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

<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> be  <a href="compts_1.html#V6">compact</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>assume </b><a NAME="E1:31"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan1c.html#E3"><i><font color="Green">Lemma3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:31"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="jordan1c.html#E1:31"><i><font color="Green">E35</font></i></a>, <a class="ref" href="jordan1c.html#T29" target="_self">Th29</a></i>;<br/>
<b>assume </b><a NAME="E5:31"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<a NAME="E6:31"/><i><font color="Green">E38</font></i>: 
 <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>2</sub></font> is <a href="jordan2c.html#V1">Bounded</a>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T114">JORDAN2C:114</a></i>;<br/>
<a NAME="E7:31"/><i><font color="Green">E39</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="txt" href="jordan1c.html#E4:31"><i><font color="Green">E36</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<a NAME="E8:31"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E4:31"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xreal_1.html#T218">XREAL_1:218</a></i>;<br/>
<a NAME="E9:31"/><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> 0
 
<b>by </b><i><a class="txt" href="jordan1c.html#E7:31"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>;<br/>
<a NAME="E10:31"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/>
<a NAME="E11:31"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E7:31"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan1c.html#E8:31"><i><font color="Green">E40</font></i></a>, <a class="ref" href="goboard6.html#T11">GOBOARD6:11</a></i>;<br/>
<a NAME="E12:31"/>
not <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1"/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:31_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E6:31"><i><font color="Green">E38</font></i></a>, <a class="ref" href="jordan1c.html#T6" target="_self">Th6</a></i>;<br/>
<b>hence </b><a NAME="E3:31_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan1c.html#E5:31"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan1c.html#E9:31"><i><font color="Green">E41</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:31"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan1c.html#E4:31"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan1c.html#E11:31"><i><font color="Green">E42</font></i></a></i>;<br/>


</div>
