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<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <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="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<b>consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:39_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E3:39_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E4:39_1_1"/><i><font color="Green">E32</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="weierstr.html#T36">WEIERSTR:36</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:39_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E7:39_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E8:39_1_1"/><i><font color="Green">E35</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="weierstr.html#T36">WEIERSTR:36</a></i>;<br/>
<a NAME="E9:39_1_1"/><i><font color="Green">E36</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="jordan1k.html#E2:39_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan1k.html#E3:39_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan1k.html#E8:39_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="weierstr.html#T40">WEIERSTR:40</a></i>;<br/>
<a NAME="E10:39_1_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="jordan1k.html#E4:39_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan1k.html#E6:39_1_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan1k.html#E7:39_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="weierstr.html#T40">WEIERSTR:40</a></i>;<br/>
<b>hence </b><a NAME="E11:39_1_1"/>
 <a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jordan1k.html#E4:39_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan1k.html#E8:39_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan1k.html#E9:39_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


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