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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="metric_1.html#NM1">MetrSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <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>assume </b><a NAME="E1:31"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="compts_1.html#V6">compact</a> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="pre_topc.html#NM2">Point</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><b> such that </b><br/><a NAME="E3:31"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="weierstr.html#K6" target="_self">dist</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="weierstr.html#K4" target="_self">upper_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K6" target="_self">dist</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="weierstr.html#E1:31"><i><font color="Green">E23</font></i></a>, <a class="ref" href="weierstr.html#T23" target="_self">Th23</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="pre_topc.html#NM2">Point</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><b> such that </b><br/><a NAME="E5:31"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="weierstr.html#K6" target="_self">dist</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="weierstr.html#K4" target="_self">upper_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K6" target="_self">dist</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="weierstr.html#E1:31"><i><font color="Green">E23</font></i></a>, <a class="ref" href="weierstr.html#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E6:31"/><i><font color="Green">E26</font></i>: 
 <a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="pcomps_1.html#K3">Family_open_set</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="pcomps_1.html#D6">PCOMPS_1:def 6</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="weierstr.html#E6:31"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E9:31"/><i><font color="Green">E27</font></i>: 
<span class="p1">(<span class="default"><a href="weierstr.html#K6" target="_self">dist</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="weierstr.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E10:31"/><i><font color="Green">E28</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="weierstr.html#K4" target="_self">upper_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K6" target="_self">dist</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="weierstr.html#E3:31"><i><font color="Green">E24</font></i></a>, <a class="ref" href="weierstr.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E11:31"/><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><a href="weierstr.html#K6" target="_self">dist</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="weierstr.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E12:31"/><i><font color="Green">E30</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="weierstr.html#K4" target="_self">upper_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K6" target="_self">dist</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="weierstr.html#E5:31"><i><font color="Green">E25</font></i></a>, <a class="ref" href="weierstr.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E13:31"/><i><font color="Green">E31</font></i>: 
(  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="metric_1.html#T4">METRIC_1:4</a></i>;<br/>
<a NAME="E14:31"/><i><font color="Green">E32</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</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>4</sub></font>
 
<b>by </b><i><a class="txt" href="weierstr.html#E1:31"><i><font color="Green">E23</font></i></a>, <a class="txt" href="weierstr.html#E3:31"><i><font color="Green">E24</font></i></a>, <a class="txt" href="weierstr.html#E5:31"><i><font color="Green">E25</font></i></a>, <a class="txt" href="weierstr.html#E11:31"><i><font color="Green">E29</font></i></a>, <a class="ref" href="weierstr.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E15:31"/><i><font color="Green">E33</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="weierstr.html#E1:31"><i><font color="Green">E23</font></i></a>, <a class="txt" href="weierstr.html#E3:31"><i><font color="Green">E24</font></i></a>, <a class="txt" href="weierstr.html#E5:31"><i><font color="Green">E25</font></i></a>, <a class="txt" href="weierstr.html#E9:31"><i><font color="Green">E27</font></i></a>, <a class="ref" href="weierstr.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E16:31"/><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="weierstr.html#E13:31"><i><font color="Green">E31</font></i></a>, <a class="txt" href="weierstr.html#E14:31"><i><font color="Green">E32</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E17:31"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="weierstr.html#E13:31"><i><font color="Green">E31</font></i></a>, <a class="txt" href="weierstr.html#E15:31"><i><font color="Green">E33</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E18:31"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E19:31"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E20:31"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K4" target="_self">upper_bound</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="weierstr.html#K6" target="_self">dist</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="weierstr.html#K4" target="_self">upper_bound</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="weierstr.html#K6" target="_self">dist</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</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="weierstr.html#E10:31"><i><font color="Green">E28</font></i></a>, <a class="txt" href="weierstr.html#E12:31"><i><font color="Green">E30</font></i></a>, <a class="txt" href="weierstr.html#E16:31"><i><font color="Green">E34</font></i></a>, <a class="ref" href="absvalue.html#T12">ABSVALUE:12</a></i>;<br/>


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