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

<b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="supinf_1.html#K3">REAL</a> , <a href="supinf_1.html#K3">REAL</a> ;<br/><b>let </b><font color="Maroon">I</font> be   <a href="measure5.html#NM1">Interval</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/>





<b>assume </b><a NAME="E1:22"/><i><font color="Green">E38</font></i>: 
(  ex <font color="Olive">x1</font>, <font color="Olive">x2</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">x1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">x2</font> ) &amp; <font color="Maroon">f</font> <a href="rfunct_3.html#R2">is_convex_on</a> <font color="Maroon">I</font> )
 ;<br/>

<a NAME="E2:22"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">I</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="rfunct_3.html#D13">RFUNCT_3:def 13</a></i>;<br/>
<b>consider </b><font color="Maroon">x1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">x2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:22"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">x1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x2</font> )
 <b>by </b><i/>;<br/>
<a NAME="E5:22"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_1_1"/>
( <font color="Maroon">I</font> is <a href="measure5.html#V1">open_interval</a> or <font color="Maroon">I</font> is <a href="measure5.html#V2">closed_interval</a> or <font color="Maroon">I</font> is <a href="measure5.html#V3">right_open_interval</a> or <font color="Maroon">I</font> is <a href="measure5.html#V4">left_open_interval</a> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D9">MEASURE5:def 9</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_1"><b>suppose </b></a><a NAME="E1:22_1_1_1_1"/>
<font color="Maroon">I</font> is <a href="measure5.html#V1">open_interval</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p</font> being   <a href="supinf_1.html#NM3">R_eal</a>, <font color="Maroon">q</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:22_1_1_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2">].</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="measure5.html#K2">.[</a></span> )
 <b>by </b><i><a class="ref" href="measure5.html#D5">MEASURE5:def 5</a></i>;<br/><b>thus </b><a NAME="E4:22_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i>, , <a class="ref" href="measure6.html#T38">MEASURE6:38</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_2"><b>suppose </b></a><a NAME="E1:22_1_1_1_2"/>
<font color="Maroon">I</font> is <a href="measure5.html#V2">closed_interval</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p</font> being   <a href="supinf_1.html#NM3">R_eal</a>, <font color="Maroon">q</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:22_1_1_1_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="measure5.html#K1">.]</a></span> )
 <b>by </b><i><a class="ref" href="measure5.html#D6">MEASURE5:def 6</a></i>;<br/><b>thus </b><a NAME="E4:22_1_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i>, , <a class="ref" href="measure6.html#T39">MEASURE6:39</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_3"><b>suppose </b></a><a NAME="E1:22_1_1_1_3"/>
<font color="Maroon">I</font> is <a href="measure5.html#V3">right_open_interval</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p</font> being   <a href="supinf_1.html#NM3">R_eal</a>, <font color="Maroon">q</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:22_1_1_1_3"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="measure5.html#K4">.[</a></span> )
 <b>by </b><i><a class="ref" href="measure5.html#D7">MEASURE5:def 7</a></i>;<br/><b>thus </b><a NAME="E4:22_1_1_1_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i>, , <a class="ref" href="measure6.html#T41">MEASURE6:41</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_4"><b>suppose </b></a><a NAME="E1:22_1_1_1_4"/>
<font color="Maroon">I</font> is <a href="measure5.html#V4">left_open_interval</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p</font> being   <a href="supinf_1.html#NM3">R_eal</a>, <font color="Maroon">q</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:22_1_1_1_4"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3">].</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="measure5.html#K3">.]</a></span> )
 <b>by </b><i><a class="ref" href="measure5.html#D8">MEASURE5:def 8</a></i>;<br/><b>thus </b><a NAME="E4:22_1_1_1_4"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i>, , <a class="ref" href="measure6.html#T40">MEASURE6:40</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:22_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:22"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x2</font> &amp; <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> holds <br/>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:22_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> )
 ;<br/>

<a NAME="E2:22_2"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_2_1_1_1"/>
( <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> or <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> )
 </span><b>by </b><i>, <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:22_2_1_1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><a NAME="E2:22_2_1_1_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_1_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_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="E1:22_2_1_1_1_1_1_1_1"/>
( <font color="Maroon">I</font> is <a href="measure5.html#V1">open_interval</a> or <font color="Maroon">I</font> is <a href="measure5.html#V2">closed_interval</a> or <font color="Maroon">I</font> is <a href="measure5.html#V3">right_open_interval</a> or <font color="Maroon">I</font> is <a href="measure5.html#V4">left_open_interval</a> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D9">MEASURE5:def 9</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:22_2_1_1_1_1_1_1_1_1"/>
<font color="Maroon">I</font> is <a href="measure5.html#V1">open_interval</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p</font> being   <a href="supinf_1.html#NM3">R_eal</a>, <font color="Maroon">q</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:22_2_1_1_1_1_1_1_1_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2">].</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="measure5.html#K2">.[</a></span> )
 <b>by </b><i><a class="ref" href="measure5.html#D5">MEASURE5:def 5</a></i>;<br/><b>thus </b><a NAME="E4:22_2_1_1_1_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i>, , , <a class="ref" href="measure6.html#T38">MEASURE6:38</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:22_2_1_1_1_1_1_1_1_2"/>
<font color="Maroon">I</font> is <a href="measure5.html#V2">closed_interval</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p</font> being   <a href="supinf_1.html#NM3">R_eal</a>, <font color="Maroon">q</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:22_2_1_1_1_1_1_1_1_2"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="measure5.html#K1">.]</a></span> )
 <b>by </b><i><a class="ref" href="measure5.html#D6">MEASURE5:def 6</a></i>;<br/><b>thus </b><a NAME="E4:22_2_1_1_1_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i>, , , <a class="ref" href="measure6.html#T39">MEASURE6:39</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_1_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:22_2_1_1_1_1_1_1_1_3"/>
<font color="Maroon">I</font> is <a href="measure5.html#V3">right_open_interval</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p</font> being   <a href="supinf_1.html#NM3">R_eal</a>, <font color="Maroon">q</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:22_2_1_1_1_1_1_1_1_3"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="measure5.html#K4">.[</a></span> )
 <b>by </b><i><a class="ref" href="measure5.html#D7">MEASURE5:def 7</a></i>;<br/><b>thus </b><a NAME="E4:22_2_1_1_1_1_1_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i>, , , <a class="ref" href="measure6.html#T41">MEASURE6:41</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_1_1_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:22_2_1_1_1_1_1_1_1_4"/>
<font color="Maroon">I</font> is <a href="measure5.html#V4">left_open_interval</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p</font> being   <a href="supinf_1.html#NM3">R_eal</a>, <font color="Maroon">q</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:22_2_1_1_1_1_1_1_1_4"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3">].</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="measure5.html#K3">.]</a></span> )
 <b>by </b><i><a class="ref" href="measure5.html#D8">MEASURE5:def 8</a></i>;<br/><b>thus </b><a NAME="E4:22_2_1_1_1_1_1_1_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i>, , , <a class="ref" href="measure6.html#T40">MEASURE6:40</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:22_2_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E3:22_2_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> )
 <b>by </b><i>, , , , </i>;<br/><a NAME="E4:22_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E5:22_2_1_1_1_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T192">XCMPLX_1:192</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_1_1_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_2_1_1_1_1_2_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or <font color="Maroon">x1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:22_2_1_1_1_1_2_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:22_2_1_1_1_1_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:22_2_1_1_1_1_2_1_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">x1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><i><font color="Green">E68</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T192">XCMPLX_1:192</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span>

;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T192">XCMPLX_1:192</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span>

;<br/><b>hence </b><a NAME="E4:22_2_1_1_1_1_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i>, , , , , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:22_2_1_1_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:22_2_1_1_1_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><a NAME="E2:22_2_1_1_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_1_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_1_1_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_2_1_1_1_2_1_1_1"/>
( <font color="Maroon">I</font> is <a href="measure5.html#V1">open_interval</a> or <font color="Maroon">I</font> is <a href="measure5.html#V2">closed_interval</a> or <font color="Maroon">I</font> is <a href="measure5.html#V3">right_open_interval</a> or <font color="Maroon">I</font> is <a href="measure5.html#V4">left_open_interval</a> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D9">MEASURE5:def 9</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_1_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:22_2_1_1_1_2_1_1_1_1"/>
<font color="Maroon">I</font> is <a href="measure5.html#V1">open_interval</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p</font> being   <a href="supinf_1.html#NM3">R_eal</a>, <font color="Maroon">q</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:22_2_1_1_1_2_1_1_1_1"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2">].</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="measure5.html#K2">.[</a></span> )
 <b>by </b><i><a class="ref" href="measure5.html#D5">MEASURE5:def 5</a></i>;<br/><b>thus </b><a NAME="E4:22_2_1_1_1_2_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i>, , , <a class="ref" href="measure6.html#T38">MEASURE6:38</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_1_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:22_2_1_1_1_2_1_1_1_2"/>
<font color="Maroon">I</font> is <a href="measure5.html#V2">closed_interval</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p</font> being   <a href="supinf_1.html#NM3">R_eal</a>, <font color="Maroon">q</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:22_2_1_1_1_2_1_1_1_2"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="measure5.html#K1">.]</a></span> )
 <b>by </b><i><a class="ref" href="measure5.html#D6">MEASURE5:def 6</a></i>;<br/><b>thus </b><a NAME="E4:22_2_1_1_1_2_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i>, , , <a class="ref" href="measure6.html#T39">MEASURE6:39</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_1_1_2_1_1_1_3"><b>suppose </b></a><a NAME="E1:22_2_1_1_1_2_1_1_1_3"/>
<font color="Maroon">I</font> is <a href="measure5.html#V3">right_open_interval</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p</font> being   <a href="supinf_1.html#NM3">R_eal</a>, <font color="Maroon">q</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:22_2_1_1_1_2_1_1_1_3"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="measure5.html#K4">.[</a></span> )
 <b>by </b><i><a class="ref" href="measure5.html#D7">MEASURE5:def 7</a></i>;<br/><b>thus </b><a NAME="E4:22_2_1_1_1_2_1_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i>, , , <a class="ref" href="measure6.html#T41">MEASURE6:41</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_1_1_2_1_1_1_4"><b>suppose </b></a><a NAME="E1:22_2_1_1_1_2_1_1_1_4"/>
<font color="Maroon">I</font> is <a href="measure5.html#V4">left_open_interval</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p</font> being   <a href="supinf_1.html#NM3">R_eal</a>, <font color="Maroon">q</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:22_2_1_1_1_2_1_1_1_4"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3">].</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="measure5.html#K3">.]</a></span> )
 <b>by </b><i><a class="ref" href="measure5.html#D8">MEASURE5:def 8</a></i>;<br/><b>thus </b><a NAME="E4:22_2_1_1_1_2_1_1_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i>, , , <a class="ref" href="measure6.html#T40">MEASURE6:40</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:22_2_1_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E3:22_2_1_1_1_2"/><b>then </b><i><font color="Green">E82</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> )
 <b>by </b><i>, , , , </i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T192">XCMPLX_1:192</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span>

;<br/><b>hence </b><a NAME="E5:22_2_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_1_1_1_2_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_1_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_2_1_1_1_2_3_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> or <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x2</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_1_1_2_3_1_1"><b>suppose </b></a><a NAME="E1:22_2_1_1_1_2_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:22_2_1_1_1_2_3_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_1_1_2_3_1_2"><b>suppose </b></a><a NAME="E1:22_2_1_1_1_2_3_1_2"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:22_2_1_1_1_2_3_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i>, , , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:22_2_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:22_2_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:22_2"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> )
 ;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">M</font> =  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E7:22"/><i><font color="Green">E83</font></i>: 
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<a NAME="E8:22"/><b>then </b><i><font color="Green">E84</font></i>: 
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></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="square_1.html#T39">SQUARE_1:39</a></i>;<br/>
<a NAME="E9:22"/><i><font color="Green">E95</font></i>: 
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E10:22"/><i><font color="Green">E96</font></i>: 
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:22"/><i><font color="Green">E97</font></i>: 
for <font color="Olive">x</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x2</font> &amp; <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</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">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:22_3"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</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="E3:22_3"/>
( <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> or <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/>
<a NAME="E4:22_3"/><b>then </b>
<font color="Maroon">x</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
;<br/>
<a NAME="E5:22_3"/><b>then </b><i><font color="Green">E99</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="complex1.html#T133">COMPLEX1:133</a></i>;<br/>
<a NAME="E6:22_3"/><i><font color="Green">E100</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:22_3"/>
(  <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a></i>;<br/>
<a NAME="E8:22_3"/><b>then </b><i><font color="Green">E101</font></i>: 
(  <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_3_1_1"/>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> or  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_3_1_1_1"><b>suppose </b></a><a NAME="E1:22_3_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:22_3_1_1_1"/><b>then </b>
 <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E3:22_3_1_1_1"/><b>then </b>
 <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E4:22_3_1_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="absvalue.html#T12">ABSVALUE:12</a></i>;<br/><a NAME="E5:22_3_1_1_1"/><b>then </b><i><font color="Green">E102</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="complex1.html#T153">COMPLEX1:153</a></i>;<br/><a NAME="E6:22_3_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="square_1.html#T46">SQUARE_1:46</a></i>;<br/><a NAME="E7:22_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>hence </b><a NAME="E8:22_3_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">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="real_2.html#T178">REAL_2:178</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_3_1_1_2"><b>suppose </b></a><a NAME="E1:22_3_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:22_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E3:22_3_1_1_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="absvalue.html#T12">ABSVALUE:12</a></i>;<br/><a NAME="E4:22_3_1_1_2"/><b>then </b><i><font color="Green">E103</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="complex1.html#T153">COMPLEX1:153</a></i>;<br/><a NAME="E5:22_3_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="square_1.html#T46">SQUARE_1:46</a></i>;<br/><a NAME="E6:22_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>hence </b><a NAME="E7:22_3_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="real_2.html#T178">REAL_2:178</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E10:22_3"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E12:22"/>
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 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> holds <br/> 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">x</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</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">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <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="22_4"><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="E1:22_4"/><i><font color="Green">E104</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 ;<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="E3:22_4"/>
 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">x</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</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">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_4_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_4_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_4_1_1_1"/>
(  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 </span><b>by </b><i>, , <a class="ref" href="square_1.html#T39">SQUARE_1:39</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_1_1_1_1"><b>suppose </b></a><a NAME="E1:22_4_1_1_1_1"/><i><font color="Green">E105</font></i>: 
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>set </b><font color="Maroon">s</font> =  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3">min</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span>;<br/><a NAME="E2:22_4_1_1_1_1"/><i><font color="Green">E106</font></i>: 
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3">min</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_4_1_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:22_4_1_1_1_1_1"/><i><font color="Green">E107</font></i>: 
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_4_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_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="22_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="E1:22_4_1_1_1_1_1_1_1_1"/>
(  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font> or  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font> )
 </span><b>by </b><i><a class="ref" href="square_1.html#T38">SQUARE_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_1_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:22_4_1_1_1_1_1_1_1_1_1"/>
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:22_4_1_1_1_1_1_1_1_1_1"/>
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_1_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:22_4_1_1_1_1_1_1_1_1_2"/>
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:22_4_1_1_1_1_1_1_1_1_2"/>
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:22_4_1_1_1_1_1_1"/>
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_4_1_1_1_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_1_1_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_4_1_1_1_1_1_2_1"/>
(  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3">min</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span> or  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3">min</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> )
 </span><b>by </b><i><a class="ref" href="square_1.html#T38">SQUARE_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_1_1_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:22_4_1_1_1_1_1_2_1_1"/>
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3">min</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:22_4_1_1_1_1_1_2_1_1"/>
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3">min</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, , <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_1_1_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:22_4_1_1_1_1_1_2_1_2"/>
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3">min</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:22_4_1_1_1_1_1_2_1_2"/>
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3">min</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:22_4_1_1_1_1_1"/>
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3">min</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>


</div><b>end;</b></div><a NAME="E3:22_4_1_1_1_1"/>
for <font color="Olive">x</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3">min</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_4_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:22_4_1_1_1_1_2"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3">min</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> )
 ;<br/>

<a NAME="E2:22_4_1_1_1_1_2"/><i><font color="Green">E109</font></i>: 
(  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3">min</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span> &amp;  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3">min</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="square_1.html#T35">SQUARE_1:35</a></i>;<br/>
<a NAME="E3:22_4_1_1_1_1_2"/>
(  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font> &amp;  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="square_1.html#T35">SQUARE_1:35</a></i>;<br/>
<a NAME="E4:22_4_1_1_1_1_2"/><b>then </b>
(  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3">min</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font> &amp;  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3">min</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E5:22_4_1_1_1_1_2"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E6:22_4_1_1_1_1_2"/><b>then </b>
(  <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font> &amp; <font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="absvalue.html#T12">ABSVALUE:12</a></i>;<br/>
<a NAME="E7:22_4_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font> &amp; <font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font> )
 
;<br/>
<a NAME="E8:22_4_1_1_1_1_2"/><b>then </b><i><font color="Green">E110</font></i>: 
( <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x2</font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_4_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_4_1_1_1_1_2_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:22_4_1_1_1_1_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><a NAME="E2:22_4_1_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E111</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_4_1_1_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_1_1_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="E1:22_4_1_1_1_1_2_1_1_1_1_1"/>
(  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_1_1_1_1_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:22_4_1_1_1_1_2_1_1_1_1_1_1"/><i><font color="Green">E112</font></i>: 
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:22_4_1_1_1_1_2_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E118</font></i>: 
<span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3">min</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p0">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p0">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3">min</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E3:22_4_1_1_1_1_2_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3">min</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p0">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p0">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3">min</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E4:22_4_1_1_1_1_2_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E5:22_4_1_1_1_1_2_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E6:22_4_1_1_1_1_2_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E7:22_4_1_1_1_1_2_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="binop_2.html#K11">*</a> 1
 <b>by </b><i><a class="ref" href="rfunct_4.html#T2" target="_self">Th2</a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/><b>hence </b><a NAME="E8:22_4_1_1_1_1_2_1_1_1_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">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="rfunct_4.html#T1" target="_self">Th1</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_1_1_1_1_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:22_4_1_1_1_1_2_1_1_1_1_1_2"/>
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:22_4_1_1_1_1_2_1_1_1_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="rfunct_4.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:22_4_1_1_1_1_2_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">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:22_4_1_1_1_1_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><a NAME="E2:22_4_1_1_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><a NAME="E3:22_4_1_1_1_1_2_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E10:22_4_1_1_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:22_4_1_1_1_1"/>
 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">x</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</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">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_1_1_1_2"><b>suppose </b></a><a NAME="E1:22_4_1_1_1_2"/><i><font color="Green">E119</font></i>: 
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:22_4_1_1_1_2"/><i><font color="Green">E120</font></i>: 
for <font color="Olive">x</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x2</font> &amp; <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</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">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_4_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:22_4_1_1_1_2_1"/><i><font color="Green">E121</font></i>: 
( <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> )
 ;<br/>

<a NAME="E2:22_4_1_1_1_2_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:22_4_1_1_1_2_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>


</div><b>end;</b></div><b>set </b><font color="Maroon">s</font> =  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span>;<br/><a NAME="E3:22_4_1_1_1_2"/><i><font color="Green">E122</font></i>: 
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_4_1_1_1_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_4_1_1_1_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_1_1_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_4_1_1_1_2_2_1_1"/>
(  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font> or  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font> )
 </span><b>by </b><i><a class="ref" href="square_1.html#T38">SQUARE_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_1_1_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:22_4_1_1_1_2_2_1_1_1"/>
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:22_4_1_1_1_2_2_1_1_1"/>
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_1_1_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:22_4_1_1_1_2_2_1_1_2"/>
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:22_4_1_1_1_2_2_1_1_2"/>
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:22_4_1_1_1_2_2"/>
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>


</div><b>end;</b></div><a NAME="E4:22_4_1_1_1_2"/>
for <font color="Olive">x</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_4_1_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:22_4_1_1_1_2_3"/><i><font color="Green">E123</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> )
 ;<br/>

<a NAME="E2:22_4_1_1_1_2_3"/>
(  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font> &amp;  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="square_1.html#T35">SQUARE_1:35</a></i>;<br/>
<a NAME="E3:22_4_1_1_1_2_3"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E4:22_4_1_1_1_2_3"/><b>then </b>
(  <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">x1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font> &amp; <font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="absvalue.html#T12">ABSVALUE:12</a></i>;<br/>
<a NAME="E5:22_4_1_1_1_2_3"/><b>then </b>
( <font color="Maroon">x1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font> &amp; <font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x2</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font> )
 
;<br/>
<a NAME="E6:22_4_1_1_1_2_3"/><b>then </b><i><font color="Green">E124</font></i>: 
( <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x2</font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_4_1_1_1_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_1_1_1_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_4_1_1_1_2_3_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_1_1_1_2_3_1_1_1"><b>suppose </b></a><a NAME="E1:22_4_1_1_1_2_3_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:22_4_1_1_1_2_3_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">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_1_1_1_2_3_1_1_2"><b>suppose </b></a><a NAME="E1:22_4_1_1_1_2_3_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><a NAME="E2:22_4_1_1_1_2_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><a NAME="E3:22_4_1_1_1_2_3_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:22_4_1_1_1_2_3"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:22_4_1_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">x</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</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">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:22_4_1"/>
 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">x</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</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">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> ) )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:22_4"/>
 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">x</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</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">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> ) )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:22"/>
<font color="Maroon">f</font> <a href="fcont_1.html#R1">is_continuous_in</a> <font color="Maroon">a</font>
 <b>by </b><i>, , <a class="ref" href="fcont_1.html#T3">FCONT_1:3</a></i>;<br/>


</div>
