<?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>, <font color="Maroon">Q</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">q1</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">q2</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><b>that </b><br/><a NAME="E1:52"/><i><font color="Green">E34</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">Q</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font> )
 <b>and </b><br/><a NAME="E2:52"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">p</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">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/>( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  )
 <b>and </b><br/><a NAME="E3:52"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">p</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">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> holds <br/>( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  )
 <b>and </b><br/><a NAME="E4:52"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">p</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">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/>( <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a>  )
 <b>and </b><br/><a NAME="E5:52"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">p</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">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> holds <br/>( <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a>  )
 ;<br/>

<a NAME="E6:52"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font>
 
<b>by </b><i>, <a class="ref" href="jordan6.html#T49">JORDAN6:49</a></i>;<br/>
<a NAME="E7:52"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">q1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q2</font>
 
<b>by </b><i>, <a class="ref" href="jordan6.html#T49">JORDAN6:49</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="E9:52"/><i><font color="Green">E41</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/>
<b>consider </b><font color="Maroon">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">Q</font></span>)</span><b> such that </b><br/><a NAME="E11:52"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">g</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> )
 <b>by </b><i>, <a class="ref" href="topreal1.html#D2">TOPREAL1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">f1</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"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E13:52"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">f</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">f1</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">f1</font> is <a href="funct_1.html#V2">one-to-one</a> )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E13:52"><i><font color="Green">E54</font></i></a>, <a class="ref" href="jordan7.html#T15">JORDAN7:15</a></i>;<br/>
<b>consider </b><font color="Maroon">g1</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"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E15:52"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">g</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">g1</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">g1</font> is <a href="funct_1.html#V2">one-to-one</a> )
 <b>by </b><i><a class="ref" href="jgraph_1.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan7.html#T15">JORDAN7:15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">P'</font> = <font color="Maroon">P</font>, <font color="Maroon">Q'</font> = <font color="Maroon">Q</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span></span>)</span> ;<br/>
<a NAME="E17:52"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">P'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">P'</font> is <a href="compts_1.html#V6">compact</a> &amp; <font color="Maroon">Q'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Q'</font> is <a href="compts_1.html#V6">compact</a> )
 
<b>by </b><i>, <a class="ref" href="jordan5a.html#T1">JORDAN5A:1</a></i>;<br/>
<b>assume </b><a NAME="E18:52"/>
<font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/>

<a NAME="E19:52"/><b>then </b>
<font color="Maroon">P</font> <a href="subset_1.html#R1">misses</a> <font color="Maroon">Q</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E20:52"/><b>then </b><i><font color="Green">E63</font></i>: 
 <a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="txt" href="jgraph_1.html#E7:52"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">e</font> = <span class="p1">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5;<br/>
<a NAME="E21:52"/><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="real_1.html#K6">/</a> 5
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E15:52"><i><font color="Green">E59</font></i></a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<b>then consider </b><font color="Maroon">h</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E23:52"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; 5 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  &amp; <font color="Maroon">h</font> is <a href="goboard1.html#V1">increasing</a> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">R</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> <br/> for <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font> &amp; <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Olive">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">R</font> holds <br/> <a href="tbsp_1.html#K2">diameter</a> <font color="Olive">W</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5 ) )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E13:52"><i><font color="Green">E54</font></i></a>, <a class="ref" href="jgraph_1.html#T10" target="_self">Th10</a>, <a class="ref" href="uniform1.html#T15">UNIFORM1:15</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E24:52"/>
 ex <font color="Olive">h1'</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3">len</a> <font color="Olive">h1'</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font></span>)</span> holds <br/><font color="Olive">h1'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</font>) ) )
 
<b>from </b><i><a class="ref" href="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i>
<b>then consider </b><font color="Maroon">h1'</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E26:52"/><i><font color="Green">E66</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1'</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font></span>)</span> holds <br/><font color="Maroon">h1'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <font color="Olive">i</font></span>)</span> ) )
 ;<br/>
<a NAME="E27:52"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">h1'</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:52_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h1'</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:52_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h1'</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:52_1"/><i><font color="Green">E68</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h1'</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="jgraph_1.html#E20:52"><i><font color="Green">E63</font></i></a></i>;<br/>
<a NAME="E6:52_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">h1'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="jgraph_1.html#E20:52"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_1.html#E21:52"><i><font color="Green">E64</font></i></a></i>;<br/>
<a NAME="E7:52_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h</font>
 
<b>by </b><i>, <a class="txt" href="jgraph_1.html#E20:52"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_1.html#E21:52"><i><font color="Green">E64</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:52_1"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> = 
 <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> 
<b>by </b><i><a class="txt" href="jgraph_1.html#E13:52"><i><font color="Green">E54</font></i></a>, <a class="ref" href="jgraph_1.html#T10" target="_self">Th10</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
;<br/>
<a NAME="E10:52_1"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">h1'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</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="jgraph_1.html#T10" target="_self">Th10</a>, , <a class="txt" href="jgraph_1.html#E23:52"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jgraph_1.html#E26:52"><i><font color="Green">E66</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
 <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="txt" href="jgraph_1.html#E13:52"><i><font color="Green">E54</font></i></a>, <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/>
<b>hence </b><a NAME="E12:52_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_1.html#E20:52"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_1.html#E3:52_1"><i><font color="Green">E67</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">h1</font> = <font color="Maroon">h1'</font> as   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E29:52"/><i><font color="Green">E72</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E30:52"/><i><font color="Green">E73</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<a NAME="E31:52"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> holds <br/><span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:52_2"/><i><font color="Green">E75</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> )
 ;<br/>

<a NAME="E2:52_2"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:52_2"/><i><font color="Green">E77</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E26:52"><i><font color="Green">E66</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:52_2"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="jgraph_1.html#E26:52"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:52_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="jgraph_1.html#E26:52"><i><font color="Green">E66</font></i></a>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:52_2"/><b>then </b><i><font color="Green">E109</font></i>: 
<font color="Maroon">h1'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E7:52_2"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E26:52"><i><font color="Green">E66</font></i></a>, , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E8:52_2"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">h1'</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:52_2"/><b>then </b><i><font color="Green">E112</font></i>: 
<font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E26:52"><i><font color="Green">E66</font></i></a>, , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E10:52_2"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h</font>
 
<b>by </b><i>, <a class="txt" href="jgraph_1.html#E26:52"><i><font color="Green">E66</font></i></a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E11:52_2"/><b>then </b><i><font color="Green">E114</font></i>: 
<font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<i><font color="Green">E115</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> = 
 <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> 
<b>by </b><i><a class="txt" href="jgraph_1.html#E13:52"><i><font color="Green">E54</font></i></a>, <a class="ref" href="jgraph_1.html#T10" target="_self">Th10</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
;<br/>
<a NAME="E13:52_2"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">h1'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</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="jgraph_1.html#T10" target="_self">Th10</a>, , , <a class="txt" href="jgraph_1.html#E2:52_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<i><font color="Green">E117</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="txt" href="jgraph_1.html#E13:52"><i><font color="Green">E54</font></i></a>, <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/>
<a NAME="E15:52_2"/><b>then </b><i><font color="Green">E118</font></i>: 
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i>, , <a class="txt" href="jgraph_1.html#E4:52_2"><i><font color="Green">E78</font></i></a></i>;<br/>
<a NAME="E16:52_2"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h</font>
 
<b>by </b><i>, <a class="txt" href="jgraph_1.html#E26:52"><i><font color="Green">E66</font></i></a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E17:52_2"/><b>then </b>
<font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E18:52_2"/><b>then </b>
<font color="Maroon">h1'</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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="jgraph_1.html#T10" target="_self">Th10</a>, , <a class="txt" href="jgraph_1.html#E30:52"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jgraph_1.html#E3:52_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E19:52_2"/><b>then </b><i><font color="Green">E119</font></i>: 
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i>, , <a class="txt" href="jgraph_1.html#E5:52_2"><i><font color="Green">E79</font></i></a></i>;<br/>
<a NAME="E20:52_2"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, <a class="txt" href="jgraph_1.html#E26:52"><i><font color="Green">E66</font></i></a>, , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E21:52_2"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h</font>
 
<b>by </b><i>, <a class="txt" href="jgraph_1.html#E26:52"><i><font color="Green">E66</font></i></a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E22:52_2"/><b>then </b><i><font color="Green">E122</font></i>: 
<font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E23:52_2"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="txt" href="jgraph_1.html#E26:52"><i><font color="Green">E66</font></i></a>, , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E24:52_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E25:52_2"/><b>then </b><i><font color="Green">E124</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i>, <a class="txt" href="jgraph_1.html#E1:52_2"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jgraph_1.html#E2:52_2"><i><font color="Green">E76</font></i></a>, , <a class="ref" href="jgraph_1.html#D3" target="_self">Def3</a>, , , <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<b>reconsider </b><font color="Maroon">W1</font> = <font color="Maroon">P</font> as   <a href="struct_0.html#NM2">Subset</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="jgraph_1.html#E21:52"><i><font color="Green">E64</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">Pa</font> = <font color="Maroon">P</font> as   <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/>
<a NAME="E28:52_2"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">Pa</font> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i>, <a class="ref" href="jordan5a.html#T1">JORDAN5A:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Pa</font> = <font color="Maroon">Pa</font> as  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>reconsider </b><font color="Maroon">R</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i>, <a class="txt" href="jgraph_1.html#E2:52_2"><i><font color="Green">E76</font></i></a>, , , , <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T16">RCOMP_1:16</a></i>;<br/>
<b>reconsider </b><font color="Maroon">w1</font> = <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>, <font color="Maroon">w2</font> = <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> 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="ref" href="jgraph_1.html#T11" target="_self">Th11</a>, , <a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<a NAME="E32:52_2"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i>, , <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E33:52_2"/>
<font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i>, , <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E34:52_2"/><b>then </b><i><font color="Green">E127</font></i>: 
( <font color="Maroon">w1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">R</font> &amp; <font color="Maroon">w2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">R</font> )
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E3:52_2"><i><font color="Green">E77</font></i></a>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">W</font> = <font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">R</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<a NAME="E36:52_2"/><i><font color="Green">E128</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</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">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> )
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E13:52"><i><font color="Green">E54</font></i></a>, <a class="ref" href="jgraph_1.html#T10" target="_self">Th10</a>, <a class="txt" href="jgraph_1.html#E5:52_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E37:52_2"/>
 <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a>  <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="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E38:52_2"/><b>then </b>
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</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="jgraph_1.html#T13" target="_self">Th13</a>, <a class="ref" href="relat_1.html#T146">RELAT_1:146</a></i>;<br/>
<a NAME="E39:52_2"/><b>then </b><i><font color="Green">E137</font></i>: 
<font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W1</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<a NAME="E40:52_2"/>
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pa</font>
 
<b>by </b><i>, <a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/>
<a NAME="E41:52_2"/><b>then </b><i><font color="Green">E138</font></i>: 
(  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">Pa</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">Pa</font> &amp;  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">Pa</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">Pa</font> )
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T12" target="_self">Th12</a>, <a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/>
<b>set </b><font color="Maroon">r1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">Pa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">Pa</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">Pa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">Pa</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1;<br/>
<a NAME="E42:52_2"/>
 <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">Pa</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">Pa</font>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T14" target="_self">Th14</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E43:52_2"/><b>then </b><i><font color="Green">E139</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">Pa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">Pa</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E44:52_2"/>
 <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">Pa</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">Pa</font>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T14" target="_self">Th14</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E45:52_2"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">Pa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">Pa</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E46:52_2"/><b>then </b><i><font color="Green">E140</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">Pa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">Pa</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">Pa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">Pa</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E47:52_2"/><i><font color="Green">E141</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">Pa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">Pa</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">Pa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">Pa</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">Pa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">Pa</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">Pa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">Pa</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E48:52_2"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <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>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> holds <br/> <a href="metric_1.html#K4">dist</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">Pa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">Pa</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">Pa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">Pa</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <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>, <font color="Maroon">y</font> be   <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>;<br/>

<b>assume </b><a NAME="E1:52_2_3"/><i><font color="Green">E142</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">pw1</font> = <font color="Maroon">x</font>, <font color="Maroon">pw2</font> = <font color="Maroon">y</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="E3:52_2_3"/><i><font color="Green">E143</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">pw1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">pw2</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:52_2_3"/><i><font color="Green">E144</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">pw1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">pw2</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">pw1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">pw2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">pw1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">pw2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:52"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E5:52_2_3"/><i><font color="Green">E145</font></i>: 
(  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">Pa</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pw1</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">pw1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">Pa</font> &amp;  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">Pa</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pw1</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">pw1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">Pa</font> )
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T12" target="_self">Th12</a>, , <a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/>
<a NAME="E6:52_2_3"/><i><font color="Green">E146</font></i>: 
(  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">Pa</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pw2</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">pw2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">Pa</font> &amp;  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">Pa</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pw2</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">pw2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">Pa</font> )
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T12" target="_self">Th12</a>, , <a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/>
<a NAME="E7:52_2_3"/><b>then </b><i><font color="Green">E147</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">pw1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">pw2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">Pa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">Pa</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T18" target="_self">Th18</a>, </i>;<br/>
<a NAME="E8:52_2_3"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">pw1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">pw2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">Pa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">Pa</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T18" target="_self">Th18</a>, <a class="ref" href="jgraph_1.html#T19" target="_self">Th19</a>, </i>;<br/>
<a NAME="E9:52_2_3"/><b>then </b><i><font color="Green">E148</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">pw1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">pw2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">pw1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">pw2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">Pa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">Pa</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">Pa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">Pa</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T20" target="_self">Th20</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E10:52_2_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">Pa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">Pa</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">Pa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">Pa</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">Pa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">Pa</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">Pa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">Pa</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E11:52_2_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">pw1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">pw2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">pw1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">pw2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">Pa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">Pa</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">Pa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">Pa</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E12:52_2_3"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">Pa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">Pa</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">Pa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">Pa</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="jgraph_1.html#D4" target="_self">Def4</a>, <a class="ref" href="jgraph_1.html#T17" target="_self">Th17</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E49:52_2"/><b>then </b>
<font color="Maroon">W1</font> is <a href="tbsp_1.html#V6">bounded</a>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T15" target="_self">Th15</a>, , <a class="ref" href="tbsp_1.html#D9">TBSP_1:def 9</a></i>;<br/>
<a NAME="E50:52_2"/><b>then </b><i><font color="Green">E149</font></i>: 
<font color="Maroon">W</font> is <a href="tbsp_1.html#V6">bounded</a>
 
<b>by </b><i>, <a class="ref" href="tbsp_1.html#T21">TBSP_1:21</a></i>;<br/>
<a NAME="E51:52_2"/><i><font color="Green">E150</font></i>: 
 <a href="tbsp_1.html#K2">diameter</a> <font color="Maroon">W</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5
 
<b>by </b><i>, , <a class="txt" href="jgraph_1.html#E26:52"><i><font color="Green">E66</font></i></a>, </i>;<br/>
<a NAME="E52:52_2"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">w1</font>,<font color="Maroon">w2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="tbsp_1.html#K2">diameter</a> <font color="Maroon">W</font>
 
<b>by </b><i>, , <a class="ref" href="tbsp_1.html#D10">TBSP_1:def 10</a></i>;<br/>
<a NAME="E53:52_2"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">w1</font>,<font color="Maroon">w2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5
 
<b>by </b><i><a class="ref" href="jgraph_1.html#D4" target="_self">Def4</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E54:52_2"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5
 <b>by </b><i><a class="txt" href="jgraph_1.html#E29:52"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_1.html#E31:52"><i><font color="Green">E74</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E32:52"/><i><font color="Green">E151</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>consider </b><font color="Maroon">hb</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E34:52"/><i><font color="Green">E152</font></i>: 
( <font color="Maroon">hb</font> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">hb</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">hb</font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; 5 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">hb</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">hb</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  &amp; <font color="Maroon">hb</font> is <a href="goboard1.html#V1">increasing</a> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">R</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> <br/> for <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">hb</font> &amp; <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">hb</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">hb</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Olive">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">R</font> holds <br/> <a href="tbsp_1.html#K2">diameter</a> <font color="Olive">W</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5 ) )
 <b>by </b><i><a class="ref" href="jgraph_1.html#T9" target="_self">Th9</a>, , <a class="txt" href="jgraph_1.html#E17:52"><i><font color="Green">E60</font></i></a>, <a class="ref" href="uniform1.html#T15">UNIFORM1:15</a></i>;<br/>
<a NAME="E35:52"/><i><font color="Green">E153</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">hb</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E36:52"/><b>then </b><i><font color="Green">E154</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">hb</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E37:52"/><i><font color="Green">E155</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">hb</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">hb</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">hb</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E38:52"/>
 ex <font color="Olive">h2'</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3">len</a> <font color="Olive">h2'</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">hb</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">hb</font></span>)</span> holds <br/><font color="Olive">h2'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">i</font>) ) )
 
<b>from </b><i><a class="ref" href="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i>
<b>then consider </b><font color="Maroon">h2'</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E40:52"/><i><font color="Green">E156</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h2'</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">hb</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">hb</font></span>)</span> holds <br/><font color="Maroon">h2'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">hb</font> <a href="goboard1.html#K1">.</a> <font color="Olive">i</font></span>)</span> ) )
 ;<br/>
<a NAME="E41:52"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">h2'</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:52_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h2'</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:52_3"/><i><font color="Green">E157</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h2'</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:52_3"/><i><font color="Green">E158</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h2'</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="jgraph_1.html#E29:52"><i><font color="Green">E72</font></i></a></i>;<br/>
<a NAME="E6:52_3"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">h2'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">hb</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="jgraph_1.html#E29:52"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_1.html#E30:52"><i><font color="Green">E73</font></i></a></i>;<br/>
<a NAME="E7:52_3"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">hb</font>
 
<b>by </b><i>, <a class="txt" href="jgraph_1.html#E29:52"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_1.html#E30:52"><i><font color="Green">E73</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:52_3"/><b>then </b><i><font color="Green">E160</font></i>: 
<font color="Maroon">hb</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">hb</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font> = 
 <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> 
<b>by </b><i><a class="ref" href="jgraph_1.html#T9" target="_self">Th9</a>, , <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
;<br/>
<a NAME="E10:52_3"/><b>then </b><i><font color="Green">E161</font></i>: 
<font color="Maroon">h2'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font>
 
<b>by </b><i>, , <a class="txt" href="jgraph_1.html#E31:52"><i><font color="Green">E74</font></i></a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</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">Q</font></span>)</span>
<b>by </b><i><a class="ref" href="jgraph_1.html#T9" target="_self">Th9</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
<font color="Maroon">Q</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<b>hence </b><a NAME="E12:52_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_1.html#E29:52"><i><font color="Green">E72</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">h2</font> = <font color="Maroon">h2'</font> as   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E43:52"/><i><font color="Green">E162</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">hb</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">hb</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E44:52"/><i><font color="Green">E163</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">hb</font> holds <br/><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">hb</font> <a href="goboard1.html#K1">.</a> <font color="Olive">i</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:52_4"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">hb</font>
 ;<br/>

<a NAME="E2:52_4"/><b>then </b><i><font color="Green">E165</font></i>: 
<font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">hb</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="jgraph_1.html#E29:52"><i><font color="Green">E72</font></i></a></i>;<br/>
<a NAME="E3:52_4"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">hb</font> )
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E29:52"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_1.html#E31:52"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E4:52_4"/>
<font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">hb</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E45:52"/><i><font color="Green">E166</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E46:52"/><i><font color="Green">E167</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h1</font> holds <br/>( <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:52_5"/><i><font color="Green">E168</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h1</font>
 ;<br/>

<a NAME="E2:52_5"/><b>then </b><i><font color="Green">E169</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:52_5"/><b>then </b><i><font color="Green">E170</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:52_5"/>
<font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:52_5"/><b>then </b><i><font color="Green">E171</font></i>: 
<font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E6:52_5"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h</font>
 
<b>by </b><i>, <a class="txt" href="jgraph_1.html#E26:52"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E7:52_5"/><b>then </b>
<font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> 1</span>)</span>
 
<b>by </b><i>, <a class="txt" href="jgraph_1.html#E20:52"><i><font color="Green">E63</font></i></a></i>;<br/>
<a NAME="E8:52_5"/><b>then </b><i><font color="Green">E172</font></i>: 
<font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E26:52"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E9:52_5"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h</font>
 
<b>by </b><i>, <a class="txt" href="jgraph_1.html#E26:52"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E10:52_5"/><b>then </b>
<font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="jgraph_1.html#E20:52"><i><font color="Green">E63</font></i></a></i>;<br/>
<a NAME="E11:52_5"/><b>then </b><i><font color="Green">E173</font></i>: 
<font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E26:52"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E12:52_5"/><i><font color="Green">E174</font></i>: 
<font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> = 
 <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> 
<b>by </b><i><a class="txt" href="jgraph_1.html#E13:52"><i><font color="Green">E54</font></i></a>, <a class="ref" href="jgraph_1.html#T10" target="_self">Th10</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
;<br/>
<a NAME="E14:52_5"/><b>then </b><i><font color="Green">E175</font></i>: 
<font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</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="jgraph_1.html#T10" target="_self">Th10</a>, , , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
 <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="txt" href="jgraph_1.html#E13:52"><i><font color="Green">E54</font></i></a>, <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/>
<b>hence </b><a NAME="E16:52_5"/>
( <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="ref" href="jgraph_1.html#D2" target="_self">Def2</a>, <a class="txt" href="jgraph_1.html#E13:52"><i><font color="Green">E54</font></i></a>, <a class="ref" href="jgraph_1.html#T10" target="_self">Th10</a>, , <a class="ref" href="jgraph_1.html#T11" target="_self">Th11</a>, , <a class="ref" href="jgraph_1.html#D3" target="_self">Def3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E47:52"/><i><font color="Green">E176</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">h1</font></span>)</span> holds <br/>( <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">h1</font></span>)</span> <a href="goboard1.html#K1">.</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">h1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">i</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">h1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">h1</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:52_6"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">h1</font></span>)</span>
 ;<br/>

<a NAME="E2:52_6"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">h1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:52_6"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E26:52"><i><font color="Green">E66</font></i></a>, <a class="ref" href="jgraph_1.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E4:52_6"/><b>then </b><i><font color="Green">E205</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h1</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:52_6"/><b>then </b><i><font color="Green">E206</font></i>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">h1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E6:52_6"/><i><font color="Green">E207</font></i>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">h1</font></span>)</span> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E26:52"><i><font color="Green">E66</font></i></a>, <a class="ref" href="jgraph_1.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E7:52_6"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">h1</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E26:52"><i><font color="Green">E66</font></i></a>, <a class="ref" href="jgraph_1.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E8:52_6"/>
( <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">h1</font></span>)</span> <a href="goboard1.html#K1">.</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">h1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">h1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">h1</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> )
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E48:52"/><i><font color="Green">E208</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h1</font> holds <br/>( <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:52_7"/><i><font color="Green">E209</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h1</font>
 ;<br/>

<a NAME="E2:52_7"/><b>then </b><i><font color="Green">E210</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:52_7"/><b>then </b><i><font color="Green">E211</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:52_7"/>
<font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:52_7"/><b>then </b><i><font color="Green">E212</font></i>: 
<font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E6:52_7"/><i><font color="Green">E213</font></i>: 
<font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T11" target="_self">Th11</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> = 
 <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> 
<b>by </b><i><a class="txt" href="jgraph_1.html#E13:52"><i><font color="Green">E54</font></i></a>, <a class="ref" href="jgraph_1.html#T10" target="_self">Th10</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
;<br/>
<a NAME="E8:52_7"/><b>then </b><i><font color="Green">E214</font></i>: 
<font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</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="jgraph_1.html#T10" target="_self">Th10</a>, , , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
 <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="txt" href="jgraph_1.html#E13:52"><i><font color="Green">E54</font></i></a>, <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/>
<b>hence </b><a NAME="E10:52_7"/>
( <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i>, <a class="ref" href="jgraph_1.html#D3" target="_self">Def3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E49:52"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">h1</font></span>)</span> holds <br/>( <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">h1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">i</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">h1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:52_8"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">h1</font></span>)</span>
 ;<br/>

<a NAME="E2:52_8"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">h1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:52_8"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E26:52"><i><font color="Green">E66</font></i></a>, <a class="ref" href="jgraph_1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E4:52_8"/><b>then </b><i><font color="Green">E215</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h1</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:52_8"/><b>then </b>
<span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">h1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:52_8"/>
( <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">h1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">h1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E50:52"/><b>then </b><i><font color="Green">E216</font></i>: 
(  <a href="goboard1.html#K2">X_axis</a> <font color="Maroon">h1</font> <a href="goboard4.html#R1">lies_between</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">h1</font></span>)</span> <a href="goboard1.html#K1">.</a> 1,<span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">h1</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> &amp;  <a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">h1</font> <a href="goboard4.html#R1">lies_between</a> <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a> ,<font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a>  )
 
<b>by </b><i>, <a class="ref" href="goboard4.html#D1">GOBOARD4:def 1</a></i>;<br/>
<a NAME="E51:52"/><i><font color="Green">E217</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h2</font> holds <br/>( <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:52_9"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h2</font>
 ;<br/>

<a NAME="E2:52_9"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:52_9"/><b>then </b><i><font color="Green">E218</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">hb</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:52_9"/><b>then </b><i><font color="Green">E219</font></i>: 
<font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">hb</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E30:52"><i><font color="Green">E73</font></i></a></i>;<br/>
<a NAME="E5:52_9"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">hb</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="jgraph_1.html#E31:52"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:52_9"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">hb</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:52_9"/><b>then </b><i><font color="Green">E220</font></i>: 
<font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">hb</font> <a href="goboard1.html#K1">.</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E30:52"><i><font color="Green">E73</font></i></a></i>;<br/>
<a NAME="E8:52_9"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">hb</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">hb</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="jgraph_1.html#E31:52"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E9:52_9"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">hb</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">hb</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:52_9"/><b>then </b><i><font color="Green">E221</font></i>: 
<font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">hb</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">hb</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="jgraph_1.html#E30:52"><i><font color="Green">E73</font></i></a></i>;<br/>
<a NAME="E11:52_9"/><i><font color="Green">E222</font></i>: 
<font color="Maroon">hb</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">hb</font>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T11" target="_self">Th11</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font> = 
 <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> 
<b>by </b><i><a class="ref" href="jgraph_1.html#T9" target="_self">Th9</a>, , <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
;<br/>
<a NAME="E13:52_9"/><b>then </b><i><font color="Green">E223</font></i>: 
<font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font>
 
<b>by </b><i>, , , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</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">Q</font></span>)</span>
<b>by </b><i><a class="ref" href="jgraph_1.html#T9" target="_self">Th9</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
<font color="Maroon">Q</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<b>hence </b><a NAME="E15:52_9"/>
( <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i>, <a class="ref" href="jgraph_1.html#T9" target="_self">Th9</a>, , , , <a class="ref" href="jgraph_1.html#D3" target="_self">Def3</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E52:52"/><i><font color="Green">E224</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">h2</font></span>)</span> holds <br/>( <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">h2</font></span>)</span> <a href="goboard1.html#K1">.</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">h2</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">i</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">h2</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">h2</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:52_10"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">h2</font></span>)</span>
 ;<br/>

<a NAME="E2:52_10"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">h2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:52_10"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E31:52"><i><font color="Green">E74</font></i></a>, <a class="ref" href="jgraph_1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E4:52_10"/><b>then </b><i><font color="Green">E225</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h2</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:52_10"/><b>then </b><i><font color="Green">E226</font></i>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">h2</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E6:52_10"/><i><font color="Green">E227</font></i>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">h2</font></span>)</span> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E31:52"><i><font color="Green">E74</font></i></a>, <a class="ref" href="jgraph_1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E7:52_10"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">h2</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E31:52"><i><font color="Green">E74</font></i></a>, <a class="ref" href="jgraph_1.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E8:52_10"/>
( <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">h2</font></span>)</span> <a href="goboard1.html#K1">.</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">h2</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">h2</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">h2</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> )
 <b>by </b><i>, , , <a class="ref" href="jgraph_1.html#D3" target="_self">Def3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E53:52"/><i><font color="Green">E228</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h2</font> holds <br/>( <font color="Maroon">p1</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">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:52_11"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h2</font>
 ;<br/>

<a NAME="E2:52_11"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:52_11"/><b>then </b><i><font color="Green">E229</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">hb</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:52_11"/><b>then </b><i><font color="Green">E230</font></i>: 
<font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">hb</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E30:52"><i><font color="Green">E73</font></i></a></i>;<br/>
<a NAME="E5:52_11"/><i><font color="Green">E231</font></i>: 
<font color="Maroon">hb</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">hb</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font> = 
 <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> 
<b>by </b><i><a class="ref" href="jgraph_1.html#T9" target="_self">Th9</a>, , <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
;<br/>
<a NAME="E7:52_11"/><b>then </b><i><font color="Green">E232</font></i>: 
<font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font>
 
<b>by </b><i>, , <a class="ref" href="jgraph_1.html#D3" target="_self">Def3</a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</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">Q</font></span>)</span>
<b>by </b><i><a class="ref" href="jgraph_1.html#T9" target="_self">Th9</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
<font color="Maroon">Q</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<b>hence </b><a NAME="E9:52_11"/>
( <font color="Maroon">p1</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">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E54:52"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">h2</font></span>)</span> holds <br/>( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">h2</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">i</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">h2</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:52_12"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">h2</font></span>)</span>
 ;<br/>

<a NAME="E2:52_12"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">h2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:52_12"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E31:52"><i><font color="Green">E74</font></i></a>, <a class="ref" href="jgraph_1.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E4:52_12"/><b>then </b><i><font color="Green">E233</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h2</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:52_12"/><b>then </b>
<span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">h2</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:52_12"/>
( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">h2</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">h2</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E55:52"/><b>then </b><i><font color="Green">E234</font></i>: 
(  <a href="goboard1.html#K2">X_axis</a> <font color="Maroon">h2</font> <a href="goboard4.html#R1">lies_between</a> <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> ,<font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">h2</font> <a href="goboard4.html#R1">lies_between</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">h2</font></span>)</span> <a href="goboard1.html#K1">.</a> 1,<span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">h2</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T11" target="_self">Th11</a>, <a class="ref" href="goboard4.html#D1">GOBOARD4:def 1</a></i>;<br/>
<a NAME="E56:52"/><i><font color="Green">E235</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font> holds <br/><span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:52_13"/><i><font color="Green">E236</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font> )
 ;<br/>

<a NAME="E2:52_13"/><b>then </b><i><font color="Green">E237</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:52_13"/><i><font color="Green">E238</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:52_13"/><b>then </b><i><font color="Green">E239</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">hb</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:52_13"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">hb</font></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:52_13"/><b>then </b><i><font color="Green">E240</font></i>: 
<font color="Maroon">h2'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">hb</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E7:52_13"/><b>then </b><i><font color="Green">E241</font></i>: 
<font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">hb</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E8:52_13"/><i><font color="Green">E242</font></i>: 
<font color="Maroon">h2'</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">hb</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="jgraph_1.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E9:52_13"/>
<font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E10:52_13"/><b>then </b><i><font color="Green">E243</font></i>: 
<font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">hb</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="jgraph_1.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E11:52_13"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">hb</font></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E12:52_13"/><b>then </b><i><font color="Green">E244</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">hb</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E13:52_13"/><b>then </b><i><font color="Green">E245</font></i>: 
<font color="Maroon">hb</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">hb</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<i><font color="Green">E246</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font> = 
 <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> 
<b>by </b><i><a class="ref" href="jgraph_1.html#T9" target="_self">Th9</a>, , <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
;<br/>
<a NAME="E15:52_13"/><b>then </b><i><font color="Green">E247</font></i>: 
<font color="Maroon">h2'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font>
 
<b>by </b><i>, , , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<i><font color="Green">E248</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</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">Q</font></span>)</span>
<b>by </b><i><a class="ref" href="jgraph_1.html#T9" target="_self">Th9</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
<font color="Maroon">Q</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<a NAME="E17:52_13"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">hb</font></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E18:52_13"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">hb</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E19:52_13"/><b>then </b>
<font color="Maroon">hb</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">hb</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E20:52_13"/><b>then </b>
<font color="Maroon">h2'</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font>
 
<b>by </b><i>, , <a class="ref" href="jgraph_1.html#T13" target="_self">Th13</a>, <a class="ref" href="jgraph_1.html#T15" target="_self">Th15</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E21:52_13"/><b>then </b><i><font color="Green">E249</font></i>: 
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">hb</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i>, <a class="ref" href="jgraph_1.html#T12" target="_self">Th12</a>, </i>;<br/>
<a NAME="E22:52_13"/><i><font color="Green">E250</font></i>: 
<font color="Maroon">hb</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">hb</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, , , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E23:52_13"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">hb</font></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E24:52_13"/><b>then </b><i><font color="Green">E251</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">hb</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E25:52_13"/><b>then </b><i><font color="Green">E252</font></i>: 
<font color="Maroon">hb</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">hb</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E26:52_13"/><i><font color="Green">E253</font></i>: 
<font color="Maroon">hb</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">hb</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E27:52_13"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E28:52_13"/><b>then </b><i><font color="Green">E254</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">hb</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">hb</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">hb</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">hb</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i>, <a class="ref" href="jgraph_1.html#T14" target="_self">Th14</a>, , <a class="ref" href="jgraph_1.html#T17" target="_self">Th17</a>, <a class="ref" href="jgraph_1.html#T18" target="_self">Th18</a>, <a class="ref" href="jgraph_1.html#T19" target="_self">Th19</a>, <a class="ref" href="jgraph_1.html#T20" target="_self">Th20</a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<b>reconsider </b><font color="Maroon">W1</font> = <font color="Maroon">Q</font> as   <a href="struct_0.html#NM2">Subset</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="jgraph_1.html#E21:52"><i><font color="Green">E64</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">Qa</font> = <font color="Maroon">Q</font> as   <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/>
<a NAME="E31:52_13"/><i><font color="Green">E255</font></i>: 
<font color="Maroon">Qa</font> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i>, <a class="ref" href="jordan5a.html#T1">JORDAN5A:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Qa</font> = <font color="Maroon">Qa</font> as  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>reconsider </b><font color="Maroon">R</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">hb</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">hb</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i>, , <a class="ref" href="jgraph_1.html#T17" target="_self">Th17</a>, <a class="ref" href="jgraph_1.html#T19" target="_self">Th19</a>, <a class="ref" href="jgraph_1.html#T20" target="_self">Th20</a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T16">RCOMP_1:16</a></i>;<br/>
<b>reconsider </b><font color="Maroon">w1</font> = <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">hb</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>, <font color="Maroon">w2</font> = <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">hb</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> 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="ref" href="jgraph_1.html#D4" target="_self">Def4</a>, <a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<a NAME="E35:52_13"/><i><font color="Green">E256</font></i>: 
<font color="Maroon">hb</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T17" target="_self">Th17</a>, , <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E36:52_13"/>
<font color="Maroon">hb</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T20" target="_self">Th20</a>, , <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E37:52_13"/><b>then </b><i><font color="Green">E257</font></i>: 
( <font color="Maroon">w1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">R</font> &amp; <font color="Maroon">w2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">R</font> )
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T15" target="_self">Th15</a>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">W</font> = <font color="Maroon">g1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">R</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<a NAME="E39:52_13"/><i><font color="Green">E258</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</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">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> )
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T9" target="_self">Th9</a>, , , <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E40:52_13"/>
 <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a>  <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="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E41:52_13"/><b>then </b>
<font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="relset_1.html#K10">.:</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="relat_1.html#T146">RELAT_1:146</a></i>;<br/>
<a NAME="E42:52_13"/><b>then </b><i><font color="Green">E259</font></i>: 
<font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W1</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<a NAME="E43:52_13"/>
<font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Qa</font>
 
<b>by </b><i>, <a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/>
<a NAME="E44:52_13"/><b>then </b><i><font color="Green">E260</font></i>: 
(  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">Qa</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">Qa</font> &amp;  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">Qa</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">Qa</font> )
 
<b>by </b><i>, <a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/>
<b>set </b><font color="Maroon">r1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">Qa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">Qa</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">Qa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">Qa</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1;<br/>
<a NAME="E45:52_13"/>
 <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">Qa</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">Qa</font>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T22" target="_self">Th22</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E46:52_13"/><b>then </b><i><font color="Green">E261</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">Qa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">Qa</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E47:52_13"/>
 <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">Qa</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">Qa</font>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T22" target="_self">Th22</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E48:52_13"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">Qa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">Qa</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E49:52_13"/><b>then </b><i><font color="Green">E262</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">Qa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">Qa</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">Qa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">Qa</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E50:52_13"/><i><font color="Green">E288</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">Qa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">Qa</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">Qa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">Qa</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">Qa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">Qa</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">Qa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">Qa</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E51:52_13"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <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>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> holds <br/> <a href="metric_1.html#K4">dist</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">Qa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">Qa</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">Qa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">Qa</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_13_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <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>, <font color="Maroon">y</font> be   <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>;<br/>

<b>assume </b><a NAME="E1:52_13_3"/><i><font color="Green">E289</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">pw1</font> = <font color="Maroon">x</font>, <font color="Maroon">pw2</font> = <font color="Maroon">y</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="E3:52_13_3"/><i><font color="Green">E290</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">pw1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">pw2</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:52_13_3"/><i><font color="Green">E291</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">pw1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">pw2</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">pw1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">pw2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">pw1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">pw2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:52"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E5:52_13_3"/><i><font color="Green">E292</font></i>: 
(  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">Qa</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pw1</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">pw1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">Qa</font> &amp;  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">Qa</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pw1</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">pw1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">Qa</font> )
 
<b>by </b><i>, , <a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/>
<a NAME="E6:52_13_3"/><i><font color="Green">E293</font></i>: 
(  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">Qa</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pw2</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">pw2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">Qa</font> &amp;  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">Qa</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pw2</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">pw2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">Qa</font> )
 
<b>by </b><i>, , <a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/>
<a NAME="E7:52_13_3"/><b>then </b><i><font color="Green">E294</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">pw1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">pw2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">Qa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">Qa</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:52_13_3"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">pw1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">pw2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">Qa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">Qa</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E9:52_13_3"/><b>then </b><i><font color="Green">E295</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">pw1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">pw2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">pw1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">pw2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">Qa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">Qa</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">Qa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">Qa</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E10:52_13_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">Qa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">Qa</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">Qa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">Qa</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">Qa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">Qa</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">Qa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">Qa</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E11:52_13_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">pw1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">pw2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">pw1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">pw2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">Qa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">Qa</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">Qa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">Qa</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E12:52_13_3"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">Qa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">Qa</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">Qa</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">Qa</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E52:52_13"/><b>then </b>
<font color="Maroon">W1</font> is <a href="tbsp_1.html#V6">bounded</a>
 
<b>by </b><i>, , <a class="ref" href="tbsp_1.html#D9">TBSP_1:def 9</a></i>;<br/>
<a NAME="E53:52_13"/><b>then </b><i><font color="Green">E296</font></i>: 
<font color="Maroon">W</font> is <a href="tbsp_1.html#V6">bounded</a>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T21" target="_self">Th21</a>, <a class="ref" href="tbsp_1.html#T21">TBSP_1:21</a></i>;<br/>
<a NAME="E54:52_13"/><i><font color="Green">E297</font></i>: 
 <a href="tbsp_1.html#K2">diameter</a> <font color="Maroon">W</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E55:52_13"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">w1</font>,<font color="Maroon">w2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="tbsp_1.html#K2">diameter</a> <font color="Maroon">W</font>
 
<b>by </b><i>, , <a class="ref" href="tbsp_1.html#D10">TBSP_1:def 10</a></i>;<br/>
<a NAME="E56:52_13"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">w1</font>,<font color="Maroon">w2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E57:52_13"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f2</font> being   <a href="struct_0.html#NM7">FinSequence</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="E58:52"/><i><font color="Green">E298</font></i>: 
( <font color="Maroon">f2</font> is <a href="topreal1.html#V1">special</a> &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> &amp;  <a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f2</font> <a href="goboard4.html#R1">lies_between</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">h1</font></span>)</span> <a href="goboard1.html#K1">.</a> 1,<span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">h1</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> &amp;  <a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f2</font> <a href="goboard4.html#R1">lies_between</a> <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a> ,<font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a>  &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f2</font> holds <br/> ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h1</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">j</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5 ) ) &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> holds <br/><span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">j</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Olive">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5 ) )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E17:52"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jgraph_1.html#E23:52"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jgraph_1.html#E26:52"><i><font color="Green">E66</font></i></a>, , <a class="txt" href="jgraph_1.html#E9:52"><i><font color="Green">E41</font></i></a></i>;<br/>
<a NAME="E59:52"/><i><font color="Green">E299</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E26:52"><i><font color="Green">E66</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>consider </b><font color="Maroon">g2</font> being   <a href="struct_0.html#NM7">FinSequence</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="E61:52"/><i><font color="Green">E300</font></i>: 
( <font color="Maroon">g2</font> is <a href="topreal1.html#V1">special</a> &amp; <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font> &amp;  <a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">g2</font> <a href="goboard4.html#R1">lies_between</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">h2</font></span>)</span> <a href="goboard1.html#K1">.</a> 1,<span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">h2</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> &amp;  <a href="goboard1.html#K2">X_axis</a> <font color="Maroon">g2</font> <a href="goboard4.html#R1">lies_between</a> <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> ,<font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font> holds <br/> ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h2</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">j</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5 ) ) &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font> holds <br/><span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">j</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Olive">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5 ) )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E17:52"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jgraph_1.html#E31:52"><i><font color="Green">E74</font></i></a>, , <a class="ref" href="jgraph_1.html#D3" target="_self">Def3</a>, <a class="txt" href="jgraph_1.html#E11:52"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E62:52"/><i><font color="Green">E301</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E31:52"><i><font color="Green">E74</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E63:52"/><i><font color="Green">E302</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h</font>
 
<b>by </b><i>, <a class="txt" href="jgraph_1.html#E26:52"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E64:52"/><i><font color="Green">E303</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h</font>
 
<b>by </b><i>, <a class="txt" href="jgraph_1.html#E26:52"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E65:52"/><i><font color="Green">E304</font></i>: 
<font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> 1</span>)</span>
 
<b>by </b><i>, <a class="txt" href="jgraph_1.html#E20:52"><i><font color="Green">E63</font></i></a>, </i>;<br/>
<a NAME="E66:52"/><i><font color="Green">E305</font></i>: 
<font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E26:52"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E67:52"/><i><font color="Green">E306</font></i>: 
<font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E26:52"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E68:52"/><i><font color="Green">E307</font></i>: 
<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> 1
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E69:52"/><i><font color="Green">E308</font></i>: 
<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E70:52"/><i><font color="Green">E309</font></i>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f2</font></span>)</span> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E13:52"><i><font color="Green">E54</font></i></a>, <a class="ref" href="jgraph_1.html#T10" target="_self">Th10</a>, , , , <a class="ref" href="jgraph_1.html#T13" target="_self">Th13</a>, , <a class="ref" href="jgraph_1.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E71:52"/>
<font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E13:52"><i><font color="Green">E54</font></i></a>, <a class="ref" href="jgraph_1.html#T10" target="_self">Th10</a>, , , <a class="txt" href="jgraph_1.html#E20:52"><i><font color="Green">E63</font></i></a>, , <a class="ref" href="jgraph_1.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E72:52"/><b>then </b><i><font color="Green">E310</font></i>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f2</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i>, , <a class="ref" href="jgraph_1.html#T14" target="_self">Th14</a>, <a class="ref" href="jgraph_1.html#T15" target="_self">Th15</a>, <a class="ref" href="jgraph_1.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E73:52"/><i><font color="Green">E311</font></i>: 
<font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T9" target="_self">Th9</a>, , , , <a class="txt" href="jgraph_1.html#E30:52"><i><font color="Green">E73</font></i></a></i>;<br/>
<a NAME="E74:52"/><i><font color="Green">E312</font></i>: 
<font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> 1
 
<b>by </b><i>, , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E75:52"/><i><font color="Green">E313</font></i>: 
<font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E76:52"/>
<font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> 1
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T12" target="_self">Th12</a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E77:52"/><b>then </b><i><font color="Green">E314</font></i>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">g2</font></span>)</span> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i>, <a class="ref" href="jgraph_1.html#T12" target="_self">Th12</a>, <a class="ref" href="jgraph_1.html#D4" target="_self">Def4</a>, <a class="ref" href="jgraph_1.html#T17" target="_self">Th17</a>, <a class="ref" href="jgraph_1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E78:52"/><i><font color="Green">E315</font></i>: 
<font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T12" target="_self">Th12</a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E79:52"/>
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T9" target="_self">Th9</a>, , , , , <a class="txt" href="jgraph_1.html#E30:52"><i><font color="Green">E73</font></i></a>, , <a class="ref" href="jgraph_1.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E80:52"/><b>then </b><i><font color="Green">E316</font></i>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">g2</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T12" target="_self">Th12</a>, <a class="ref" href="jgraph_1.html#T20" target="_self">Th20</a>, <a class="ref" href="jgraph_1.html#T4" target="_self">Th4</a></i>;<br/>
<i><font color="Green">E317</font></i>: <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f2</font></span>)</span> <a href="goboard1.html#K1">.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i>, , , , <a class="ref" href="jgraph_1.html#T3" target="_self">Th3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">h1</font></span>)</span> <a href="goboard1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="jgraph_1.html#E26:52"><i><font color="Green">E66</font></i></a>, <a class="ref" href="jgraph_1.html#T3" target="_self">Th3</a></i>
;<br/>
<i><font color="Green">E318</font></i>: <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f2</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i>, , <a class="ref" href="jgraph_1.html#T14" target="_self">Th14</a>, <a class="ref" href="jgraph_1.html#T15" target="_self">Th15</a>, <a class="ref" href="jgraph_1.html#T3" target="_self">Th3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">h1</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_1.html#E26:52"><i><font color="Green">E66</font></i></a>, <a class="ref" href="jgraph_1.html#T3" target="_self">Th3</a></i>
;<br/>
<a NAME="E83:52"/>
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> 1
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T12" target="_self">Th12</a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<b>then </b><i><font color="Green">E319</font></i>: <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">g2</font></span>)</span> <a href="goboard1.html#K1">.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, <a class="ref" href="jgraph_1.html#T12" target="_self">Th12</a>, <a class="ref" href="jgraph_1.html#T17" target="_self">Th17</a>, <a class="ref" href="jgraph_1.html#T4" target="_self">Th4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">h2</font></span>)</span> <a href="goboard1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="jgraph_1.html#E31:52"><i><font color="Green">E74</font></i></a>, <a class="ref" href="jgraph_1.html#T4" target="_self">Th4</a></i>
;<br/>
<a NAME="E85:52"/>
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T12" target="_self">Th12</a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<b>then </b><i><font color="Green">E320</font></i>: <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">g2</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, <a class="ref" href="jgraph_1.html#T12" target="_self">Th12</a>, <a class="ref" href="jgraph_1.html#T18" target="_self">Th18</a>, <a class="ref" href="jgraph_1.html#T4" target="_self">Th4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">h2</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_1.html#E31:52"><i><font color="Green">E74</font></i></a>, <a class="ref" href="jgraph_1.html#T4" target="_self">Th4</a></i>
;<br/>
<a NAME="E87:52"/>
5 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font>
 
<b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E88:52"/><b>then </b><i><font color="Green">E321</font></i>: 
2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E89:52"/>
5 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font>
 
<b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E90:52"/><b>then </b><i><font color="Green">E322</font></i>: 
2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E91:52"/><i><font color="Green">E323</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E92:52"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E93:52"/><b>then </b><i><font color="Green">E324</font></i>: 
<font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> 1</span>)</span>
 
<b>by </b><i>, <a class="txt" href="jgraph_1.html#E20:52"><i><font color="Green">E63</font></i></a>, </i>;<br/>
<a NAME="E94:52"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E95:52"/><b>then </b><i><font color="Green">E325</font></i>: 
<font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="jgraph_1.html#E13:52"><i><font color="Green">E54</font></i></a>, <a class="ref" href="jgraph_1.html#T10" target="_self">Th10</a>, , , <a class="txt" href="jgraph_1.html#E20:52"><i><font color="Green">E63</font></i></a>, , , , <a class="ref" href="jgraph_1.html#T15" target="_self">Th15</a>, </i>;<br/>
<a NAME="E96:52"/><i><font color="Green">E326</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">hb</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E97:52"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">hb</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E98:52"/><b>then </b><i><font color="Green">E327</font></i>: 
<font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">hb</font> <a href="goboard1.html#K1">.</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E30:52"><i><font color="Green">E73</font></i></a></i>;<br/>
<a NAME="E99:52"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">hb</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">hb</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E100:52"/><b>then </b>
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T8" target="_self">Th8</a>, <a class="ref" href="jgraph_1.html#T9" target="_self">Th9</a>, , , , <a class="txt" href="jgraph_1.html#E30:52"><i><font color="Green">E73</font></i></a>, , <a class="ref" href="jgraph_1.html#T17" target="_self">Th17</a>, <a class="ref" href="jgraph_1.html#T18" target="_self">Th18</a>, </i>;<br/>
<a NAME="E101:52"/><b>then </b>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">f2</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g2</font>
 
<b>by </b><i>, , , <a class="ref" href="jgraph_1.html#T15" target="_self">Th15</a>, , , <a class="ref" href="jgraph_1.html#T19" target="_self">Th19</a>, , , , , , <a class="ref" href="jgraph_1.html#T21" target="_self">Th21</a>, <a class="ref" href="jgraph_1.html#T22" target="_self">Th22</a>, , </i>;<br/>
<b>then consider </b><font color="Maroon">s</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E103:52"/><i><font color="Green">E329</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f2</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g2</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">ps</font> = <font color="Maroon">s</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> <b>by </b><i/>;<br/>
<a NAME="E105:52"/>
<font color="Maroon">ps</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f2</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> ) </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="topreal1.html#D6">TOPREAL1:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E107:52"/><i><font color="Green">E331</font></i>: 
( <font color="Maroon">ps</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f2</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E109:52"/><i><font color="Green">E332</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f2</font>,<font color="Maroon">i</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> )
 <b>by </b><i/>;<br/>
<a NAME="E110:52"/><i><font color="Green">E333</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f2</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E111:52"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E112:52"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f2</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E114:52"/><i><font color="Green">E334</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h1</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5 )
 <b>by </b><i/>;<br/>
<a NAME="E115:52"/><i><font color="Green">E335</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ps</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i>, , , <a class="txt" href="jgraph_1.html#E5:52"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E116:52"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5
 
<b>by </b><i>, </i>;<br/>
<a NAME="E117:52"/><b>then </b><i><font color="Green">E336</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ps</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E118:52"/><i><font color="Green">E337</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ps</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ps</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i><a class="ref" href="toprns_1.html#T35">TOPRNS_1:35</a></i>;<br/>
<a NAME="E119:52"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ps</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5</span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E120:52"/><b>then </b><i><font color="Green">E338</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ps</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E121:52"/><i><font color="Green">E339</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E122:52"/><b>then </b><i><font color="Green">E340</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E123:52"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E124:52"/><b>then </b>
<font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E125:52"/><b>then </b><i><font color="Green">E341</font></i>: 
<font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E126:52"/><i><font color="Green">E342</font></i>: 
<font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<i><font color="Green">E343</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> = 
 <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> 
<b>by </b><i><a class="txt" href="jgraph_1.html#E13:52"><i><font color="Green">E54</font></i></a>, <a class="ref" href="jgraph_1.html#T10" target="_self">Th10</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
;<br/>
 <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="txt" href="jgraph_1.html#E13:52"><i><font color="Green">E54</font></i></a>, <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/>
<a NAME="E129:52"/><b>then </b><i><font color="Green">E344</font></i>: 
<font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T10" target="_self">Th10</a>, , , , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p11</font> = <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</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="E131:52"/><i><font color="Green">E346</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p11</font> <a href="euclid.html#K20">-</a> <font color="Maroon">ps</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5</span>)</span>
 
<b>by </b><i>, <a class="ref" href="toprns_1.html#T28">TOPRNS_1:28</a></i>;<br/>
<a NAME="E132:52"/>
<font color="Maroon">ps</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g2</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font> ) </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="topreal1.html#D6">TOPREAL1:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E134:52"/><i><font color="Green">E347</font></i>: 
( <font color="Maroon">ps</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g2</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E136:52"/><i><font color="Green">E348</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g2</font>,<font color="Maroon">j</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font> )
 <b>by </b><i/>;<br/>
<a NAME="E137:52"/><i><font color="Green">E349</font></i>: 
(  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g2</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">ps</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g2</font>,<font color="Maroon">j</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g2</font>,<font color="Maroon">j</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font> )
 
<b>by </b><i>, <a class="ref" href="jgraph_1.html#T26" target="_self">Th26</a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E138:52"/>
( <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font> &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g2</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">ps</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g2</font>,<font color="Maroon">j</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g2</font>,<font color="Maroon">j</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font> )
 
<b>by </b><i>, <a class="ref" href="jgraph_1.html#T26" target="_self">Th26</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E139:52"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E140:52"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">k'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E142:52"/><i><font color="Green">E351</font></i>: 
( <font color="Maroon">k'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h2</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k'</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5 )
 <b>by </b><i/>;<br/>
<a NAME="E143:52"/><i><font color="Green">E352</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ps</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T27" target="_self">Th27</a>, <a class="txt" href="jgraph_1.html#E5:52"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E144:52"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5
 
<b>by </b><i>, <a class="ref" href="jgraph_1.html#T26" target="_self">Th26</a></i>;<br/>
<a NAME="E145:52"/><b>then </b><i><font color="Green">E353</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ps</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T29" target="_self">Th29</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E146:52"/><i><font color="Green">E354</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ps</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k'</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ps</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k'</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i><a class="ref" href="toprns_1.html#T35">TOPRNS_1:35</a></i>;<br/>
<a NAME="E147:52"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ps</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k'</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5</span>)</span>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T28" target="_self">Th28</a>, <a class="ref" href="jgraph_1.html#T30" target="_self">Th30</a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E148:52"/><b>then </b><i><font color="Green">E355</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ps</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k'</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E149:52"/>
<font color="Maroon">k'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T28" target="_self">Th28</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E150:52"/><b>then </b>
<font color="Maroon">k'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">hb</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E151:52"/><b>then </b><i><font color="Green">E356</font></i>: 
( <font color="Maroon">hb</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k'</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">hb</font> &amp; <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">hb</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k'</font></span>)</span> &amp; <font color="Maroon">k'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">hb</font> )
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E30:52"><i><font color="Green">E73</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font> = 
 <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> 
<b>by </b><i><a class="ref" href="jgraph_1.html#T9" target="_self">Th9</a>, , <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
;<br/>
<a NAME="E153:52"/><b>then </b><i><font color="Green">E357</font></i>: 
( <font color="Maroon">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> &amp; <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k'</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> &amp; <font color="Maroon">hb</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k'</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font> )
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<i><font color="Green">E358</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</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">Q</font></span>)</span>
<b>by </b><i><a class="ref" href="jgraph_1.html#T9" target="_self">Th9</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
<font color="Maroon">Q</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">q11</font> = <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k'</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/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">p11</font>, <font color="Maroon">x2</font> = <font color="Maroon">q11</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="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<a NAME="E157:52"/>
 <a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>
 
<b>by </b><i>, , , , <a class="ref" href="weierstr.html#T40">WEIERSTR:40</a></i>;<br/>
<a NAME="E158:52"/><b>then </b><i><font color="Green">E360</font></i>: 
( <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p11</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q11</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p11</font> <a href="euclid.html#K20">-</a> <font color="Maroon">ps</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ps</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q11</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> &amp;  <a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p11</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q11</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> )
 
<b>by </b><i>, <a class="ref" href="toprns_1.html#T35">TOPRNS_1:35</a></i>;<br/>
<a NAME="E159:52"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p11</font> <a href="euclid.html#K20">-</a> <font color="Maroon">ps</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ps</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q11</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5</span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5</span>)</span></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"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5</span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T25" target="_self">Th25</a>, , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E160:52"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p11</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q11</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5</span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E161:52"/><b>then </b>
 <a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 4 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E162:52"/><b>then </b>
<span class="p1">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">5 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5</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="E163:52"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="real_1.html#K5">-</a> 5</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P'</font>,<font color="Maroon">Q'</font></span>)</span> <a href="real_1.html#K6">/</a> 5</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E17:52"><i><font color="Green">E60</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<a NAME="E164:52"/><b>then </b>
4 <a href="real_1.html#K5">-</a> 5 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E17:52"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/>
<b>hence </b><a NAME="E165:52"/>
contradiction
 ;<br/>


</div>
