<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">A</font> be  <a href="integra1.html#V1">closed-interval</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>, <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:26"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">f</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E2:26"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">f</font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E3:26"/>
 <a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font> is <a href="partfun1.html#V1">total</a>
 
<b>by </b><i/>;<br/>
<a NAME="E4:26"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>, <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T131">FUNCT_2:131</a></i>;<br/>
<a NAME="E5:26"/>
<font color="Maroon">f</font> <a href="rfunct_1.html#R1">is_bounded_above_on</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="rfunct_1.html#D11">RFUNCT_1:def 11</a></i>;<br/>
<a NAME="E6:26"/><b>then </b>
(  <a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font> <a href="rfunct_1.html#R1">is_bounded_above_on</a> <font color="Maroon">A</font> &amp;  <a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font> <a href="rfunct_1.html#R2">is_bounded_below_on</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:26"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#D11">RFUNCT_1:def 11</a></i>;<br/>
<a NAME="E8:26"/>
for <font color="Olive">T</font> being  <a href="integra2.html#NM1">DivSequence</a> of <font color="Maroon">A</font>  st  <a href="integra2.html#K2">delta</a> <font color="Olive">T</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Olive">T</font></span>)</span> <a href="hidden.html#R1">=</a> 0 holds <br/><span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span>,<font color="Olive">T</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span>,<font color="Olive">T</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">T</font> be   <a href="integra2.html#NM1">DivSequence</a> of <font color="Maroon">A</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:26_1"/><i><font color="Green">E42</font></i>: 
 <a href="integra2.html#K2">delta</a> <font color="Maroon">T</font> is <a href="seq_2.html#V4">convergent</a>
 <b>and </b><br/><a NAME="E2:26_1"/><i><font color="Green">E43</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E3:26_1"/><i><font color="Green">E44</font></i>: 
(  <a href="integra2.html#K3">upper_sum</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">T</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="integra2.html#K4">lower_sum</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">T</font> is <a href="seq_2.html#V4">convergent</a> )
 
<b>by </b><i>, , , , , </i>;<br/>
<a NAME="E4:26_1"/><i><font color="Green">E48</font></i>: 
(  <a href="integra2.html#K3">upper_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font> is <a href="seq_2.html#V4">convergent</a> )
 
<b>by </b><i>, , , , </i>;<br/>
<a NAME="E5:26_1"/><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, , , , </i>;<br/>
<a NAME="E6:26_1"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p1">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, <a class="ref" href="seq_2.html#T25">SEQ_2:25</a></i>;<br/>
<b>reconsider </b><font color="Maroon">osc</font> = <span class="p1">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p1">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> as   <a href="seq_1.html#NM1">Real_Sequence</a> ;<br/>
<b>reconsider </b><font color="Maroon">osc1</font> = <span class="p1">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p1">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">T</font></span>)</span> as   <a href="seq_1.html#NM1">Real_Sequence</a> ;<br/>
<a NAME="E9:26_1"/><i><font color="Green">E51</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, , <a class="ref" href="seq_2.html#T26">SEQ_2:26</a></i>;<br/>
<a NAME="E10:26_1"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">a</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">a</font> holds <br/> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">osc1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:26_1_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font>
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:26_1_1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">osc</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="integra4.html#T1" target="_self">Th1</a>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>
<a NAME="E4:26_1_1"/>
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">osc1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:26_1_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E2:26_1_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">osc</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="txt" href="integra4.html#E1:26"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> as    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font> ;<br/>
<a NAME="E4:26_1_1_1"/>
<font color="Maroon">osc</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><a href="seq_1.html#K17">-</a> <span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T15">SEQ_1:15</a></i>;<br/>
<b>then </b><i><font color="Green">E55</font></i>: <font color="Maroon">osc</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K17">-</a> <span class="p3">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T14">SEQ_1:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K6">upper_sum</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="integra2.html#D4">INTEGRA2:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K6">upper_sum</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra2.html#D5">INTEGRA2:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D9">INTEGRA1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D10">INTEGRA1:def 10</a></i>
;<br/>
<a NAME="E6:26_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">UV</font> =  <a href="integra1.html#K14">upper_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="catalg_1.html#T5">CATALG_1:5</a></i>;<br/>
<a NAME="E8:26_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">LV</font> =  <a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="catalg_1.html#T5">CATALG_1:5</a></i>;<br/>
<a NAME="E10:26_1_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">osc</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">UV</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E4:26"><i><font color="Green">E40</font></i></a>, <a class="ref" href="rvsum_1.html#T120">RVSUM_1:120</a></i>;<br/>
<a NAME="E11:26_1_1_1"/>
<font color="Maroon">osc1</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><a href="seq_1.html#K17">-</a> <span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">T</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T15">SEQ_1:15</a></i>;<br/>
<b>then </b><i><font color="Green">E57</font></i>: <font color="Maroon">osc1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K17">-</a> <span class="p3">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <span class="p4">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <span class="p4">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T14">SEQ_1:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K6">upper_sum</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="integra2.html#D4">INTEGRA2:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K6">upper_sum</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra2.html#D5">INTEGRA2:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D9">INTEGRA1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">D</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D10">INTEGRA1:def 10</a></i>
;<br/>
<a NAME="E13:26_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">UV1</font> =  <a href="integra1.html#K14">upper_volume</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">D</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="catalg_1.html#T5">CATALG_1:5</a></i>;<br/>
<a NAME="E15:26_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">LV1</font> =  <a href="integra1.html#K15">lower_volume</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">D</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="catalg_1.html#T5">CATALG_1:5</a></i>;<br/>
<a NAME="E17:26_1_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">osc1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">UV1</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E1:26_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="rvsum_1.html#T120">RVSUM_1:120</a></i>;<br/>
<a NAME="E18:26_1_1_1"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">UV1</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">UV</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:26_1_1_1_3"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">x</font> = <span class="p1">(<span class="default"><font color="Maroon">UV1</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>;<br/>
<b>set </b><font color="Maroon">y</font> = <span class="p1">(<span class="default"><font color="Maroon">UV</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>;<br/>
<a NAME="E2:26_1_1_1_3"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">UV1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E3:26_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/>
<a NAME="E3:26_1_1_1_3"/>
<font color="Maroon">LV1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E3:26_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<b>then </b><i><font color="Green">E61</font></i>: <span class="p1">(<span class="default"><font color="Maroon">UV1</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra4.html#T2" target="_self">Th2</a>, <a class="ref" href="rvsum_1.html#T48">RVSUM_1:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>

;<br/>
<a NAME="E5:26_1_1_1_3"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">UV</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E3:26_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/>
<a NAME="E6:26_1_1_1_3"/>
<font color="Maroon">LV</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E3:26_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<b>then </b><i><font color="Green">E63</font></i>: <span class="p1">(<span class="default"><font color="Maroon">UV</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="rvsum_1.html#T48">RVSUM_1:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>

;<br/>
<a NAME="E8:26_1_1_1_3"/><i><font color="Green">E64</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_3_3"><b>proof </b></a><div class="add">

<a NAME="E1:26_1_1_1_3_3"/>
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> is <a href="seq_4.html#V3">bounded</a> &amp;  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_3_3_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E2:26_1_1_1_3_3_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/>
<a NAME="E3:26_1_1_1_3_3_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="txt" href="integra4.html#E3:26_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:26_1_1_1_3_3_1"/><b>then </b>
 <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="integra1.html#T10">INTEGRA1:10</a></i>;<br/>
<a NAME="E5:26_1_1_1_3_3_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="integra4.html#E6:26_1"><i><font color="Green">E50</font></i></a></i>;<br/>
<a NAME="E6:26_1_1_1_3_3_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:26_1_1_1_3_3_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E6:26_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="funct_1.html#T73">FUNCT_1:73</a></i>;<br/>
<a NAME="E8:26_1_1_1_3_3_1"/><i><font color="Green">E67</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T99">RELAT_1:99</a></i>;<br/>
<a NAME="E9:26_1_1_1_3_3_1"/>
( <font color="Maroon">f</font> <a href="rfunct_1.html#R1">is_bounded_above_on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">f</font> <a href="rfunct_1.html#R2">is_bounded_below_on</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, <a class="ref" href="rfunct_1.html#D11">RFUNCT_1:def 11</a></i>;<br/>
<a NAME="E10:26_1_1_1_3_3_1"/><b>then </b>
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> is <a href="seq_4.html#V1">bounded_above</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> is <a href="seq_4.html#V2">bounded_below</a> )
 
<b>by </b><i><a class="ref" href="integra1.html#T13">INTEGRA1:13</a>, <a class="ref" href="integra1.html#T15">INTEGRA1:15</a></i>;<br/>
<a NAME="E11:26_1_1_1_3_3_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> is <a href="seq_4.html#V3">bounded</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#D3">SEQ_4:def 3</a></i>;<br/>
<b>hence </b><a NAME="E12:26_1_1_1_3_3_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> is <a href="seq_4.html#V3">bounded</a>
 <b>by </b><i><a class="txt" href="integra4.html#E10:26_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="rcomp_1.html#T5">RCOMP_1:5</a></i>;<br/>

<b>thus </b><a NAME="E13:26_1_1_1_3_3_1"/>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra4.html#E9:26_1"><i><font color="Green">E51</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:26_1_1_1_3_3"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="seq_4.html#T24">SEQ_4:24</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:26_1_1_1_3"/><i><font color="Green">E89</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_3_4"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">M</font> =  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">m</font> =  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">M1</font> =  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">m1</font> =  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E1:26_1_1_1_3_4"/><i><font color="Green">E90</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E2:26_1_1_1_3_4"/><i><font color="Green">E91</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:26_1_1_1_3_4"/><i><font color="Green">E92</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E9:26_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:26_1_1_1_3_4"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="txt" href="integra4.html#E3:26_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:26_1_1_1_3_4"/><b>then </b><i><font color="Green">E93</font></i>: 
 <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="integra1.html#T10">INTEGRA1:10</a></i>;<br/>
<a NAME="E6:26_1_1_1_3_4"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="integra4.html#E3:26_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E7:26_1_1_1_3_4"/><b>then </b><i><font color="Green">E94</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/>
<a NAME="E8:26_1_1_1_3_4"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">f</font> <a href="rfunct_1.html#R1">is_bounded_above_on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">f</font> <a href="rfunct_1.html#R2">is_bounded_below_on</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, <a class="ref" href="rfunct_1.html#D11">RFUNCT_1:def 11</a></i>;<br/>
<a NAME="E9:26_1_1_1_3_4"/><b>then </b><i><font color="Green">E96</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> is <a href="seq_4.html#V1">bounded_above</a> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> is <a href="seq_4.html#V2">bounded_below</a> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:26_1_1_1_3_4"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_3.html#D10">RFUNCT_3:def 10</a></i>;<br/>
<a NAME="E11:26_1_1_1_3_4"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E10:26_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E12:26_1_1_1_3_4"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="integra4.html#E2:26_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E13:26_1_1_1_3_4"/><b>then </b><i><font color="Green">E97</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/>
<a NAME="E14:26_1_1_1_3_4"/>
(  <a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font> <a href="rfunct_1.html#R1">is_bounded_above_on</a> <font color="Maroon">A</font> &amp;  <a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font> <a href="rfunct_1.html#R2">is_bounded_below_on</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="txt" href="integra4.html#E10:26_1_1_1"><i><font color="Green">E56</font></i></a>, , </i>;<br/>
<a NAME="E15:26_1_1_1_3_4"/><b>then </b><i><font color="Green">E98</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> is <a href="seq_4.html#V1">bounded_above</a> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> is <a href="seq_4.html#V2">bounded_below</a> )
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_3_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_3_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1_1_1_3_4_1_1"/>
( (  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) or (  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 ) or (  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp;  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 ) )
 </span><b>by </b><i><a class="txt" href="integra4.html#E5:26_1"><i><font color="Green">E49</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_1"><b>suppose </b></a><a NAME="E1:26_1_1_1_3_4_1_1_1"/><i><font color="Green">E99</font></i>: 
(  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 ;<br/><div class="add"><a NAME="E2:26_1_1_1_3_4_1_1_1"/>
(  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:26_1_1_1_3_4_1_1_1_1"/>
( ( for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> holds <br/><font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> ) &amp; ( for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> holds <br/> ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:26_1_1_1_3_4_1_1_1_1_1"/><i><font color="Green">E101</font></i>: 
for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> holds <br/><font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:26_1_1_1_3_4_1_1_1_1_1_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:26_1_1_1_3_4_1_1_1_1_1_1"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E4:26_1_1_1_3_4_1_1_1_1_1_1"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="integra4.html#E5:26_1_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="ref" href="rfunct_3.html#T47">RFUNCT_3:47</a></i>;<br/>
<a NAME="E5:26_1_1_1_3_4_1_1_1_1_1_1"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E5:26_1_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="ref" href="rfunct_3.html#T47">RFUNCT_3:47</a></i>;<br/>
<a NAME="E6:26_1_1_1_3_4_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_3.html#D10">RFUNCT_3:def 10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1_1_1_3_4_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span><b>by </b><i><a class="txt" href="integra4.html#E7:26_1_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="rfunct_3.html#T40">RFUNCT_3:40</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:26_1_1_1_3_4_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:26_1_1_1_3_4_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra4.html#E2:26_1_1_1_3"><i><font color="Green">E60</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:26_1_1_1_3_4_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:26_1_1_1_3_4_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="rfunct_3.html#K2">max+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra4.html#E7:26_1_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="txt" href="integra4.html#E8:26_1_1_1_3"><i><font color="Green">E64</font></i></a>, <a class="ref" href="rfunct_3.html#D10">RFUNCT_3:def 10</a></i>;<br/><a NAME="E3:26_1_1_1_3_4_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E107</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>,0
 <b>by </b><i><a class="ref" href="rfunct_3.html#D1">RFUNCT_3:def 1</a></i>;<br/><a NAME="E4:26_1_1_1_3_4_1_1_1_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>;<br/><a NAME="E5:26_1_1_1_3_4_1_1_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>;<br/><a NAME="E6:26_1_1_1_3_4_1_1_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E7:26_1_1_1_3_4_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra4.html#E12:26_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:26_1_1_1_3_4_1_1_1_1_1_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:26_1_1_1_3_4_1_1_1_1_1"/>
for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> holds <br/> ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:26_1_1_1_3_4_1_1_1_1_1_2"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font>
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:26_1_1_1_3_4_1_1_1_1_1_2"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> )
 <b>by </b><i><a class="txt" href="integra4.html#E5:26_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="integra4.html#E12:26_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>
<a NAME="E4:26_1_1_1_3_4_1_1_1_1_1_2"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E2:26_1_1_1_3_4_1_1_1_1_1_2_1"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="txt" href="integra4.html#E5:26_1_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E3:26_1_1_1_3_4_1_1_1_1_1_2_1"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E12:26_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="integra4.html#E5:26_1_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r1</font> = <font color="Maroon">r</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E5:26_1_1_1_3_4_1_1_1_1_1_2_1"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E7:26_1_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="rfunct_3.html#D10">RFUNCT_3:def 10</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
 <a href="rfunct_3.html#K2">max+</a> <font color="Maroon">r1</font>
<b>by </b><i><a class="txt" href="integra4.html#E7:26_1_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="rfunct_3.html#D10">RFUNCT_3:def 10</a></i>
<br/>.= 
 <a href="xxreal_0.html#K4">max</a> <font color="Maroon">r</font>,0
<b>by </b><i><a class="ref" href="rfunct_3.html#D1">RFUNCT_3:def 1</a></i>
<br/>.= 
<font color="Maroon">r</font>
<b>by </b><i><a class="txt" href="integra4.html#E2:26_1_1_1_3"><i><font color="Green">E60</font></i></a>, <a class="txt" href="integra4.html#E8:26_1_1_1_3"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>
;<br/>
<a NAME="E7:26_1_1_1_3_4_1_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E8:26_1_1_1_3_4_1_1_1_1_1_2_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rfunct_3.html#T47">RFUNCT_3:47</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:26_1_1_1_3_4_1_1_1_1_1_2"/>
 ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> )
 <b>by </b><i><a class="txt" href="integra4.html#E5:26_1_1_1_3"><i><font color="Green">E62</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:26_1_1_1_3_4_1_1_1_1_1"/>
( ( for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> holds <br/><font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> ) &amp; ( for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> holds <br/> ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> ) ) )
 <b>by </b><i><a class="txt" href="integra4.html#E4:26_1_1_1_3"><i><font color="Green">E61</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:26_1_1_1_3_4_1_1_1_1"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra4.html#E17:26_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="integra4.html#E1:26_1_1_1_3"><i><font color="Green">E59</font></i></a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>

<a NAME="E3:26_1_1_1_3_4_1_1_1_1"/>
( ( for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> holds <br/> <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> ) &amp; ( for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> holds <br/> ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Olive">s</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:26_1_1_1_3_4_1_1_1_1_2"/><i><font color="Green">E136</font></i>: 
for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> holds <br/> <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:26_1_1_1_3_4_1_1_1_1_2_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:26_1_1_1_3_4_1_1_1_1_2_1"/><i><font color="Green">E137</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E4:26_1_1_1_3_4_1_1_1_1_2_1"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E5:26_1_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="ref" href="rfunct_3.html#T47">RFUNCT_3:47</a></i>;<br/>
<i><font color="Green">E139</font></i>: <font color="Maroon">r</font> = 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="integra4.html#E5:26_1_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="ref" href="rfunct_3.html#T47">RFUNCT_3:47</a></i>
<br/>.= 
 <a href="rfunct_3.html#K2">max+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="integra4.html#E7:26_1_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="rfunct_3.html#D10">RFUNCT_3:def 10</a></i>
<br/>.= 
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>,0
<b>by </b><i><a class="ref" href="rfunct_3.html#D1">RFUNCT_3:def 1</a></i>
;<br/>
<a NAME="E6:26_1_1_1_3_4_1_1_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E5:26_1_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="ref" href="rfunct_3.html#T47">RFUNCT_3:47</a></i>;<br/>
<a NAME="E7:26_1_1_1_3_4_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_3.html#D10">RFUNCT_3:def 10</a></i>;<br/>
<a NAME="E8:26_1_1_1_3_4_1_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E9:26_1_1_1_3_4_1_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E12:26_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>
<b>hence </b><a NAME="E10:26_1_1_1_3_4_1_1_1_1_2_1"/>
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="integra4.html#E2:26_1_1_1_3"><i><font color="Green">E60</font></i></a>, <a class="txt" href="integra4.html#E8:26_1_1_1_3"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:26_1_1_1_3_4_1_1_1_1_2"/>
for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> holds <br/> ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Olive">s</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:26_1_1_1_3_4_1_1_1_1_2_2"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font>
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:26_1_1_1_3_4_1_1_1_1_2_2"/><i><font color="Green">E140</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font> )
 <b>by </b><i><a class="txt" href="integra4.html#E5:26_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="integra4.html#E12:26_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E5:26_1_1_1_3_4_1_1_1_1_2_2"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_1_1_2_2_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E2:26_1_1_1_3_4_1_1_1_1_2_2_1"/><i><font color="Green">E141</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="txt" href="integra4.html#E5:26_1_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E3:26_1_1_1_3_4_1_1_1_1_2_2_1"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E12:26_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="integra4.html#E5:26_1_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>
<a NAME="E4:26_1_1_1_3_4_1_1_1_1_2_2_1"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E7:26_1_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="rfunct_3.html#D10">RFUNCT_3:def 10</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
 <a href="rfunct_3.html#K2">max+</a> <font color="Maroon">r</font>
<b>by </b><i><a class="txt" href="integra4.html#E7:26_1_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="rfunct_3.html#D10">RFUNCT_3:def 10</a></i>
<br/>.= 
 <a href="xxreal_0.html#K4">max</a> <font color="Maroon">r</font>,0
<b>by </b><i><a class="ref" href="rfunct_3.html#D1">RFUNCT_3:def 1</a></i>
<br/>.= 
<font color="Maroon">r</font>
<b>by </b><i><a class="txt" href="integra4.html#E2:26_1_1_1_3"><i><font color="Green">E60</font></i></a>, <a class="txt" href="integra4.html#E8:26_1_1_1_3"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>
;<br/>
<a NAME="E6:26_1_1_1_3_4_1_1_1_1_2_2_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E7:26_1_1_1_3_4_1_1_1_1_2_2_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rfunct_3.html#T47">RFUNCT_3:47</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:26_1_1_1_3_4_1_1_1_1_2_2"/>
 ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font> )
 <b>by </b><i><a class="txt" href="integra4.html#E5:26_1_1_1_3"><i><font color="Green">E62</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:26_1_1_1_3_4_1_1_1_1_2"/>
( ( for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> holds <br/> <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> ) &amp; ( for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> holds <br/> ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Olive">s</font> ) ) )
 <b>by </b><i><a class="txt" href="integra4.html#E4:26_1_1_1_3"><i><font color="Green">E61</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:26_1_1_1_3_4_1_1_1_1"/>
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra4.html#E17:26_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="integra4.html#E1:26_1_1_1_3"><i><font color="Green">E59</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:26_1_1_1_3_4_1_1_1"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_2"><b>suppose </b></a><a NAME="E1:26_1_1_1_3_4_1_1_2"/><i><font color="Green">E144</font></i>: 
(  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 ;<br/><div class="add"><a NAME="E2:26_1_1_1_3_4_1_1_2"/>
(  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:26_1_1_1_3_4_1_1_2_1"/>
( ( for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> holds <br/><font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> ) &amp; ( for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> holds <br/> ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:26_1_1_1_3_4_1_1_2_1_1"/><i><font color="Green">E145</font></i>: 
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> holds <br/><font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:26_1_1_1_3_4_1_1_2_1_1_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:26_1_1_1_3_4_1_1_2_1_1_1"/><i><font color="Green">E146</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E4:26_1_1_1_3_4_1_1_2_1_1_1"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="integra4.html#E5:26_1_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="ref" href="rfunct_3.html#T47">RFUNCT_3:47</a></i>;<br/>
<a NAME="E5:26_1_1_1_3_4_1_1_2_1_1_1"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E5:26_1_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="ref" href="rfunct_3.html#T47">RFUNCT_3:47</a></i>;<br/>
<a NAME="E6:26_1_1_1_3_4_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E149</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_3.html#D10">RFUNCT_3:def 10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1_1_1_3_4_1_1_2_1_1_1_1_1"/>
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span><b>by </b><i><a class="txt" href="integra4.html#E7:26_1_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="rfunct_3.html#T40">RFUNCT_3:40</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:26_1_1_1_3_4_1_1_2_1_1_1_1_1_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:26_1_1_1_3_4_1_1_2_1_1_1_1_1_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra4.html#E2:26_1_1_1_3"><i><font color="Green">E60</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:26_1_1_1_3_4_1_1_2_1_1_1_1_1_2"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:26_1_1_1_3_4_1_1_2_1_1_1_1_1_2"/>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="rfunct_3.html#K2">max+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra4.html#E7:26_1_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="txt" href="integra4.html#E8:26_1_1_1_3"><i><font color="Green">E64</font></i></a>, <a class="ref" href="rfunct_3.html#D10">RFUNCT_3:def 10</a></i>;<br/><a NAME="E3:26_1_1_1_3_4_1_1_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E151</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>,0
 <b>by </b><i><a class="ref" href="rfunct_3.html#D1">RFUNCT_3:def 1</a></i>;<br/><a NAME="E4:26_1_1_1_3_4_1_1_2_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>;<br/><a NAME="E5:26_1_1_1_3_4_1_1_2_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>;<br/><a NAME="E6:26_1_1_1_3_4_1_1_2_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E7:26_1_1_1_3_4_1_1_2_1_1_1_1_1_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra4.html#E12:26_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:26_1_1_1_3_4_1_1_2_1_1_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:26_1_1_1_3_4_1_1_2_1_1"/>
for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> holds <br/> ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_2_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:26_1_1_1_3_4_1_1_2_1_1_2"/>
 ex <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; ( for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   holds <br/>(  not <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> or  not <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> ) ) )
 ;<br/>

<b>then consider </b><font color="Maroon">s</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:26_1_1_1_3_4_1_1_2_1_1_2"/><i><font color="Green">E152</font></i>: 
( <font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">r</font> ) )
 ;<br/>
<b>consider </b><font color="Maroon">r1</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E5:26_1_1_1_3_4_1_1_2_1_1_2"/><i><font color="Green">E153</font></i>: 
( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> )
 <b>by </b><i><a class="txt" href="integra4.html#E5:26_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="integra4.html#E12:26_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="integra4.html#E5:26_1_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E7:26_1_1_1_3_4_1_1_2_1_1_2"/><i><font color="Green">E154</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="txt" href="integra4.html#E7:26_1_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E8:26_1_1_1_3_4_1_1_2_1_1_2"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E8:26_1_1_1_3"><i><font color="Green">E64</font></i></a>, <a class="ref" href="rfunct_3.html#D10">RFUNCT_3:def 10</a></i>;<br/>
<a NAME="E9:26_1_1_1_3_4_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="rfunct_3.html#T40">RFUNCT_3:40</a></i>;<br/>
<a NAME="E10:26_1_1_1_3_4_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E156</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="rfunct_3.html#T47">RFUNCT_3:47</a></i>;<br/>
<a NAME="E11:26_1_1_1_3_4_1_1_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="rfunct_3.html#T47">RFUNCT_3:47</a></i>;<br/>
<a NAME="E12:26_1_1_1_3_4_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E157</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E13:26_1_1_1_3_4_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E158</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="integra4.html#E5:26_1_1_1_3"><i><font color="Green">E62</font></i></a>, </i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="rfunct_3.html#T47">RFUNCT_3:47</a></i>
<br/>.= 
 <a href="rfunct_3.html#K2">max+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rfunct_3.html#D10">RFUNCT_3:def 10</a></i>
<br/>.= 
 <a href="xxreal_0.html#K4">max</a> <font color="Maroon">r1</font>,0
<b>by </b><i><a class="txt" href="integra4.html#E8:26_1_1_1_3"><i><font color="Green">E64</font></i></a>, <a class="ref" href="rfunct_3.html#D1">RFUNCT_3:def 1</a></i>
<br/>.= 
<font color="Maroon">r1</font>
<b>by </b><i><a class="txt" href="integra4.html#E7:26_1_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="integra4.html#T3" target="_self">Th3</a>, <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>
;<br/>
<b>hence </b><a NAME="E15:26_1_1_1_3_4_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="integra4.html#E5:26_1_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="integra4.html#E7:26_1_1_1_3"><i><font color="Green">E63</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:26_1_1_1_3_4_1_1_2_1_1"/>
( ( for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> holds <br/><font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> ) &amp; ( for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> holds <br/> ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> ) ) )
 <b>by </b><i><a class="txt" href="integra4.html#E4:26_1_1_1_3"><i><font color="Green">E61</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:26_1_1_1_3_4_1_1_2_1"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra4.html#E17:26_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="integra4.html#E1:26_1_1_1_3"><i><font color="Green">E59</font></i></a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>

<a NAME="E3:26_1_1_1_3_4_1_1_2_1"/>
( ( for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> ) &amp; ( for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> holds <br/> ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="xcmplx_0.html#K2">+</a> <font color="Olive">s</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_2_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:26_1_1_1_3_4_1_1_2_1_2"/><i><font color="Green">E159</font></i>: 
for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_2_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:26_1_1_1_3_4_1_1_2_1_2_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:26_1_1_1_3_4_1_1_2_1_2_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_3.html#T47">RFUNCT_3:47</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E4:26_1_1_1_3_4_1_1_2_1_2_1"/><i><font color="Green">E160</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<b>thus </b><a NAME="E5:26_1_1_1_3_4_1_1_2_1_2_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="integra4.html#E5:26_1_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="ref" href="rfunct_3.html#T40">RFUNCT_3:40</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:26_1_1_1_3_4_1_1_2_1_2"/>
for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> holds <br/> ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="xcmplx_0.html#K2">+</a> <font color="Olive">s</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_2_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:26_1_1_1_3_4_1_1_2_1_2_2"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:26_1_1_1_3_4_1_1_2_1_2_2"/><i><font color="Green">E162</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font> )
 <b>by </b><i><a class="txt" href="integra4.html#E5:26_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="integra4.html#E12:26_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>
<a NAME="E4:26_1_1_1_3_4_1_1_2_1_2_2"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E7:26_1_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E6:26_1_1_1_3_4_1_1_2_1_2_2"/><i><font color="Green">E164</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="txt" href="integra4.html#E7:26_1_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E7:26_1_1_1_3_4_1_1_2_1_2_2"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="rfunct_3.html#D10">RFUNCT_3:def 10</a></i>;<br/>
<a NAME="E8:26_1_1_1_3_4_1_1_2_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E9:26_1_1_1_3_4_1_1_2_1_2_2"/><b>then </b><i><font color="Green">E166</font></i>: 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_3.html#T47">RFUNCT_3:47</a></i>;<br/>
<i><font color="Green">E167</font></i>: <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
 <a href="rfunct_3.html#K2">max+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rfunct_3.html#D10">RFUNCT_3:def 10</a></i>
<br/>.= 
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>,0
<b>by </b><i><a class="ref" href="rfunct_3.html#D1">RFUNCT_3:def 1</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_2_1_2_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_2_1_2_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1_1_1_3_4_1_1_2_1_2_2_2_1"/>
( <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or <font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_2_1_2_2_2_1_1"><b>suppose </b></a><a NAME="E1:26_1_1_1_3_4_1_1_2_1_2_2_2_1_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:26_1_1_1_3_4_1_1_2_1_2_2_2_1_1"/><b>then </b>
0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="integra4.html#T3" target="_self">Th3</a>, <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>;<br/><b>hence </b><a NAME="E3:26_1_1_1_3_4_1_1_2_1_2_2_2_1_1"/>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font> )
 <b>by </b><i><a class="txt" href="integra4.html#E5:26_1_1_1_3"><i><font color="Green">E62</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_2_1_2_2_2_1_2"><b>suppose </b></a><a NAME="E1:26_1_1_1_3_4_1_1_2_1_2_2_2_1_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:26_1_1_1_3_4_1_1_2_1_2_2_2_1_2"/><b>then </b>
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font> )
 <b>by </b><i><a class="txt" href="integra4.html#E2:26_1_1_1_3"><i><font color="Green">E60</font></i></a>, <a class="txt" href="integra4.html#E8:26_1_1_1_3"><i><font color="Green">E64</font></i></a>, , , <a class="ref" href="integra4.html#T3" target="_self">Th3</a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a>, <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>;<br/><b>hence </b><a NAME="E3:26_1_1_1_3_4_1_1_2_1_2_2_2_1_2"/>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E12:26_1_1_1_3_4_1_1_2_1_2_2"/>
 ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:26_1_1_1_3_4_1_1_2_1_2"/>
( ( for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> ) &amp; ( for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> holds <br/> ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="xcmplx_0.html#K2">+</a> <font color="Olive">s</font> ) ) )
 <b>by </b><i><a class="txt" href="integra4.html#E4:26_1_1_1_3"><i><font color="Green">E61</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:26_1_1_1_3_4_1_1_2_1"/>
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="integra4.html#E17:26_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="integra4.html#E1:26_1_1_1_3"><i><font color="Green">E59</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:26_1_1_1_3_4_1_1_2"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra4.html#E2:26_1_1_1_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_3"><b>suppose </b></a><a NAME="E1:26_1_1_1_3_4_1_1_3"/><i><font color="Green">E168</font></i>: 
(  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp;  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 ;<br/><div class="add"><a NAME="E2:26_1_1_1_3_4_1_1_3"/>
(  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:26_1_1_1_3_4_1_1_3_1"/>
( ( for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> holds <br/><font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 ) &amp; ( for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> holds <br/> ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; 0 <a href="xcmplx_0.html#K6">-</a> <font color="Olive">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_3_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:26_1_1_1_3_4_1_1_3_1_1"/><i><font color="Green">E169</font></i>: 
for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> holds <br/><font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_3_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:26_1_1_1_3_4_1_1_3_1_1_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:26_1_1_1_3_4_1_1_3_1_1_1"/><i><font color="Green">E170</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E4:26_1_1_1_3_4_1_1_3_1_1_1"/><i><font color="Green">E171</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="integra4.html#E5:26_1_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="ref" href="rfunct_3.html#T47">RFUNCT_3:47</a></i>;<br/>
<a NAME="E5:26_1_1_1_3_4_1_1_3_1_1_1"/><i><font color="Green">E172</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E5:26_1_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="ref" href="rfunct_3.html#T47">RFUNCT_3:47</a></i>;<br/>
<a NAME="E6:26_1_1_1_3_4_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_3.html#D10">RFUNCT_3:def 10</a></i>;<br/>
<a NAME="E7:26_1_1_1_3_4_1_1_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:26_1_1_1_3_4_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E173</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E12:26_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>
<font color="Maroon">r</font> = 
 <a href="rfunct_3.html#K2">max+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="integra4.html#E7:26_1_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="txt" href="integra4.html#E8:26_1_1_1_3"><i><font color="Green">E64</font></i></a>, <a class="ref" href="rfunct_3.html#D10">RFUNCT_3:def 10</a></i>
<br/>.= 
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>,0
<b>by </b><i><a class="ref" href="rfunct_3.html#D1">RFUNCT_3:def 1</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="integra4.html#E2:26_1_1_1_3"><i><font color="Green">E60</font></i></a>, , <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>
;<br/>
<b>hence </b><a NAME="E10:26_1_1_1_3_4_1_1_3_1_1_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:26_1_1_1_3_4_1_1_3_1_1"/>
for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> holds <br/> ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; 0 <a href="xcmplx_0.html#K6">-</a> <font color="Olive">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_3_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:26_1_1_1_3_4_1_1_3_1_1_2"/>
<font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:26_1_1_1_3_4_1_1_3_1_1_2"/><b>then </b>
0 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E3:26_1_1_1_3_4_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E174</font></i>: 
0 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<b>consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:26_1_1_1_3_4_1_1_3_1_1_2"/><i><font color="Green">E175</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra4.html#E17:26_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E7:26_1_1_1_3_4_1_1_3_1_1_2"/><i><font color="Green">E176</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="txt" href="integra4.html#E7:26_1_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E8:26_1_1_1_3_4_1_1_3_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> )
 
<b>by </b><i><a class="txt" href="integra4.html#E8:26_1_1_1_3"><i><font color="Green">E64</font></i></a>, <a class="ref" href="rfunct_3.html#T47">RFUNCT_3:47</a></i>;<br/>
<a NAME="E9:26_1_1_1_3_4_1_1_3_1_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> )
 
<b>by </b><i><a class="ref" href="rfunct_3.html#D10">RFUNCT_3:def 10</a></i>;<br/>
<a NAME="E10:26_1_1_1_3_4_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="rfunct_3.html#T40">RFUNCT_3:40</a></i>;<br/>
<b>hence </b><a NAME="E11:26_1_1_1_3_4_1_1_3_1_1_2"/>
 ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; 0 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> )
 <b>by </b><i><a class="txt" href="integra4.html#E5:26_1_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="integra4.html#E7:26_1_1_1_3"><i><font color="Green">E63</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:26_1_1_1_3_4_1_1_3_1_1"/>
( ( for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> holds <br/><font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 ) &amp; ( for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> holds <br/> ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; 0 <a href="xcmplx_0.html#K6">-</a> <font color="Olive">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> ) ) )
 <b>by </b><i><a class="txt" href="integra4.html#E4:26_1_1_1_3"><i><font color="Green">E61</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:26_1_1_1_3_4_1_1_3_1"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="integra4.html#E17:26_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="integra4.html#E1:26_1_1_1_3"><i><font color="Green">E59</font></i></a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>

<a NAME="E3:26_1_1_1_3_4_1_1_3_1"/>
( ( for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> ) &amp; ( for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> holds <br/> ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="xcmplx_0.html#K2">+</a> <font color="Olive">s</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_3_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:26_1_1_1_3_4_1_1_3_1_2"/><i><font color="Green">E177</font></i>: 
for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_3_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:26_1_1_1_3_4_1_1_3_1_2_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:26_1_1_1_3_4_1_1_3_1_2_1"/><i><font color="Green">E178</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E4:26_1_1_1_3_4_1_1_3_1_2_1"/><i><font color="Green">E179</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="integra4.html#E5:26_1_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="ref" href="rfunct_3.html#T47">RFUNCT_3:47</a></i>;<br/>
<b>thus </b><a NAME="E5:26_1_1_1_3_4_1_1_3_1_2_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="integra4.html#E7:26_1_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="rfunct_3.html#T40">RFUNCT_3:40</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:26_1_1_1_3_4_1_1_3_1_2"/>
for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> holds <br/> ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="xcmplx_0.html#K2">+</a> <font color="Olive">s</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_3_4_1_1_3_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:26_1_1_1_3_4_1_1_3_1_2_2"/><i><font color="Green">E180</font></i>: 
<font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:26_1_1_1_3_4_1_1_3_1_2_2"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra4.html#E17:26_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E5:26_1_1_1_3_4_1_1_3_1_2_2"/><i><font color="Green">E182</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="txt" href="integra4.html#E7:26_1_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E6:26_1_1_1_3_4_1_1_3_1_2_2"/><i><font color="Green">E183</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> )
 
<b>by </b><i><a class="txt" href="integra4.html#E8:26_1_1_1_3"><i><font color="Green">E64</font></i></a>, <a class="ref" href="rfunct_3.html#T47">RFUNCT_3:47</a></i>;<br/>
<a NAME="E7:26_1_1_1_3_4_1_1_3_1_2_2"/><b>then </b><i><font color="Green">E184</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> )
 
<b>by </b><i><a class="ref" href="rfunct_3.html#D10">RFUNCT_3:def 10</a></i>;<br/>
<i><font color="Green">E185</font></i>: <font color="Maroon">r</font> = 
 <a href="rfunct_3.html#K2">max+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rfunct_3.html#D10">RFUNCT_3:def 10</a></i>
<br/>.= 
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>,0
<b>by </b><i><a class="ref" href="rfunct_3.html#D1">RFUNCT_3:def 1</a></i>
;<br/>
<a NAME="E9:26_1_1_1_3_4_1_1_3_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E10:26_1_1_1_3_4_1_1_3_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E12:26_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>
<a NAME="E11:26_1_1_1_3_4_1_1_3_1_2_2"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="integra4.html#E2:26_1_1_1_3"><i><font color="Green">E60</font></i></a>, , <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>;<br/>
<b>hence </b><a NAME="E12:26_1_1_1_3_4_1_1_3_1_2_2"/>
 ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font> )
 <b>by </b><i><a class="txt" href="integra4.html#E5:26_1_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="integra4.html#E7:26_1_1_1_3"><i><font color="Green">E63</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:26_1_1_1_3_4_1_1_3_1_2"/>
( ( for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> ) &amp; ( for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> holds <br/> ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="xcmplx_0.html#K2">+</a> <font color="Olive">s</font> ) ) )
 <b>by </b><i><a class="txt" href="integra4.html#E4:26_1_1_1_3"><i><font color="Green">E61</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:26_1_1_1_3_4_1_1_3_1"/>
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="integra4.html#E17:26_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="integra4.html#E1:26_1_1_1_3"><i><font color="Green">E59</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:26_1_1_1_3_4_1_1_3"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra4.html#E5:26_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E17:26_1_1_1_3_4"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:26_1_1_1_3"/>
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="integra1.html#T11">INTEGRA1:11</a></i>;<br/>
<b>hence </b><a NAME="E11:26_1_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">UV1</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">UV</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="txt" href="integra4.html#E4:26_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="integra4.html#E6:26_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:26_1_1_1"/><b>then </b><i><font color="Green">E186</font></i>: 
<font color="Maroon">osc1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">osc</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="txt" href="integra4.html#E7:26"><i><font color="Green">E41</font></i></a>, <a class="txt" href="integra4.html#E2:26_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="rvsum_1.html#T112">RVSUM_1:112</a></i>;<br/>
<b>reconsider </b><font color="Maroon">F</font> = <font color="Maroon">UV1</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV1</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E21:26_1_1_1"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">osc1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_4"><b>proof </b></a><div class="add">

<a NAME="E1:26_1_1_1_4"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">F</font> <a href="goboard1.html#K1">.</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:26_1_1_1_4_1"/><i><font color="Green">E188</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 ;<br/>

<b>set </b><font color="Maroon">r</font> = <font color="Maroon">F</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>;<br/>
<a NAME="E2:26_1_1_1_4_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:26_1_1_1_4_1"/><b>then </b><i><font color="Green">E189</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E4:26_1_1_1_4_1"/><b>then </b>
( <font color="Maroon">UV1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp; <font color="Maroon">LV1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<b>then </b><i><font color="Green">E190</font></i>: <font color="Maroon">F</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="rvsum_1.html#T47">RVSUM_1:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>

;<br/>
<a NAME="E6:26_1_1_1_4_1"/><i><font color="Green">E191</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_4_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:26_1_1_1_4_1_2"/>
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> is <a href="seq_4.html#V3">bounded</a> &amp;  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_4_1_2_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E2:26_1_1_1_4_1_2_1"/><i><font color="Green">E192</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/>
<a NAME="E3:26_1_1_1_4_1_2_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:26_1_1_1_4_1_2_1"/><b>then </b>
 <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="integra1.html#T10">INTEGRA1:10</a></i>;<br/>
<a NAME="E5:26_1_1_1_4_1_2_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="integra4.html#E6:26_1"><i><font color="Green">E50</font></i></a></i>;<br/>
<a NAME="E6:26_1_1_1_4_1_2_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:26_1_1_1_4_1_2_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E6:26_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:26_1_1_1_4_1_2_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E9:26_1_1_1_4_1_2_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_3.html#D10">RFUNCT_3:def 10</a></i>;<br/>
<a NAME="E10:26_1_1_1_4_1_2_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_3.html#T47">RFUNCT_3:47</a></i>;<br/>
<a NAME="E11:26_1_1_1_4_1_2_1"/><b>then </b><i><font color="Green">E193</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E12:26_1_1_1_4_1_2_1"/><i><font color="Green">E194</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T99">RELAT_1:99</a></i>;<br/>
<a NAME="E13:26_1_1_1_4_1_2_1"/>
( <font color="Maroon">f</font> <a href="rfunct_1.html#R1">is_bounded_above_on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">f</font> <a href="rfunct_1.html#R2">is_bounded_below_on</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, <a class="ref" href="rfunct_1.html#D11">RFUNCT_1:def 11</a></i>;<br/>
<a NAME="E14:26_1_1_1_4_1_2_1"/><b>then </b>
(  <a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font> <a href="rfunct_1.html#R1">is_bounded_above_on</a> <font color="Maroon">A</font> &amp;  <a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font> <a href="rfunct_1.html#R2">is_bounded_below_on</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E15:26_1_1_1_4_1_2_1"/><b>then </b>
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> is <a href="seq_4.html#V1">bounded_above</a> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> is <a href="seq_4.html#V2">bounded_below</a> )
 
<b>by </b><i><a class="ref" href="integra1.html#T13">INTEGRA1:13</a>, <a class="ref" href="integra1.html#T15">INTEGRA1:15</a></i>;<br/>
<a NAME="E16:26_1_1_1_4_1_2_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> is <a href="seq_4.html#V3">bounded</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#D3">SEQ_4:def 3</a></i>;<br/>
<b>hence </b><a NAME="E17:26_1_1_1_4_1_2_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span> is <a href="seq_4.html#V3">bounded</a>
 <b>by </b><i><a class="txt" href="integra4.html#E10:26_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="rcomp_1.html#T5">RCOMP_1:5</a></i>;<br/>

<b>thus </b><a NAME="E18:26_1_1_1_4_1_2_1"/>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra4.html#E9:26_1"><i><font color="Green">E51</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:26_1_1_1_4_1_2"/><b>then </b>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_4.html#T24">SEQ_4:24</a></i>;<br/>
<b>hence </b><a NAME="E3:26_1_1_1_4_1_2"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:26_1_1_1_4_1"/>
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="integra1.html#T11">INTEGRA1:11</a></i>;<br/>
<b>hence </b><a NAME="E8:26_1_1_1_4_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">F</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="integra4.html#E4:26_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="integra4.html#E5:26_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="real_2.html#T121">REAL_2:121</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:26_1_1_1_4"/>
<font color="Maroon">osc1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="integra4.html#E2:26_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="rvsum_1.html#T114">RVSUM_1:114</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:26_1_1_1"/><b>then </b><i><font color="Green">E195</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">osc1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">osc1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E23:26_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">osc</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">osc</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="txt" href="integra4.html#E3:26_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="integra4.html#T2" target="_self">Th2</a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<b>hence </b><a NAME="E24:26_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">osc1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="integra4.html#E2:26"><i><font color="Green">E39</font></i></a>, <a class="txt" href="integra4.html#E3:26_1"><i><font color="Green">E44</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:26_1_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">osc1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E11:26_1"/><b>then </b>
<font color="Maroon">osc1</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#D6">SEQ_2:def 6</a></i>;<br/>
<a NAME="E12:26_1"/><b>then </b>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>
<b>hence </b><a NAME="E13:26_1"/>
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="seq_2.html#T26">SEQ_2:26</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:26"/>
 <a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">A</font>
 <b>by </b><i>, , </i>;<br/>


</div>
