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

<b>let </b><font color="Maroon">P</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">p1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">p2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">e</font> be   <a href="real_1.html#NM1">Real</a>;<br/>





<b>assume </b><a NAME="E1:11"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">P</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> &amp; <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">e</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_1"/><i><font color="Green">E28</font></i>: 
not <font color="Maroon">p</font> <a href="jordan20.html#R6" target="_self">is_OSout</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">e</font>
 ;<br/><a NAME="E2:11_1"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Maroon">p</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>by </b><i>, <a class="ref" href="jordan5c.html#T10">JORDAN5C:10</a></i>;<br/><a NAME="E3:11_1"/><b>then </b><i><font color="Green">E30</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; ( for <font color="Olive">g</font> being  <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span><br/> for <font color="Olive">s1</font>, <font color="Olive">s2</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">g</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> &amp; <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s1</font> &amp; <font color="Olive">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s2</font> &amp; <font color="Olive">s2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 holds <br/><font color="Olive">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s2</font> ) )
 <b>by </b><i><a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/><b>consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span><b> such that </b><br/><a NAME="E5:11_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 <b>by </b><i>, <a class="ref" href="topreal1.html#D2">TOPREAL1:def 2</a></i>;<br/><a NAME="E6:11_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><a NAME="E7:11_1"/><i><font color="Green">E37</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a>  &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><a NAME="E8:11_1"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/><a NAME="E9:11_1"/><i><font color="Green">E43</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/><a NAME="E10:11_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/><b>then consider </b><font color="Maroon">xs</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:11_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">xs</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">xs</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">s2</font> = <font color="Maroon">xs</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, </i>;<br/><a NAME="E14:11_1"/><i><font color="Green">E45</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">s2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><b>reconsider </b><font color="Maroon">pr1</font> =  <a href="pscomp_1.html#K13">proj1</a>  as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/><b>reconsider </b><font color="Maroon">pro1</font> = <font color="Maroon">pr1</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">P</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="pre_topc.html#T30">PRE_TOPC:30</a></i>;<br/><a NAME="E17:11_1"/>
for <font color="Olive">p7</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> holds  <font color="Maroon">pro1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p7</font> <a href="hidden.html#R1">=</a> <a href="pscomp_1.html#K13">proj1</a>  <a href="funct_1.html#K1">.</a> <font color="Olive">p7</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p7</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span>;<br/>

<a NAME="E1:11_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<b>hence </b><a NAME="E2:11_1_1"/>
<font color="Maroon">pro1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p7</font> <a href="hidden.html#R1">=</a> <a href="pscomp_1.html#K13">proj1</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">p7</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>


</div><b>end;</b></div><a NAME="E18:11_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">pro1</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T39">JGRAPH_2:39</a></i>;<br/><b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">pro1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K3">R^1</a>  ;<br/><a NAME="E20:11_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">h</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i>, <a class="ref" href="jordan20.html#T1" target="_self">Th1</a>, <a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/><a NAME="E21:11_1"/><i><font color="Green">E48</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E22:11_1"/>
for <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 holds <br/><font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, </i>;<br/><a NAME="E23:11_1"/><b>then </b><i><font color="Green">E71</font></i>: 
1 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s2</font> &amp; ( for <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> holds <br/><font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e</font> ) ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E24:11_1"/>
<span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s2</font> &amp; ( for <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> holds <br/><font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e</font> ) ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:11_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s2</font> &amp; ( for <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> holds <br/><font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e</font> ) ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">s</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:11_1_2"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">s</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s2</font> &amp; ( for <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> holds <br/><font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e</font> ) )
 ;<br/>
<b>thus </b><a NAME="E4:11_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">R</font> = <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s2</font> &amp; ( for <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> holds <br/><font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e</font> ) ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">s0</font> =  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">R</font> as   <a href="real_1.html#NM1">Real</a> ;<br/><a NAME="E27:11_1"/>
<font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:11_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 ;<br/>

<b>then consider </b><font color="Maroon">s</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:11_1_3"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">s</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s2</font> &amp; ( for <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> holds <br/><font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e</font> ) )
 ;<br/>
<b>thus </b><a NAME="E4:11_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, , <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>


</div><b>end;</b></div><a NAME="E28:11_1"/><b>then </b><i><font color="Green">E78</font></i>: 
1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s0</font>
 <b>by </b><i>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="borsuk_4.html#T51">BORSUK_4:51</a></i>;<br/><a NAME="E29:11_1"/><i><font color="Green">E79</font></i>: 
for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> holds <br/><font color="Olive">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:11_1_4"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 ;<br/>

<b>then consider </b><font color="Maroon">s3</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:11_1_4"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">s3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s3</font> &amp; <font color="Maroon">s3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s2</font> &amp; ( for <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s3</font> holds <br/><font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e</font> ) )
 ;<br/>
<a NAME="E4:11_1_4"/>
<font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">s2</font>
 
<b>by </b><i>, , <a class="txt" href="jordan20.html#E1:11_1"><i><font color="Green">E28</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:11_1_4"/>
<font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:11_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E30:11_1"/><b>then </b>
for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> holds <br/><font color="Olive">s</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s2</font>
 ;<br/><a NAME="E31:11_1"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">s0</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s2</font>
 <b>by </b><i><a class="ref" href="seq_4.html#T60">SEQ_4:60</a></i>;<br/><a NAME="E32:11_1"/><b>then </b>
<font color="Maroon">s0</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i>, , , <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E33:11_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s0</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E34:11_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/><b>then reconsider </b><font color="Maroon">p9</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s0</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E36:11_1"/>
for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> holds <br/><font color="Olive">s</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:11"><i><font color="Green">E27</font></i></a></i>;<br/><a NAME="E37:11_1"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">R</font> is <a href="seq_4.html#V2">bounded_below</a>
 <b>by </b><i><a class="ref" href="seq_4.html#D2">SEQ_4:def 2</a></i>;<br/><a NAME="E38:11_1"/><i><font color="Green">E85</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">p9</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>by </b><i>, , , , , <a class="txt" href="jordan20.html#E1:11_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jordan5c.html#T8">JORDAN5C:8</a></i>;<br/><a NAME="E39:11_1"/><i><font color="Green">E86</font></i>: 
for <font color="Olive">p8</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p8</font>,<font color="Maroon">p9</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Olive">p8</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> holds <br/><font color="Olive">p8</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p8</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:11_1_5"/><i><font color="Green">E87</font></i>: 
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p8</font>,<font color="Maroon">p9</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">p8</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 ;<br/>

<a NAME="E2:11_1_5"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">p8</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">x8</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:11_1_5"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">x8</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">p8</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x8</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">s8</font> = <font color="Maroon">x8</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, <a class="ref" href="jordan20.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E6:11_1_5"/><i><font color="Green">E91</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s8</font> &amp; <font color="Maroon">s8</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s8</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p8</font> )
 
<b>by </b><i><a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E7:11_1_5"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">s0</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s8</font>
 
<b>by </b><i>, , , <a class="txt" href="jordan20.html#E1:11_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E5:11_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/>
<a NAME="E8:11_1_5"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">s8</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s2</font>
 
<b>by </b><i>, , , <a class="txt" href="jordan20.html#E5:11_1"><i><font color="Green">E32</font></i></a>, , <a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_1_5_1"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e</font>
 ;<br/><a NAME="E2:11_1_5_1"/><b>then </b><i><font color="Green">E96</font></i>: 
<font color="Maroon">s8</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s2</font>
 <b>by </b><i>, , <a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, <a class="txt" href="jordan20.html#E7:11_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">ee</font> = <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">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 as   <a href="real_1.html#NM1">Real</a> ;<br/><a NAME="E4:11_1_5_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="jordan20.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E5:11_1_5_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">e</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="E6:11_1_5_1"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">ee</font> <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/><b>reconsider </b><font color="Maroon">w</font> = <font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a>  as   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/><b>reconsider </b><font color="Maroon">B</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">w</font>,<font color="Maroon">ee</font> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/><a NAME="E9:11_1_5_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">w</font>,<font color="Maroon">ee</font>
 <b>by </b><i>, <a class="ref" href="goboard6.html#T4">GOBOARD6:4</a></i>;<br/><i><font color="Green">E102</font></i>: <font color="Maroon">B</font> = 
<span class="p1">{<span class="default"> <font color="Olive">s7</font> where <font color="Olive">s7</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">ee</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s7</font> &amp; <font color="Olive">s7</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">ee</font> ) </span>}</span> 
<b>by </b><i>, <a class="ref" href="jordan2b.html#T21">JORDAN2B:21</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">ee</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">ee</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
<b>by </b><i><a class="ref" href="rcomp_1.html#D2">RCOMP_1:def 2</a></i>
;<br/><a NAME="E11:11_1_5_1"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">B</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a>, <a class="ref" href="uniform1.html#T3">UNIFORM1:3</a></i>;<br/><a NAME="E12:11_1_5_1"/><i><font color="Green">E105</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="topmetr.html#T14">TOPMETR:14</a></i>;<br/><b>reconsider </b><font color="Maroon">s8m</font> = <font color="Maroon">s8</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span> <b>by </b><i><a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="topmetr.html#T14">TOPMETR:14</a></i>;<br/><b>reconsider </b><font color="Maroon">s8n</font> = <font color="Maroon">s8</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/><a NAME="E15:11_1_5_1"/><i><font color="Green">E106</font></i>: 
 <a href="topmetr.html#K5">I[01]</a>  <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span>
 <b>by </b><i><a class="ref" href="topmetr.html#T27">TOPMETR:27</a>, <a class="ref" href="topmetr.html#D8">TOPMETR:def 8</a></i>;<br/><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s8</font> = 
<font color="Maroon">pro1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s8</font></span>)</span>
<b>by </b><i>, <a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<a href="pscomp_1.html#K13">proj1</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">p8</font>
<b>by </b><i><a class="txt" href="jordan20.html#E6:11_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="ref" href="pscomp_1.html#D28">PSCOMP_1:def 28</a></i>
;<br/><a NAME="E17:11_1_5_1"/><b>then </b>
<font color="Maroon">s8</font> <a href="hidden.html#R2">in</a> <font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, <a class="ref" href="jordan20.html#T6" target="_self">Th6</a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><b>then consider </b><font color="Maroon">r0</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E19:11_1_5_1"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">r0</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">s8m</font>,<font color="Maroon">r0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E9:11_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="jordan20.html#E14:11_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="topmetr.html#T22">TOPMETR:22</a></i>;<br/><b>reconsider </b><font color="Maroon">r0</font> = <font color="Maroon">r0</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/><b>reconsider </b><font color="Maroon">r01</font> =  <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">s8</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span>,<font color="Maroon">r0</font> as   <a href="real_1.html#NM1">Real</a> ;<br/><a NAME="E22:11_1_5_1"/>
<font color="Maroon">s8</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="jordan20.html#E8:11_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><a NAME="E23:11_1_5_1"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">r01</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="jordan20.html#E18:11_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="square_1.html#T113">SQUARE_1:113</a></i>;<br/><a NAME="E24:11_1_5_1"/><b>then </b>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">s8n</font>,<font color="Maroon">r01</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s8</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r01</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s8</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r01</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="frechet.html#T7">FRECHET:7</a></i>;<br/><a NAME="E25:11_1_5_1"/><b>then </b><i><font color="Green">E113</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">s8m</font>,<font color="Maroon">r01</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s8</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r01</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s8</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r01</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E12:11_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="topmetr.html#T13">TOPMETR:13</a></i>;<br/><a NAME="E26:11_1_5_1"/>
<font color="Maroon">s8</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r01</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/><a NAME="E27:11_1_5_1"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">s8</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">r01</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E28:11_1_5_1"/><b>then </b><i><font color="Green">E114</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s8</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">s8</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r01</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">s8</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E29:11_1_5_1"/>
<font color="Maroon">r01</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r0</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/><a NAME="E30:11_1_5_1"/><b>then </b><i><font color="Green">E115</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">s8m</font>,<font color="Maroon">r01</font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">s8m</font>,<font color="Maroon">r0</font>
 <b>by </b><i><a class="ref" href="pcomps_1.html#T1">PCOMPS_1:1</a></i>;<br/><b>reconsider </b><font color="Maroon">Q</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E32:11_1_5_1"/><i><font color="Green">E116</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s8</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r01</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s8</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r01</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E18:11_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan20.html#E21:11_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="jordan20.html#T7" target="_self">Th7</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E33:11_1_5_1"/>
<font color="Maroon">r01</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">r01</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="jordan20.html#E20:11_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/><a NAME="E34:11_1_5_1"/><b>then </b><i><font color="Green">E118</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">r01</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">r01</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">r01</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">r01</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E35:11_1_5_1"/><b>then </b><i><font color="Green">E119</font></i>: 
<font color="Maroon">s8</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">r01</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s8</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r01</font>
 <b>by </b><i><a class="ref" href="real_2.html#T105">REAL_2:105</a></i>;<br/><a NAME="E36:11_1_5_1"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">r01</font> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="jordan20.html#E20:11_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/><a NAME="E37:11_1_5_1"/>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">r01</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="jordan20.html#E20:11_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/><a NAME="E38:11_1_5_1"/><b>then </b><i><font color="Green">E121</font></i>: 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">r01</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><a NAME="E39:11_1_5_1"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">s8</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">r01</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s8</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r01</font>
 <b>by </b><i><a class="ref" href="jordan20.html#D1" target="_self">Def1</a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E40:11_1_5_1"/>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">r01</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r01</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i>, <a class="ref" href="jordan20.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E41:11_1_5_1"/><b>then </b>
<font color="Maroon">s8</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">r01</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s8</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">r01</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E42:11_1_5_1"/><b>then </b><i><font color="Green">E123</font></i>: 
( <font color="Maroon">s8</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r01</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s8</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">r01</font> <a href="real_1.html#K6">/</a> 2</span>)</span> &amp; <font color="Maroon">s8</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">r01</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s8</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r01</font> )
 <b>by </b><i><a class="ref" href="jordan20.html#D1" target="_self">Def1</a>, <a class="ref" href="jordan20.html#D3" target="_self">Def3</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="real_2.html#T105">REAL_2:105</a></i>;<br/><a NAME="E43:11_1_5_1"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">s8</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">r01</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s8</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r01</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s8</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r01</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E44:11_1_5_1"/><i><font color="Green">E125</font></i>: 
1 <a href="real_1.html#K5">-</a> 0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s8</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">r01</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i>, , <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E45:11_1_5_1"/>
<font color="Maroon">s8</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">r01</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, , </i>;<br/><a NAME="E46:11_1_5_1"/><b>then </b>
<font color="Maroon">s8</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">r01</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E47:11_1_5_1"/><b>then </b>
<font color="Maroon">s8</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">r01</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s8</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r01</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s8</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r01</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="jordan20.html#D5" target="_self">Def5</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E48:11_1_5_1"/><b>then </b><i><font color="Green">E127</font></i>: 
( <font color="Maroon">s8</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">r01</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">s8</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">r01</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><b>reconsider </b><font color="Maroon">s70</font> = <font color="Maroon">s8</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">r01</font> <a href="real_1.html#K6">/</a> 2</span>)</span> as   <a href="real_1.html#NM1">Real</a> ;<br/><a NAME="E50:11_1_5_1"/>
<font color="Maroon">s8</font> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s8</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">r01</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan20.html#D2" target="_self">Def2</a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E51:11_1_5_1"/><b>then </b><i><font color="Green">E128</font></i>: 
( <font color="Maroon">s8</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s70</font> &amp; 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s70</font> &amp; <font color="Maroon">s70</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s2</font> )
 <b>by </b><i>, , <a class="ref" href="jordan20.html#D4" target="_self">Def4</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E52:11_1_5_1"/>
for <font color="Olive">p7</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">p7</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s70</font> holds <br/><font color="Olive">p7</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_5_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p7</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:11_1_5_1_3"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">p7</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s70</font>
 ;<br/>

<a NAME="E2:11_1_5_1_3"/>
<font color="Maroon">s70</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, , <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E3:11_1_5_1_3"/><b>then </b><i><font color="Green">E130</font></i>: 
<font color="Maroon">p7</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="jordan20.html#D6" target="_self">Def6</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<i><font color="Green">E131</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span>
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
<font color="Maroon">P</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s70</font> = 
<font color="Maroon">pro1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s70</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon">pr1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p7</font>
<b>by </b><i><a class="ref" href="jordan20.html#D6" target="_self">Def6</a>, , , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">p7</font> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="ref" href="pscomp_1.html#D28">PSCOMP_1:def 28</a></i>
;<br/>
<a NAME="E6:11_1_5_1_3"/><b>then </b><i><font color="Green">E132</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">ee</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p7</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p7</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">ee</font> )
 
<b>by </b><i>, , <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_5_1_3_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_1_5_1_3_3"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">p7</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_5_1_3_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_5_1_3_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_1_5_1_3_3_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">e</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <span class="p1">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">e</font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_5_1_3_3_1_1_1"><b>case </b></a><a NAME="E1:11_1_5_1_3_3_1_1_1"/><i><font color="Green">E134</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">e</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:11_1_5_1_3_3_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">e</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> &amp; <font color="Maroon">e</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">e</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> )
 <b>by </b><i>, , <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E3:11_1_5_1_3_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 ;<br/><a NAME="E4:11_1_5_1_3_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E5:11_1_5_1_3_3_1_1_1"/><b>then </b><i><font color="Green">E135</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E6:11_1_5_1_3_3_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">e</font></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="real_1.html#K6">/</a> 2
 <b>by </b><i>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><b>hence </b><a NAME="E7:11_1_5_1_3_3_1_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_5_1_3_3_1_1_2"><b>case </b></a><a NAME="E1:11_1_5_1_3_3_1_1_2"/><i><font color="Green">E136</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">e</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:11_1_5_1_3_3_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> &amp; <font color="Maroon">e</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> )
 <b>by </b><i>, , <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E3:11_1_5_1_3_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 ;<br/><a NAME="E4:11_1_5_1_3_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E5:11_1_5_1_3_3_1_1_2"/><b>then </b><i><font color="Green">E137</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E6:11_1_5_1_3_3_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">e</font></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> 0 <a href="real_1.html#K6">/</a> 2
 <b>by </b><i>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><b>hence </b><a NAME="E7:11_1_5_1_3_3_1_1_2"/>
contradiction
 <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="E3:11_1_5_1_3_3"/>
contradiction
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:11_1_5_1_3"/>
<font color="Maroon">p7</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e</font>
 ;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">s7</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E54:11_1_5_1"/><i><font color="Green">E138</font></i>: 
( <font color="Maroon">s8</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s7</font> &amp; 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s7</font> &amp; <font color="Maroon">s7</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s2</font> &amp; ( for <font color="Olive">p7</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">p7</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s7</font> holds <br/><font color="Olive">p7</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e</font> ) )
 <b>by </b><i/>;<br/><a NAME="E55:11_1_5_1"/>
<font color="Maroon">s7</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="jordan20.html#D6" target="_self">Def6</a></i>;<br/><a NAME="E56:11_1_5_1"/><b>then </b>
<font color="Maroon">s7</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s0</font>
 <b>by </b><i>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/><b>hence </b><a NAME="E57:11_1_5_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jordan20.html#D6" target="_self">Def6</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:11_1_5"/>
<font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 ;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">p4</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E41:11_1"/><i><font color="Green">E139</font></i>: 
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p9</font>,<font color="Maroon">p4</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; <font color="Maroon">p4</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p9</font> &amp; ( for <font color="Olive">p5</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p5</font>,<font color="Maroon">p4</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p9</font>,<font color="Olive">p5</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> holds <br/><font color="Olive">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> or for <font color="Olive">p6</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p6</font>,<font color="Maroon">p4</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p9</font>,<font color="Olive">p6</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> holds <br/><font color="Olive">p6</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">e</font> ) )
 <b>by </b><i>, , <a class="txt" href="jordan20.html#E3:11_1"><i><font color="Green">E30</font></i></a>, </i>;<br/><a NAME="E42:11_1"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">p9</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:11_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_6"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_1_6_1"/>
( for <font color="Olive">p5</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p5</font>,<font color="Maroon">p4</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p9</font>,<font color="Olive">p5</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> holds <br/><font color="Olive">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> or for <font color="Olive">p6</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p6</font>,<font color="Maroon">p4</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p9</font>,<font color="Olive">p6</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> holds <br/><font color="Olive">p6</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">e</font> )
 </span><b>by </b><i><a class="txt" href="jordan20.html#E5:11_1"><i><font color="Green">E32</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_6_1_1"><b>case </b></a><a NAME="E1:11_1_6_1_1"/><i><font color="Green">E141</font></i>: 
for <font color="Olive">p5</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p5</font>,<font color="Maroon">p4</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p9</font>,<font color="Olive">p5</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> holds <br/><font color="Olive">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font>
 ;<br/><div class="add"><div><i><font color="Green">E143</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_6_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_1_6_1_1_1"/>
 ex <font color="Olive">p51</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>(  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p51</font>,<font color="Maroon">p4</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p9</font>,<font color="Olive">p51</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; <font color="Olive">p51</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> )
 ;<br/><b>then consider </b><font color="Maroon">p51</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:11_1_6_1_1_1"/><i><font color="Green">E144</font></i>: 
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p51</font>,<font color="Maroon">p4</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p9</font>,<font color="Maroon">p51</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; <font color="Maroon">p51</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> )
 ;<br/><a NAME="E4:11_1_6_1_1_1"/><i><font color="Green">E145</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">p51</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E3:11_1"><i><font color="Green">E30</font></i></a>, , <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/><a NAME="E5:11_1_6_1_1_1"/>
for <font color="Olive">p5</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p5</font>,<font color="Maroon">p51</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Olive">p5</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> holds <br/><font color="Olive">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_6_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p5</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:11_1_6_1_1_1_1"/><i><font color="Green">E146</font></i>: 
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p5</font>,<font color="Maroon">p51</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">p5</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 ;<br/>

<a NAME="E2:11_1_6_1_1_1_1"/><b>then </b><i><font color="Green">E147</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p5</font>,<font color="Maroon">p4</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 
<b>by </b><i>, <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/>
<a NAME="E3:11_1_6_1_1_1_1"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">p5</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jordan20.html#T3" target="_self">Th3</a>, <a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/>
<a NAME="E4:11_1_6_1_1_1_1"/><b>then </b><i><font color="Green">E149</font></i>: 
( <font color="Maroon">p5</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p9</font> implies  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p9</font>,<font color="Maroon">p5</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 
<b>by </b><i><a class="ref" href="jordan5c.html#T9">JORDAN5C:9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_6_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_6_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="E1:11_1_6_1_1_1_1_1_1"/>
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p5</font>,<font color="Maroon">p9</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> or  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p9</font>,<font color="Maroon">p5</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 </span><b>by </b><i>, <a class="txt" href="jordan20.html#E6:11_1"><i><font color="Green">E33</font></i></a>, , <a class="ref" href="jordan20.html#T6" target="_self">Th6</a>, <a class="ref" href="jordan5c.html#T14">JORDAN5C:14</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_6_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:11_1_6_1_1_1_1_1_1_1"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p5</font>,<font color="Maroon">p9</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:11_1_6_1_1_1_1_1_1_1"/>
<font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="jordan20.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_6_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:11_1_6_1_1_1_1_1_1_2"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p9</font>,<font color="Maroon">p5</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:11_1_6_1_1_1_1_1_1_2"/>
<font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, <a class="txt" href="jordan20.html#E8:11_1"><i><font color="Green">E39</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:11_1_6_1_1_1_1"/>
<font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:11_1_6_1_1_1"/>
<font color="Maroon">p</font> <a href="jordan20.html#R3" target="_self">is_Lout</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">e</font>
 <b>by </b><i>, , <a class="txt" href="jordan20.html#E7:11_1"><i><font color="Green">E37</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_6_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_1_6_1_1_2"/><i><font color="Green">E150</font></i>: 
for <font color="Olive">p51</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  holds <br/>(  not  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p51</font>,<font color="Maroon">p4</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> or  not  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p9</font>,<font color="Olive">p51</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> or  not <font color="Olive">p51</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> )
 ;<br/><a NAME="E2:11_1_6_1_1_2"/><i><font color="Green">E151</font></i>: 
for <font color="Olive">p51</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p51</font>,<font color="Maroon">p4</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p9</font>,<font color="Olive">p51</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> holds <br/><font color="Olive">p51</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_6_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p51</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:11_1_6_1_1_2_1"/><i><font color="Green">E152</font></i>: 
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p51</font>,<font color="Maroon">p4</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p9</font>,<font color="Maroon">p51</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 ;<br/>

<a NAME="E2:11_1_6_1_1_2_1"/><b>then </b><i><font color="Green">E154</font></i>: 
<font color="Maroon">p51</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">e</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:11_1_6_1_1_2_1"/>
<font color="Maroon">p51</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font>
 
<b>by </b><i><a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, <a class="ref" href="jordan20.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E4:11_1_6_1_1_2_1"/>
<font color="Maroon">p51</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:11_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:11_1_6_1_1_2"/>
<font color="Maroon">p4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:11_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/><a NAME="E4:11_1_6_1_1_2"/><b>then </b>
<font color="Maroon">p4</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/><b>then consider </b><font color="Maroon">xs4</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:11_1_6_1_1_2"/><i><font color="Green">E156</font></i>: 
( <font color="Maroon">xs4</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">p4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">xs4</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">s4</font> = <font color="Maroon">xs4</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, <a class="ref" href="jordan20.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E8:11_1_6_1_1_2"/><i><font color="Green">E157</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s4</font> &amp; <font color="Maroon">s4</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="ref" href="jordan20.html#T3" target="_self">Th3</a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E9:11_1_6_1_1_2"/><b>then </b><i><font color="Green">E158</font></i>: 
<font color="Maroon">s4</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s0</font>
 <b>by </b><i>, , <a class="txt" href="jordan20.html#E5:11_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="jordan20.html#T3" target="_self">Th3</a>, <a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_6_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_1_6_1_1_2_2"/>
<font color="Maroon">s4</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s0</font>
 ;<br/><a NAME="E2:11_1_6_1_1_2_2"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">s4</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span></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/><a NAME="E3:11_1_6_1_1_2_2"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">s4</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><a NAME="E4:11_1_6_1_1_2_2"/><b>then </b><i><font color="Green">E159</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s4</font></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i><a class="ref" href="real_2.html#T128">REAL_2:128</a></i>;<br/><a NAME="E5:11_1_6_1_1_2_2"/><b>then </b><i><font color="Green">E160</font></i>: 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s4</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><a NAME="E6:11_1_6_1_1_2_2"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">s4</font> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s4</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s4</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="jordan20.html#T6" target="_self">Th6</a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>consider </b><font color="Maroon">r</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E8:11_1_6_1_1_2_2"/><i><font color="Green">E162</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s0</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s4</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/><a NAME="E9:11_1_6_1_1_2_2"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s0</font>
 <b>by </b><i>, <a class="txt" href="jordan20.html#E12:11_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/><a NAME="E10:11_1_6_1_1_2_2"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">s4</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E9:11_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="jordan20.html#E12:11_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E11:11_1_6_1_1_2_2"/><b>then </b><i><font color="Green">E165</font></i>: 
1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:11_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>consider </b><font color="Maroon">s7</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E13:11_1_6_1_1_2_2"/><i><font color="Green">E166</font></i>: 
( <font color="Maroon">s7</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp; 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s7</font> &amp; <font color="Maroon">s7</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s2</font> &amp; ( for <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s7</font> holds <br/><font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e</font> ) )
 <b>by </b><i><a class="txt" href="jordan20.html#E12:11_1"><i><font color="Green">E44</font></i></a></i>;<br/><a NAME="E14:11_1_6_1_1_2_2"/><i><font color="Green">E169</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="txt" href="jordan20.html#E21:11_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><b>reconsider </b><font color="Maroon">rss</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="E16:11_1_6_1_1_2_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">rss</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E17:11_1_6_1_1_2_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">rss</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/><b>then reconsider </b><font color="Maroon">pss</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">rss</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E19:11_1_6_1_1_2_2"/>
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">pss</font>,<font color="Maroon">p4</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p9</font>,<font color="Maroon">pss</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 <b>by </b><i>, , , , <a class="txt" href="jordan20.html#E1:11_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jordan20.html#T3" target="_self">Th3</a>, <a class="txt" href="jordan20.html#E8:11_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan20.html#E14:11_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan20.html#E18:11_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan20.html#E20:11_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="jordan5c.html#T8">JORDAN5C:8</a></i>;<br/><a NAME="E20:11_1_6_1_1_2_2"/><b>then </b>
<font color="Maroon">pss</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E7:11_1"><i><font color="Green">E37</font></i></a></i>;<br/><b>hence </b><a NAME="E21:11_1_6_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E21:11_1"><i><font color="Green">E48</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E11:11_1_6_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E5:11_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="jordan20.html#T3" target="_self">Th3</a>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:11_1_6_1_1"/>
( <font color="Maroon">p</font> <a href="jordan20.html#R3" target="_self">is_Lout</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">e</font> or <font color="Maroon">p</font> <a href="jordan20.html#R4" target="_self">is_Rout</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">e</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_6_1_2"><b>case </b></a><a NAME="E1:11_1_6_1_2"/><i><font color="Green">E170</font></i>: 
for <font color="Olive">p6</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p6</font>,<font color="Maroon">p4</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p9</font>,<font color="Olive">p6</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> holds <br/><font color="Olive">p6</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">e</font>
 ;<br/><div class="add"><div><i><font color="Green">E171</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_6_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_1_6_1_2_1"/>
 ex <font color="Olive">p51</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>(  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p51</font>,<font color="Maroon">p4</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p9</font>,<font color="Olive">p51</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; <font color="Olive">p51</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">e</font> )
 ;<br/><b>then consider </b><font color="Maroon">p51</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:11_1_6_1_2_1"/><i><font color="Green">E172</font></i>: 
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p51</font>,<font color="Maroon">p4</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p9</font>,<font color="Maroon">p51</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; <font color="Maroon">p51</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">e</font> )
 ;<br/><a NAME="E4:11_1_6_1_2_1"/><i><font color="Green">E173</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">p51</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E3:11_1"><i><font color="Green">E30</font></i></a>, , <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/><a NAME="E5:11_1_6_1_2_1"/>
for <font color="Olive">p5</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p5</font>,<font color="Maroon">p51</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Olive">p5</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> holds <br/><font color="Olive">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_6_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p5</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:11_1_6_1_2_1_1"/><i><font color="Green">E174</font></i>: 
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p5</font>,<font color="Maroon">p51</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">p5</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 ;<br/>

<a NAME="E2:11_1_6_1_2_1_1"/><b>then </b><i><font color="Green">E175</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p5</font>,<font color="Maroon">p4</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 
<b>by </b><i>, <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/>
<a NAME="E3:11_1_6_1_2_1_1"/><i><font color="Green">E176</font></i>: 
<font color="Maroon">p5</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jordan20.html#T3" target="_self">Th3</a>, <a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/>
<a NAME="E4:11_1_6_1_2_1_1"/><b>then </b><i><font color="Green">E177</font></i>: 
( <font color="Maroon">p5</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p9</font> implies  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p9</font>,<font color="Maroon">p5</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 
<b>by </b><i><a class="ref" href="jordan5c.html#T9">JORDAN5C:9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_6_1_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_6_1_2_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="E1:11_1_6_1_2_1_1_1_1"/>
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p9</font>,<font color="Maroon">p5</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> or  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p5</font>,<font color="Maroon">p9</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 </span><b>by </b><i>, <a class="txt" href="jordan20.html#E6:11_1"><i><font color="Green">E33</font></i></a>, , <a class="ref" href="jordan20.html#T6" target="_self">Th6</a>, <a class="ref" href="jordan5c.html#T14">JORDAN5C:14</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_6_1_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:11_1_6_1_2_1_1_1_1_1"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p9</font>,<font color="Maroon">p5</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:11_1_6_1_2_1_1_1_1_1"/>
<font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, <a class="txt" href="jordan20.html#E8:11_1"><i><font color="Green">E39</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_6_1_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:11_1_6_1_2_1_1_1_1_2"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p5</font>,<font color="Maroon">p9</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:11_1_6_1_2_1_1_1_1_2"/>
<font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="jordan20.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:11_1_6_1_2_1_1"/>
<font color="Maroon">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">e</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:11_1_6_1_2_1"/>
<font color="Maroon">p</font> <a href="jordan20.html#R4" target="_self">is_Rout</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">e</font>
 <b>by </b><i>, , <a class="txt" href="jordan20.html#E7:11_1"><i><font color="Green">E37</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_6_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_1_6_1_2_2"/><i><font color="Green">E178</font></i>: 
for <font color="Olive">p51</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  holds <br/>(  not  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p51</font>,<font color="Maroon">p4</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> or  not  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p9</font>,<font color="Olive">p51</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> or  not <font color="Olive">p51</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">e</font> )
 ;<br/><a NAME="E2:11_1_6_1_2_2"/><i><font color="Green">E179</font></i>: 
for <font color="Olive">p51</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p51</font>,<font color="Maroon">p4</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p9</font>,<font color="Olive">p51</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> holds <br/><font color="Olive">p51</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_6_1_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p51</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:11_1_6_1_2_2_1"/><i><font color="Green">E180</font></i>: 
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p51</font>,<font color="Maroon">p4</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p9</font>,<font color="Maroon">p51</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 ;<br/>

<a NAME="E2:11_1_6_1_2_2_1"/><b>then </b><i><font color="Green">E181</font></i>: 
<font color="Maroon">p51</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:11_1_6_1_2_2_1"/>
<font color="Maroon">p51</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">e</font>
 
<b>by </b><i><a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, <a class="ref" href="jordan20.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E4:11_1_6_1_2_2_1"/>
<font color="Maroon">p51</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:11_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:11_1_6_1_2_2"/>
<font color="Maroon">p4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:11_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/><a NAME="E4:11_1_6_1_2_2"/><b>then </b>
<font color="Maroon">p4</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/><b>then consider </b><font color="Maroon">xs4</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:11_1_6_1_2_2"/><i><font color="Green">E182</font></i>: 
( <font color="Maroon">xs4</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">p4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">xs4</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">s4</font> = <font color="Maroon">xs4</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, <a class="ref" href="jordan20.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E8:11_1_6_1_2_2"/><i><font color="Green">E183</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s4</font> &amp; <font color="Maroon">s4</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="ref" href="jordan20.html#T3" target="_self">Th3</a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E9:11_1_6_1_2_2"/><b>then </b><i><font color="Green">E184</font></i>: 
<font color="Maroon">s4</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s0</font>
 <b>by </b><i>, , <a class="txt" href="jordan20.html#E5:11_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="jordan20.html#T3" target="_self">Th3</a>, <a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_6_1_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_1_6_1_2_2_2"/>
<font color="Maroon">s4</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s0</font>
 ;<br/><a NAME="E2:11_1_6_1_2_2_2"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">s4</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span></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/><a NAME="E3:11_1_6_1_2_2_2"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">s4</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><a NAME="E4:11_1_6_1_2_2_2"/><b>then </b><i><font color="Green">E185</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s4</font></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i><a class="ref" href="real_2.html#T128">REAL_2:128</a></i>;<br/><a NAME="E5:11_1_6_1_2_2_2"/><b>then </b><i><font color="Green">E186</font></i>: 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s4</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><a NAME="E6:11_1_6_1_2_2_2"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">s4</font> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s4</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s4</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="jordan20.html#T6" target="_self">Th6</a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>consider </b><font color="Maroon">r</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E8:11_1_6_1_2_2_2"/><i><font color="Green">E188</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s0</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s4</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/><a NAME="E9:11_1_6_1_2_2_2"/><i><font color="Green">E189</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s0</font>
 <b>by </b><i>, <a class="txt" href="jordan20.html#E12:11_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/><a NAME="E10:11_1_6_1_2_2_2"/><i><font color="Green">E190</font></i>: 
<font color="Maroon">s4</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E9:11_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="jordan20.html#E12:11_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E11:11_1_6_1_2_2_2"/><b>then </b><i><font color="Green">E191</font></i>: 
1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:11_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>consider </b><font color="Maroon">s7</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E13:11_1_6_1_2_2_2"/><i><font color="Green">E192</font></i>: 
( <font color="Maroon">s7</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp; 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s7</font> &amp; <font color="Maroon">s7</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s2</font> &amp; ( for <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s7</font> holds <br/><font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e</font> ) )
 <b>by </b><i><a class="txt" href="jordan20.html#E12:11_1"><i><font color="Green">E44</font></i></a></i>;<br/><a NAME="E14:11_1_6_1_2_2_2"/><i><font color="Green">E193</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="txt" href="jordan20.html#E21:11_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><b>reconsider </b><font color="Maroon">rss</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="E16:11_1_6_1_2_2_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">rss</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E17:11_1_6_1_2_2_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">rss</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/><b>then reconsider </b><font color="Maroon">pss</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">rss</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E19:11_1_6_1_2_2_2"/>
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">pss</font>,<font color="Maroon">p4</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p9</font>,<font color="Maroon">pss</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 <b>by </b><i>, , , , <a class="txt" href="jordan20.html#E1:11_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jordan20.html#T3" target="_self">Th3</a>, <a class="txt" href="jordan20.html#E8:11_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan20.html#E14:11_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan20.html#E18:11_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan20.html#E20:11_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="jordan5c.html#T8">JORDAN5C:8</a></i>;<br/><a NAME="E20:11_1_6_1_2_2_2"/><b>then </b>
<font color="Maroon">pss</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E7:11_1"><i><font color="Green">E37</font></i></a></i>;<br/><b>hence </b><a NAME="E21:11_1_6_1_2_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E21:11_1"><i><font color="Green">E48</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E11:11_1_6_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E5:11_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="jordan20.html#T3" target="_self">Th3</a>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:11_1_6_1_2"/>
( <font color="Maroon">p</font> <a href="jordan20.html#R3" target="_self">is_Lout</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">e</font> or <font color="Maroon">p</font> <a href="jordan20.html#R4" target="_self">is_Rout</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">e</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="E44:11_1"/>
( <font color="Maroon">p</font> <a href="jordan20.html#R3" target="_self">is_Lout</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">e</font> or <font color="Maroon">p</font> <a href="jordan20.html#R4" target="_self">is_Rout</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">e</font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:11"/>
( <font color="Maroon">p</font> <a href="jordan20.html#R3" target="_self">is_Lout</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">e</font> or <font color="Maroon">p</font> <a href="jordan20.html#R4" target="_self">is_Rout</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">e</font> or <font color="Maroon">p</font> <a href="jordan20.html#R6" target="_self">is_OSout</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">e</font> )
 ;<br/>


</div>
