3"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_4_3_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="E1:35_4_3_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="c5">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="c7">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="c5">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="c7">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="c5">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="c7">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="35_4_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_4_3_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="c5">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="c7">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="c5">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="c7">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="E2:35_4_3_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="c5">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="c7">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="#E22:35_4"><span class="lab"><font color="Green" title="E54">A643</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="35_4_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_4_3_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="c5">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="c7">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="c5">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="c7">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="c5">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="c7">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="E2:35_4_3_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="c5">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="c7">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="#E1:35_4"><span class="lab"><font color="Green" title="E42">A576</font></span></a>, <a class="txt" href="#E24:35_4"><span class="lab"><font color="Green" title="E56">A645</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="E27:35_4"/><span class="lab"><font color="Green" title="E57">A646</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="c5">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="c7">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="#E9:35"><span class="lab"><font color="Green" title="E40">A567</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#K3" title="INTEGRA2:func.3">upper_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#K10" title="INTEGRA1:func.10">upper_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="c4">e</font></span><br/>

<a NAME="E28:35_4"/><span class="lab"><font color="Green" title="E58">A647</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">D</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:35_4"><span class="lab"><font color="Green" title="E47">A632</font></span></a>, <a class="ref" href="integra1.html#D10" title="INTEGRA1:def.10">INTEGRA1:def 10</a></span>;<br/>
<a NAME="E29:35_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#K3" title="INTEGRA2:func.3">upper_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#K10" title="INTEGRA1:func.10">upper_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="c5">e</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_4_4"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:35_4_4"/><span class="lab"><font color="Green" title="E59">A648</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="35_4_4_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:35_4_4_1"/><span class="lab"><font color="Green" title="E60">A649</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="c7">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="c7">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="35_4_4_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="c7">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="c7">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="E1:35_4_4_1_1"/><span class="lab"><font color="Green" title="E61">A650</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="c7">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="c7">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="E2:35_4_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A651</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="35_4_4_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_4_4_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="#E2:35_4_4_1_1"><span class="lab"><font color="Green" title="E62">A651</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="35_4_4_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_4_4_1_1_1_1"/><span class="lab"><font color="Green" title="E63">A652</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="c7">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="E2:35_4_4_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A653</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="c7">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="c7">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="#E1:35_4_4_1_1"><span class="lab"><font color="Green" title="E61">A650</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E3:35_4_4_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="c7">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="#E1:35_4_4_1_1"><span class="lab"><font color="Green" title="E61">A650</font></span></a>, <a class="txt" href="#E1:35_4_4_1_1_1_1"><span class="lab"><font color="Green" title="E63">A652</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E4:35_4_4_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="c7">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="c7">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="#E2:35_4_4_1_1_1_1"><span class="lab"><font color="Green" title="E64">A653</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="E5:35_4_4_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="c7">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="#E13:35_4"><span class="lab"><font color="Green" title="E48">A633</font></span></a>, <a class="txt" href="#E1:35_4_4_1_1_1_1"><span class="lab"><font color="Green" title="E63">A652</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="35_4_4_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_4_4_1_1_1_2"/><span class="lab"><font color="Green" title="E63">A654</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="c7">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="E2:35_4_4_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A655</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="c7">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="c7">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="#E1:35_4_4_1_1"><span class="lab"><font color="Green" title="E61">A650</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E3:35_4_4_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="c7">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="c7">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="#E1:35_4_4_1_1"><span class="lab"><font color="Green" title="E61">A650</font></span></a>, <a class="txt" href="#E1:35_4_4_1_1_1_2"><span class="lab"><font color="Green" title="E63">A654</font></span></a>, <a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></span>;<br/><a NAME="E4:35_4_4_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A656</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="c7">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="c7">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="c7">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="#E2:35_4_4_1_1_1_2"><span class="lab"><font color="Green" title="E64">A655</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><a NAME="E5:35_4_4_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="E6:35_4_4_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A657</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="E7:35_4_4_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="c7">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_4_4_1_1"><span class="lab"><font color="Green" title="E61">A650</font></span></a>, <a class="txt" href="#E1:35_4_4_1_1_1_2"><span class="lab"><font color="Green" title="E63">A654</font></span></a>, <a class="ref" href="integra1.html#T7" title="INTEGRA1:th.7">INTEGRA1:7</a></span>;<br/><a NAME="E8:35_4_4_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">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="c7">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="#E1:35_4_4_1_1"><span class="lab"><font color="Green" title="E61">A650</font></span></a>, <a class="txt" href="#E6:35_4_4_1_1_1_2"><span class="lab"><font color="Green" title="E66">A657</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="E9:35_4_4_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="c7">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="#E4:35_4_4_1_1_1_2"><span class="lab"><font color="Green" title="E65">A656</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="E2:35_4_4_1"/><span class="lab"><font color="Green" title="E61">A658</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="#E22:35_4"><span class="lab"><font color="Green" title="E54">A643</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E3:35_4_4_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="#E22:35_4"><span class="lab"><font color="Green" title="E54">A643</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E4:35_4_4_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="#E2:35_4_4_1"><span class="lab"><font color="Green" title="E61">A658</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E5:35_4_4_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="E6:35_4_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A659</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="E7:35_4_4_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="#E18:35_4"><span class="lab"><font color="Green" title="E51">A636</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="E9:35_4_4_1"/><span class="lab"><font color="Green" title="E63">A660</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="E10:35_4_4_1"/><span class="lab"><font color="Green" title="E64">A661</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="#E6:35_4_4_1"><span class="lab"><font color="Green" title="E62">A659</font></span></a>, <a class="ref" href="partfun1.html#T3" title="PARTFUN1:th.3">PARTFUN1:3</a></span>;<br/>
<a NAME="E11:35_4_4_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="c7">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_4"><span class="lab"><font color="Green" title="E49">A634</font></span></a>, <a class="txt" href="#E9:35_4_4_1"><span class="lab"><font color="Green" title="E63">A660</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="E12:35_4_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A662</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="c7">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="E13:35_4_4_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="c7">D</font>,<font color="Maroon" title="c15">n1</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_4"><span class="lab"><font color="Green" title="E49">A634</font></span></a>, <a class="txt" href="#E9:35_4_4_1"><span class="lab"><font color="Green" title="E63">A660</font></span></a>, <a class="txt" href="#E10:35_4_4_1"><span class="lab"><font color="Green" title="E64">A661</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:35_4_4_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="#E1:35_4_4_1"><span class="lab"><font color="Green" title="E60">A649</font></span></a>, <a class="txt" href="#E12:35_4_4_1"><span class="lab"><font color="Green" title="E65">A662</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="E2:35_4_4"/><span class="lab"><font color="Green" title="E60">A663</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="c7">D</font>)</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_4_4_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:35_4_4_2"/><span class="lab"><font color="Green" title="E61">A664</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="35_4_4_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:35_4_4_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="#E18:35_4"><span class="lab"><font color="Green" title="E51">A636</font></span></a>, <a class="ref" href="rfinseq.html#T3" title="RFINSEQ:th.3">RFINSEQ:3</a></span>;<br/>
<a NAME="E2:35_4_4_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="#E22:35_4"><span class="lab"><font color="Green" title="E54">A643</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="E3:35_4_4_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A665</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="E4:35_4_4_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="#E22:35_4"><span class="lab"><font color="Green" title="E54">A643</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="E5:35_4_4_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="E6:35_4_4_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="#E3:35_4_4_2_1"><span class="lab"><font color="Green" title="E62">A665</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E7:35_4_4_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A666</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="E8:35_4_4_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="E9:35_4_4_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="#E3:35_4_4_2_1"><span class="lab"><font color="Green" title="E62">A665</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="E10:35_4_4_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:35_4"><span class="lab"><font color="Green" title="E54">A643</font></span></a>, <a class="txt" href="#E1:35_4_4"><span class="lab"><font color="Green" title="E59">A648</font></span></a>, <a class="txt" href="#E7:35_4_4_2_1"><span class="lab"><font color="Green" title="E63">A666</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="E2:35_4_4_2"/><span class="lab"><font color="Green" title="E62">A667</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="#E18:35_4"><span class="lab"><font color="Green" title="E51">A636</font></span></a>, <a class="ref" href="classes1.html#T75" title="CLASSES1:th.75">CLASSES1:75</a></span>;<br/>
<a NAME="E3:35_4_4_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> <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="c7">D</font>)</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_4_4_2_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:35_4_4_2_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="c11">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:35_4"><span class="lab"><font color="Green" title="E54">A643</font></span></a>, <a class="txt" href="#E2:35_4_4_2"><span class="lab"><font color="Green" title="E62">A667</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="c15">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="E3:35_4_4_2_2"/><span class="lab"><font color="Green" title="E63">A668</font></span>: 
<font color="Maroon" title="c15">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="E4:35_4_4_2_2"/><span class="lab"><font color="Green" title="E64">A669</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="c15">k2</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="E5:35_4_4_2_2"/><span class="lab"><font color="Green" title="E65">A670</font></span>: 
<font color="Maroon" title="c15">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="c7">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_4"><span class="lab"><font color="Green" title="E49">A634</font></span></a>, <a class="txt" href="#E3:35_4_4_2_2"><span class="lab"><font color="Green" title="E63">A668</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="E6:35_4_4_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A671</font></span>: 
<font color="Maroon" title="c15">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="c7">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="E7:35_4_4_2_2"/>
<font color="Maroon" title="c15">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="c7">D</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_4_4_2_2"><span class="lab"><font color="Green" title="E65">A670</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/>
<a NAME="E8:35_4_4_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A672</font></span>: 
<font color="Maroon" title="c15">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="c7">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="E9:35_4_4_2_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="c7">D</font>,<font color="Maroon" title="c15">k2</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_4"><span class="lab"><font color="Green" title="E49">A634</font></span></a>, <a class="txt" href="#E3:35_4_4_2_2"><span class="lab"><font color="Green" title="E63">A668</font></span></a>, <a class="txt" href="#E4:35_4_4_2_2"><span class="lab"><font color="Green" title="E64">A669</font></span></a></span>;<br/>
<a NAME="E10:35_4_4_2_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="c7">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">k2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_4_4_2_2"><span class="lab"><font color="Green" title="E66">A671</font></span></a>, <a class="ref" href="integra1.html#T20" title="INTEGRA1:th.20">INTEGRA1:20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:35_4_4_2_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> <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="c7">D</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_4_4_2_2"><span class="lab"><font color="Green" title="E67">A672</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="E4:35_4_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A673</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="c7">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="E5:35_4_4_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="c7">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="c7">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="E7:35_4_4_2"/><span class="lab"><font color="Green" title="E64">A674</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="c7">D</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:35_4_4_2"/><span class="lab"><font color="Green" title="E65">A675</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="c7">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="c7">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="E9:35_4_4_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="c7">D</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_4_4_2"><span class="lab"><font color="Green" title="E64">A674</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="E10:35_4_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A676</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="c7">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="E11:35_4_4_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="c7">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="E12:35_4_4_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="c7">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="c7">D</font>,<font color="Maroon" title="c15">m</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_4_4_2"><span class="lab"><font color="Green" title="E65">A675</font></span></a>, <a class="ref" href="integra1.html#T20" title="INTEGRA1:th.20">INTEGRA1:20</a></span>;<br/>
<a NAME="E13:35_4_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A677</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="c7">D</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_4"><span class="lab"><font color="Green" title="E49">A634</font></span></a>, <a class="txt" href="#E19:35_4"><span class="lab"><font color="Green" title="E52">A637</font></span></a>, <a class="txt" href="#E10:35_4_4_2"><span class="lab"><font color="Green" title="E66">A676</font></span></a></span>;<br/>
<a NAME="E14:35_4_4_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="#E15:35_4"><span class="lab"><font color="Green" title="E49">A634</font></span></a>, <a class="txt" href="#E10:35_4_4_2"><span class="lab"><font color="Green" title="E66">A676</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="E15:35_4_4_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="c7">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="#E13:35_4_4_2"><span class="lab"><font color="Green" title="E67">A677</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="E17:35_4_4_2"/><span class="lab"><font color="Green" title="E68">A678</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="E18:35_4_4_2"/><span class="lab"><font color="Green" title="E69">A679</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="c7">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="#E2:35_4_4_2"><span class="lab"><font color="Green" title="E62">A667</font></span></a>, <a class="ref" href="partfun1.html#T3" title="PARTFUN1:th.3">PARTFUN1:3</a></span>;<br/>
<a NAME="E19:35_4_4_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="#E17:35_4_4_2"><span class="lab"><font color="Green" title="E68">A678</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E20:35_4_4_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="c7">D</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:35_4"><span class="lab"><font color="Green" title="E54">A643</font></span></a>, <a class="txt" href="#E1:35_4_4_2"><span class="lab"><font color="Green" title="E61">A664</font></span></a>, <a class="txt" href="#E17:35_4_4_2"><span class="lab"><font color="Green" title="E68">A678</font></span></a>, <a class="txt" href="#E18:35_4_4_2"><span class="lab"><font color="Green" title="E69">A679</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="E21:35_4_4_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="c7">D</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_4_4_2"><span class="lab"><font color="Green" title="E61">A664</font></span></a>, <a class="txt" href="#E4:35_4_4_2"><span class="lab"><font color="Green" title="E63">A673</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="E3:35_4_4"/><span class="lab"><font color="Green" title="E61">A680</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="c5">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="c7">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/>
<span class="kw">set </span><font color="Maroon" title="c15">s</font> =  <a href="integra1.html#K10" title="INTEGRA1:func.10">upper_integral</a> <font color="Maroon" title="c2">f</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">sD</font> =  <a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">D</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#K3" title="INTEGRA2:func.3">upper_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#K10" title="INTEGRA1:func.10">upper_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="c5">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="E5:35_4_4"/><span class="lab"><font color="Green" title="E62">A681</font></span>: 
 <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/>
<span class="kw">consider </span><font color="Maroon" title="c19">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="E7:35_4_4"/><span class="lab"><font color="Green" title="E63">A682</font></span>: 
<font color="Maroon" title="c7">D</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Maroon" title="c19">D2</font>
 <span class="kw">and </span><br/><a NAME="E8:35_4_4"/>
<font color="Maroon" title="c18">D1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Maroon" title="c19">D2</font>
 <span class="kw">and </span><br/><a NAME="E9:35_4_4"/><span class="lab"><font color="Green" title="E64">A683</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c19">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="c7">D</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E10:35_4_4"/>
 <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#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">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#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c19">D2</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E11:35_4_4"/>
 <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#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c18">D1</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c19">D2</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E33">A2</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c20">sD1</font> =  <a href="integra1.html#K6" title="INTEGRA1:func.6">upper_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">set </span><font color="Maroon" title="c21">sD2</font> =  <a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c19">D2</font>);<br/>
<a NAME="E12:35_4_4"/>
 <a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c19">D2</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">D</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E7:35_4_4"><span class="lab"><font color="Green" title="E63">A682</font></span></a>, <a class="ref" href="integra1.html#T45" title="INTEGRA1:th.45">INTEGRA1:45</a></span>;<br/>
<a NAME="E13:35_4_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A684</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_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="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">D</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_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="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c19">D2</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/>
<a NAME="E14:35_4_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">D</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<span class="p4">(<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="p2">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_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"><a href="integra1.html#K10" title="INTEGRA1:func.10">upper_integral</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">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="#E8:35_4"><span class="lab"><font color="Green" title="E46">A580</font></span></a>, <a class="txt" href="#E28:35_4"><span class="lab"><font color="Green" title="E58">A647</font></span></a>, <a class="ref" href="xreal_1.html#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/>
<a NAME="E15:35_4_4"/><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="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<span class="p4">(<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="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">upper_integral</a> <font color="Maroon" title="c2">f</font></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#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">D</font>)</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_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="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
;<br/>
<a NAME="E16:35_4_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_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> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">upper_integral</a> <font color="Maroon" title="c2">f</font></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#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">D</font>)</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#K6" title="INTEGRA1:func.6">upper_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="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">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#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/>
<a NAME="E17:35_4_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A685</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_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="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">upper_integral</a> <font color="Maroon" title="c2">f</font></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#K6" title="INTEGRA1:func.6">upper_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> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">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#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">D</font>)</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="E18:35_4_4"/>
<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="E19:35_4_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A686</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#K8" title="INTEGRA1:func.8">upper_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="E20:35_4_4"/>
<span class="p1">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_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#K6" title="INTEGRA1:func.6">upper_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#D2" title="INTEGRA2:def.2">INTEGRA2:def 2</a></span>;<br/>
<a NAME="E21:35_4_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_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#K8" title="INTEGRA1:func.8">upper_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#D10" title="INTEGRA1:def.10">INTEGRA1:def 10</a></span>;<br/>
<a NAME="E22:35_4_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_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#K8" title="INTEGRA1:func.8">upper_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:35_4_4"><span class="lab"><font color="Green" title="E67">A686</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="E23:35_4_4"/><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"><a href="integra1.html#K8" title="INTEGRA1:func.8">upper_sum_set</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"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_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="#E4:35_4"><span class="lab"><font color="Green" title="E44">A578</font></span></a>, <a class="ref" href="seq_4.html#D2" title="SEQ_4:def.2">SEQ_4:def 2</a></span>;<br/>
<a NAME="E24:35_4_4"/><span class="kw">then </span>
 <a href="integra1.html#K10" title="INTEGRA1:func.10">upper_integral</a> <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_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#D14" title="INTEGRA1:def.14">INTEGRA1:def 14</a></span>;<br/>
<a NAME="E25:35_4_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A687</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> (<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="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">upper_integral</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="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="E26:35_4_4"/>
<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="E27:35_4_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A688</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="c7">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="c7">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/>
<a NAME="E28:35_4_4"/><span class="lab"><font color="Green" title="E70">A689</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="c5">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="c7">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="c5">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="c7">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="E29:35_4_4"/><span class="lab"><font color="Green" title="E71">A690</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#K3" title="INTEGRA2:func.3">upper_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#K10" title="INTEGRA1:func.10">upper_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="c5">e</font></span><br/>

<a NAME="E30:35_4_4"/><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="c5">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="c7">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="#E27:35_4"><span class="lab"><font color="Green" title="E57">A646</font></span></a></span>;<br/>
<a NAME="E31:35_4_4"/><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="c5">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="c7">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="#E28:35_4_4"><span class="lab"><font color="Green" title="E70">A689</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E32:35_4_4"/><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="c7">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="c5">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:35_4"><span class="lab"><font color="Green" title="E50">A635</font></span></a>, <a class="txt" href="#E23:35_4"><span class="lab"><font color="Green" title="E55">A644</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="E33:35_4_4"/><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="c7">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="c5">e</font>
 
;<br/>
<a NAME="E34:35_4_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A691</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="c7">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="c5">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="E35:35_4_4"/>
<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="c5">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="c7">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="#E27:35_4"><span class="lab"><font color="Green" title="E57">A646</font></span></a>, <a class="txt" href="#E29:35_4_4"><span class="lab"><font color="Green" title="E71">A690</font></span></a></span>;<br/>
<a NAME="E36:35_4_4"/><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="#E5:35_4_4"><span class="lab"><font color="Green" title="E62">A681</font></span></a>, <a class="txt" href="#E3:35_4_4"><span class="lab"><font color="Green" title="E61">A680</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E37:35_4_4"/><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="c7">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="c7">D</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c18">D1</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">D3</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="c7">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="#E3:35"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E2:35_4_4"><span class="lab"><font color="Green" title="E60">A663</font></span></a></span>;<br/>
<a NAME="E38:35_4_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A692</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c18">D1</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c19">D2</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="c7">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="#E9:35_4_4"><span class="lab"><font color="Green" title="E64">A683</font></span></a>, <a class="ref" href="integra3.html#T6" target="_self" title="INTEGRA3:th.6">Th6</a></span>;<br/>
<a NAME="E39:35_4_4"/>
 <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="#E27:35_4"><span class="lab"><font color="Green" title="E57">A646</font></span></a>, <a class="txt" href="#E29:35_4_4"><span class="lab"><font color="Green" title="E71">A690</font></span></a></span>;<br/>
<a NAME="E40:35_4_4"/><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="c7">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="c7">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="#E27:35_4_4"><span class="lab"><font color="Green" title="E69">A688</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/>
<a NAME="E41:35_4_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_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="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c19">D2</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="c7">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="#E5:35_4_4"><span class="lab"><font color="Green" title="E62">A681</font></span></a>, <a class="txt" href="#E38:35_4_4"><span class="lab"><font color="Green" title="E73">A692</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E42:35_4_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_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="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">D</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="c7">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="#E13:35_4_4"><span class="lab"><font color="Green" title="E65">A684</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E43:35_4_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_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="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">D</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">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="#E34:35_4_4"><span class="lab"><font color="Green" title="E72">A691</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E44:35_4_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_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> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">D</font>)</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="c5">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="c5">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="c5">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="E45:35_4_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_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="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">upper_integral</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:35_4_4"><span class="lab"><font color="Green" title="E66">A685</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E46:35_4_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> (<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="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">upper_integral</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra2.html#D2" title="INTEGRA2:def.2">INTEGRA2:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E47:35_4_4"/>
 <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#K3" title="INTEGRA2:func.3">upper_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#K10" title="INTEGRA1:func.10">upper_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="c5">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:35_4_4"><span class="lab"><font color="Green" title="E68">A687</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E30:35_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#K3" title="INTEGRA2:func.3">upper_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#K10" title="INTEGRA1:func.10">upper_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="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>
<span class="kw">hence </span><a NAME="E11:35"/>
 <a href="integra2.html#K3" title="INTEGRA2:func.3">upper_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#K3" title="INTEGRA2:func.3">upper_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#K10" title="INTEGRA1:func.10">upper_integral</a> <font color="Maroon" title="c2">f</font></span><br/>

<span class="kw">hence </span><a NAME="E12:35"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_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#K10" title="INTEGRA1:func.10">upper_integral</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:35"><span class="lab"><font color="Green" title="E41">A574</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>
