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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:52"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 <b>and </b><br/><a NAME="E2:52"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  )
 <b>and </b><br/><a NAME="E3:52"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  )
 <b>and </b><br/><a NAME="E4:52"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K22">`2</a>  )
 <b>and </b><br/><a NAME="E5:52"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K22">`2</a>  )
 ;<br/>

<a NAME="E6:52"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:52"><i><font color="Green">E47</font></i></a>, <a class="ref" href="jordan6.html#T49">JORDAN6:49</a></i>;<br/>
<a NAME="E7:52"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:52"><i><font color="Green">E47</font></i></a>, <a class="ref" href="jordan6.html#T49">JORDAN6:49</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM7">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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E9:52"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E1:52"><i><font color="Green">E47</font></i></a>, <a class="ref" href="topreal1.html#D2">TOPREAL1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM7">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">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E11:52"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E1:52"><i><font color="Green">E47</font></i></a>, <a class="ref" href="topreal1.html#D2">TOPREAL1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="struct_0.html#NM7">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">E56</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">c<sub>9</sub></font> is <a href="funct_1.html#V2">one-to-one</a> )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E9:52"><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">c<sub>10</sub></font> being   <a href="struct_0.html#NM7">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">E57</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">c<sub>10</sub></font> is <a href="funct_1.html#V2">one-to-one</a> )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E11:52"><i><font color="Green">E55</font></i></a>, <a class="ref" href="jordan7.html#T15">JORDAN7:15</a></i>;<br/>
<a NAME="E16:52"/><i><font color="Green">E58</font></i>: 
 <a href="euclid.html#K15">TOP-REAL</a> 2 <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K4">TopSpaceMetr</a> <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#D8">EUCLID:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>2</sub></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> <b>by </b><i><a class="txt" href="jgraph_1.html#E16:52"><i><font color="Green">E58</font></i></a></i>;<br/>
<a NAME="E18:52"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>11</sub></font> is <a href="compts_1.html#V6">compact</a> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>12</sub></font> is <a href="compts_1.html#V6">compact</a> )
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:52"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jgraph_1.html#E16:52"><i><font color="Green">E58</font></i></a>, <a class="ref" href="jordan5a.html#T1">JORDAN5A:1</a></i>;<br/>
<b>assume </b><a NAME="E19:52"/>
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></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="E20:52"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#R1">misses</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E21:52"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E18:52"><i><font color="Green">E59</font></i></a>, <a class="ref" href="jgraph_1.html#T55" target="_self">Th55</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 5;<br/>
<a NAME="E22:52"/><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></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#E21:52"><i><font color="Green">E60</font></i></a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>14</sub></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="E24:52"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>14</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font> is <a href="goboard1.html#V1">increasing</a> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> <br/>for <font color="Olive">b<sub>3</sub></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">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Olive">b<sub>2</sub></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">c<sub>14</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></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">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Olive">b<sub>2</sub></font> holds <br/> <a href="tbsp_1.html#K2">diameter</a> <font color="Olive">b<sub>3</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 5 ) )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E9:52"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jgraph_1.html#E13:52"><i><font color="Green">E56</font></i></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="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E25:52"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></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">c<sub>14</sub></font></span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></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">c<sub>15</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E27:52"/><i><font color="Green">E63</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></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">c<sub>14</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) )
 ;<br/>
<a NAME="E28:52"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></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">c<sub>16</sub></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">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>17</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:52_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></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">E65</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></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">c<sub>15</sub></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">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="jgraph_1.html#E3:52_1"><i><font color="Green">E64</font></i></a></i>;<br/>
<a NAME="E6:52_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E27:52"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_1.html#E3:52_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jgraph_1.html#E4:52_1"><i><font color="Green">E65</font></i></a></i>;<br/>
<a NAME="E7:52_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E27:52"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_1.html#E3:52_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jgraph_1.html#E4:52_1"><i><font color="Green">E65</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">E67</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>14</sub></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">c<sub>9</sub></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#E9:52"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jgraph_1.html#E13:52"><i><font color="Green">E56</font></i></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">E68</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E13:52"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jgraph_1.html#E24:52"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jgraph_1.html#E6:52_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jgraph_1.html#E8:52_1"><i><font color="Green">E67</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">c<sub>7</sub></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">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_1.html#E9:52"><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">c<sub>1</sub></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">c<sub>16</sub></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#E3:52_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jgraph_1.html#E10:52_1"><i><font color="Green">E68</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="struct_0.html#NM8">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="E30:52"/><i><font color="Green">E64</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E31:52"/><i><font color="Green">E65</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="E32:52"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>16</sub></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">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Olive">b<sub>1</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></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">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:52_2"/><i><font color="Green">E67</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>17</sub></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">c<sub>16</sub></font> )
 ;<br/>

<a NAME="E2:52_2"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>16</sub></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">E69</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:52_2"><i><font color="Green">E67</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">E70</font></i>: 
<font color="Maroon">c<sub>17</sub></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">c<sub>14</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E27:52"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_1.html#E1:52_2"><i><font color="Green">E67</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">E71</font></i>: 
<font color="Maroon">c<sub>17</sub></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">c<sub>14</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E27:52"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_1.html#E1:52_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jgraph_1.html#E2:52_2"><i><font color="Green">E68</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">E72</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E27:52"><i><font color="Green">E63</font></i></a></i>;<br/>
<a NAME="E7:52_2"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:52_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jgraph_1.html#E2:52_2"><i><font color="Green">E68</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">E74</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E27:52"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_1.html#E4:52_2"><i><font color="Green">E70</font></i></a></i>;<br/>
<a NAME="E9:52_2"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:52_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jgraph_1.html#E3:52_2"><i><font color="Green">E69</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">E76</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E27:52"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_1.html#E1:52_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jgraph_1.html#E2:52_2"><i><font color="Green">E68</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">E77</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<i><font color="Green">E78</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></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#E9:52"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jgraph_1.html#E13:52"><i><font color="Green">E56</font></i></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">E79</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E13:52"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jgraph_1.html#E24:52"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jgraph_1.html#E6:52_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_1.html#E11:52_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<i><font color="Green">E80</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_1.html#E9:52"><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">c<sub>1</sub></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">E81</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></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#E27:52"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_1.html#E5:52_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jgraph_1.html#E13:52_2"><i><font color="Green">E79</font></i></a></i>;<br/>
<a NAME="E16:52_2"/>
<font color="Maroon">c<sub>17</sub></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">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E27:52"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_1.html#E1:52_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jgraph_1.html#E3:52_2"><i><font color="Green">E69</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">c<sub>14</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></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">c<sub>14</sub></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">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></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">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E13:52"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jgraph_1.html#E24:52"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jgraph_1.html#E8:52_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jgraph_1.html#E12:52_2"><i><font color="Green">E78</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">E82</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></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#E27:52"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_1.html#E4:52_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jgraph_1.html#E14:52_2"><i><font color="Green">E80</font></i></a></i>;<br/>
<a NAME="E20:52_2"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E27:52"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_1.html#E1:52_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jgraph_1.html#E2:52_2"><i><font color="Green">E68</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">E84</font></i>: 
<font color="Maroon">c<sub>17</sub></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">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E27:52"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_1.html#E1:52_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jgraph_1.html#E3:52_2"><i><font color="Green">E69</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">E85</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></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">c<sub>14</sub></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">E86</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E27:52"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_1.html#E1:52_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jgraph_1.html#E3:52_2"><i><font color="Green">E69</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">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>17</sub></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">E87</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></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#E24:52"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jgraph_1.html#E10:52_2"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jgraph_1.html#E11:52_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jgraph_1.html#E20:52_2"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jgraph_1.html#E21:52_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jgraph_1.html#E22:52_2"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jgraph_1.html#E23:52_2"><i><font color="Green">E86</font></i></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">c<sub>18</sub></font> = <font color="Maroon">c<sub>1</sub></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#E31:52"><i><font color="Green">E65</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>1</sub></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">E88</font></i>: 
<font color="Maroon">c<sub>19</sub></font> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:52"><i><font color="Green">E47</font></i></a>, <a class="ref" href="jordan5a.html#T1">JORDAN5A:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</sub></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">c<sub>21</sub></font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>17</sub></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#E24:52"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jgraph_1.html#E11:52_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jgraph_1.html#E20:52_2"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jgraph_1.html#E22:52_2"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jgraph_1.html#E23:52_2"><i><font color="Green">E86</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">c<sub>22</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>, <font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></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="txt" href="jgraph_1.html#E15:52_2"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jgraph_1.html#E19:52_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<a NAME="E32:52_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E20:52_2"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jgraph_1.html#E25:52_2"><i><font color="Green">E87</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E33:52_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E23:52_2"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_1.html#E25:52_2"><i><font color="Green">E87</font></i></a>, <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">E90</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>21</sub></font> )
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E12:52_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="jgraph_1.html#E32:52_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>21</sub></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">E91</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E9:52"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jgraph_1.html#E13:52"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jgraph_1.html#E14:52_2"><i><font color="Green">E80</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">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></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="txt" href="jgraph_1.html#E36:52_2"><i><font color="Green">E91</font></i></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">E92</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>18</sub></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">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:52"><i><font color="Green">E47</font></i></a>, <a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/>
<a NAME="E41:52_2"/><b>then </b><i><font color="Green">E93</font></i>: 
(  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c<sub>3</sub></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">c<sub>20</sub></font> &amp;  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>3</sub></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">c<sub>20</sub></font> )
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E28:52_2"><i><font color="Green">E88</font></i></a>, <a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>25</sub></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">c<sub>20</sub></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">c<sub>20</sub></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">c<sub>20</sub></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">c<sub>20</sub></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">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E41:52_2"><i><font color="Green">E93</font></i></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">E94</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">c<sub>20</sub></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">c<sub>20</sub></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">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E41:52_2"><i><font color="Green">E93</font></i></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">c<sub>20</sub></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">c<sub>20</sub></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">E95</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">c<sub>20</sub></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">c<sub>20</sub></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">c<sub>20</sub></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">c<sub>20</sub></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="txt" href="jgraph_1.html#E43:52_2"><i><font color="Green">E94</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E47:52_2"/><i><font color="Green">E96</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">c<sub>20</sub></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">c<sub>20</sub></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">c<sub>20</sub></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">c<sub>20</sub></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">c<sub>20</sub></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">c<sub>20</sub></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">c<sub>20</sub></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">c<sub>20</sub></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">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> holds <br/> <a href="metric_1.html#K4">dist</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">c<sub>20</sub></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">c<sub>20</sub></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">c<sub>20</sub></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">c<sub>20</sub></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">c<sub>26</sub></font>, <font color="Maroon">c<sub>27</sub></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">E97</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>26</sub></font>, <font color="Maroon">c<sub>29</sub></font> = <font color="Maroon">c<sub>27</sub></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">E98</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>27</sub></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">c<sub>28</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T45" target="_self">Th45</a></i>;<br/>
<a NAME="E4:52_2_3"/><i><font color="Green">E99</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>29</sub></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">c<sub>28</sub></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">c<sub>29</sub></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">c<sub>28</sub></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">c<sub>29</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T49" target="_self">Th49</a></i>;<br/>
<a NAME="E5:52_2_3"/><i><font color="Green">E100</font></i>: 
(  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>28</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c<sub>28</sub></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">c<sub>20</sub></font> &amp;  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>28</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>28</sub></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">c<sub>20</sub></font> )
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E28:52_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="jgraph_1.html#E1:52_2_3"><i><font color="Green">E97</font></i></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">E101</font></i>: 
(  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>29</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c<sub>29</sub></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">c<sub>20</sub></font> &amp;  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>29</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>29</sub></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">c<sub>20</sub></font> )
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E28:52_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="jgraph_1.html#E1:52_2_3"><i><font color="Green">E97</font></i></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">E102</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>28</sub></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">c<sub>29</sub></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">c<sub>20</sub></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">c<sub>20</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E5:52_2_3"><i><font color="Green">E100</font></i></a>, <a class="ref" href="jgraph_1.html#T31" target="_self">Th31</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">c<sub>28</sub></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">c<sub>29</sub></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">c<sub>20</sub></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">c<sub>20</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E5:52_2_3"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jgraph_1.html#E6:52_2_3"><i><font color="Green">E101</font></i></a>, <a class="ref" href="jgraph_1.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E9:52_2_3"/><b>then </b><i><font color="Green">E103</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">c<sub>28</sub></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">c<sub>29</sub></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">c<sub>28</sub></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">c<sub>29</sub></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">c<sub>20</sub></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">c<sub>20</sub></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">c<sub>20</sub></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">c<sub>20</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E7:52_2_3"><i><font color="Green">E102</font></i></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">c<sub>20</sub></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">c<sub>20</sub></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">c<sub>20</sub></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">c<sub>20</sub></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">c<sub>20</sub></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">c<sub>20</sub></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">c<sub>20</sub></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">c<sub>20</sub></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">c<sub>28</sub></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">c<sub>29</sub></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">c<sub>28</sub></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">c<sub>29</sub></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">c<sub>20</sub></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">c<sub>20</sub></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">c<sub>20</sub></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">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E9:52_2_3"><i><font color="Green">E103</font></i></a>, <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">c<sub>26</sub></font>,<font color="Maroon">c<sub>27</sub></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">c<sub>20</sub></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">c<sub>20</sub></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">c<sub>20</sub></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">c<sub>20</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="txt" href="jgraph_1.html#E3:52_2_3"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jgraph_1.html#E4:52_2_3"><i><font color="Green">E99</font></i></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">c<sub>18</sub></font> is <a href="tbsp_1.html#V6">bounded</a>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E46:52_2"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jgraph_1.html#E47:52_2"><i><font color="Green">E96</font></i></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">E97</font></i>: 
<font color="Maroon">c<sub>24</sub></font> is <a href="tbsp_1.html#V6">bounded</a>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E39:52_2"><i><font color="Green">E92</font></i></a>, <a class="ref" href="tbsp_1.html#T21">TBSP_1:21</a></i>;<br/>
<a NAME="E51:52_2"/><i><font color="Green">E98</font></i>: 
 <a href="tbsp_1.html#K2">diameter</a> <font color="Maroon">c<sub>24</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 5
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E24:52"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jgraph_1.html#E27:52"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_1.html#E1:52_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jgraph_1.html#E2:52_2"><i><font color="Green">E68</font></i></a></i>;<br/>
<a NAME="E52:52_2"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="tbsp_1.html#K2">diameter</a> <font color="Maroon">c<sub>24</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E34:52_2"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jgraph_1.html#E50:52_2"><i><font color="Green">E97</font></i></a>, <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">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 5
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E51:52_2"><i><font color="Green">E98</font></i></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">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>17</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 5
 <b>by </b><i><a class="txt" href="jgraph_1.html#E7:52_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jgraph_1.html#E9:52_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="jgraph_1.html#T45" target="_self">Th45</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E33:52"/><i><font color="Green">E67</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E24:52"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jgraph_1.html#E27:52"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></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="E35:52"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></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">c<sub>17</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>17</sub></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">c<sub>17</sub></font> is <a href="goboard1.html#V1">increasing</a> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> <br/>for <font color="Olive">b<sub>3</sub></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">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Olive">b<sub>2</sub></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">c<sub>17</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></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">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Olive">b<sub>2</sub></font> holds <br/> <a href="tbsp_1.html#K2">diameter</a> <font color="Olive">b<sub>3</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 5 ) )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E11:52"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jgraph_1.html#E15:52"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jgraph_1.html#E22:52"><i><font color="Green">E61</font></i></a>, <a class="ref" href="uniform1.html#T15">UNIFORM1:15</a></i>;<br/>
<a NAME="E36:52"/><i><font color="Green">E69</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E35:52"><i><font color="Green">E68</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E37:52"/><b>then </b><i><font color="Green">E70</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E38:52"/><i><font color="Green">E71</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E36:52"><i><font color="Green">E69</font></i></a>, <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="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E39:52"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></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">c<sub>17</sub></font></span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>2</sub></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">c<sub>18</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E41:52"/><i><font color="Green">E72</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></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">c<sub>17</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) )
 ;<br/>
<a NAME="E42:52"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>18</sub></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">c<sub>19</sub></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">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>20</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:52_3"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></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">E74</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>18</sub></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">c<sub>18</sub></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">c<sub>21</sub></font> = <font color="Maroon">c<sub>20</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="jgraph_1.html#E3:52_3"><i><font color="Green">E73</font></i></a></i>;<br/>
<a NAME="E6:52_3"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E41:52"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_1.html#E3:52_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jgraph_1.html#E4:52_3"><i><font color="Green">E74</font></i></a></i>;<br/>
<a NAME="E7:52_3"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E41:52"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_1.html#E3:52_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jgraph_1.html#E4:52_3"><i><font color="Green">E74</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">E76</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>17</sub></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">c<sub>10</sub></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#E11:52"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jgraph_1.html#E15:52"><i><font color="Green">E57</font></i></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">E77</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E15:52"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jgraph_1.html#E35:52"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jgraph_1.html#E6:52_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jgraph_1.html#E8:52_3"><i><font color="Green">E76</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">c<sub>8</sub></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">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_1.html#E11:52"><i><font color="Green">E55</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></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">c<sub>19</sub></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#E3:52_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jgraph_1.html#E10:52_3"><i><font color="Green">E77</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as   <a href="struct_0.html#NM8">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="E44:52"/><i><font color="Green">E73</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></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">c<sub>17</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E45:52"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font> holds <br/><font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>20</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:52_4"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/>

<a NAME="E2:52_4"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E41:52"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_1.html#E44:52"><i><font color="Green">E73</font></i></a></i>;<br/>
<a NAME="E3:52_4"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font> )
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E44:52"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jgraph_1.html#E1:52_4"><i><font color="Green">E75</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">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_1.html#E41:52"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_1.html#E2:52_4"><i><font color="Green">E76</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E46:52"/><i><font color="Green">E75</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E35:52"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jgraph_1.html#E41:52"><i><font color="Green">E72</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E47:52"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>16</sub></font> holds <br/>( <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></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">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></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">c<sub>16</sub></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">c<sub>16</sub></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">c<sub>20</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:52_5"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/>

<a NAME="E2:52_5"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>20</sub></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">c<sub>14</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E27:52"><i><font color="Green">E63</font></i></a>, <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">E79</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></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">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E27:52"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_1.html#E2:52_5"><i><font color="Green">E78</font></i></a></i>;<br/>
<a NAME="E5:52_5"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:52_5"><i><font color="Green">E77</font></i></a>, <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">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E27:52"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_1.html#E33:52"><i><font color="Green">E67</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">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="goboard1.html#K1">.</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E27:52"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_1.html#E30:52"><i><font color="Green">E64</font></i></a></i>;<br/>
<a NAME="E8:52_5"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="goboard1.html#K1">.</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E33:52"><i><font color="Green">E67</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">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E27:52"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_1.html#E33:52"><i><font color="Green">E67</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">c<sub>16</sub></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">c<sub>16</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E27:52"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_1.html#E30:52"><i><font color="Green">E64</font></i></a></i>;<br/>
<a NAME="E11:52_5"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>16</sub></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">c<sub>16</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E33:52"><i><font color="Green">E67</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">E83</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E3:52_5"><i><font color="Green">E79</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">c<sub>9</sub></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#E9:52"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jgraph_1.html#E13:52"><i><font color="Green">E56</font></i></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">E84</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E13:52"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jgraph_1.html#E24:52"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jgraph_1.html#E5:52_5"><i><font color="Green">E80</font></i></a>, <a class="txt" href="jgraph_1.html#E12:52_5"><i><font color="Green">E83</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">c<sub>7</sub></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">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_1.html#E9:52"><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">c<sub>1</sub></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">c<sub>16</sub></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">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>20</sub></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">c<sub>16</sub></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">c<sub>16</sub></font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E2:52"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jgraph_1.html#E9:52"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jgraph_1.html#E13:52"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jgraph_1.html#E24:52"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jgraph_1.html#E8:52_5"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jgraph_1.html#E11:52_5"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jgraph_1.html#E14:52_5"><i><font color="Green">E84</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E48:52"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></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">c<sub>16</sub></font></span>)</span> holds <br/>( <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c<sub>16</sub></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">c<sub>16</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>16</sub></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">c<sub>16</sub></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">c<sub>20</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:52_6"/>
<font color="Maroon">c<sub>20</sub></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">c<sub>16</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:52_6"/><b>then </b>
<font color="Maroon">c<sub>20</sub></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">c<sub>16</sub></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">c<sub>20</sub></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">c<sub>16</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E33:52"><i><font color="Green">E67</font></i></a>, <a class="ref" href="jgraph_1.html#T59" target="_self">Th59</a></i>;<br/>
<a NAME="E4:52_6"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>16</sub></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">E79</font></i>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T61" target="_self">Th61</a></i>;<br/>
<a NAME="E6:52_6"/><i><font color="Green">E80</font></i>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c<sub>16</sub></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">c<sub>16</sub></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#E33:52"><i><font color="Green">E67</font></i></a>, <a class="ref" href="jgraph_1.html#T59" target="_self">Th59</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">c<sub>16</sub></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">c<sub>16</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></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">c<sub>16</sub></font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E33:52"><i><font color="Green">E67</font></i></a>, <a class="ref" href="jgraph_1.html#T59" target="_self">Th59</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">c<sub>16</sub></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">c<sub>16</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></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">c<sub>16</sub></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">c<sub>16</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E47:52"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jgraph_1.html#E4:52_6"><i><font color="Green">E78</font></i></a>, <a class="txt" href="jgraph_1.html#E5:52_6"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jgraph_1.html#E6:52_6"><i><font color="Green">E80</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E49:52"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>16</sub></font> holds <br/>( <font color="Maroon">c<sub>5</sub></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">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></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">c<sub>20</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:52_7"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/>

<a NAME="E2:52_7"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>20</sub></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">c<sub>16</sub></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">E81</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E27:52"><i><font color="Green">E63</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:52_7"/>
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E27:52"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_1.html#E2:52_7"><i><font color="Green">E80</font></i></a></i>;<br/>
<a NAME="E5:52_7"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:52_7"><i><font color="Green">E79</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E6:52_7"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E3:52_7"><i><font color="Green">E81</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">c<sub>9</sub></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#E9:52"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jgraph_1.html#E13:52"><i><font color="Green">E56</font></i></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">E84</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E13:52"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jgraph_1.html#E24:52"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jgraph_1.html#E5:52_7"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jgraph_1.html#E6:52_7"><i><font color="Green">E83</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">c<sub>7</sub></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">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_1.html#E9:52"><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">c<sub>1</sub></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">c<sub>5</sub></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">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E4:52"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jgraph_1.html#E8:52_7"><i><font color="Green">E84</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E50:52"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></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">c<sub>16</sub></font></span>)</span> holds <br/>( <font color="Maroon">c<sub>5</sub></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">c<sub>16</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></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">c<sub>20</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:52_8"/>
<font color="Maroon">c<sub>20</sub></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">c<sub>16</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:52_8"/><b>then </b>
<font color="Maroon">c<sub>20</sub></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">c<sub>16</sub></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">c<sub>20</sub></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">c<sub>16</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E33:52"><i><font color="Green">E67</font></i></a>, <a class="ref" href="jgraph_1.html#T60" target="_self">Th60</a></i>;<br/>
<a NAME="E4:52_8"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>16</sub></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">c<sub>16</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T61" target="_self">Th61</a></i>;<br/>
<b>hence </b><a NAME="E6:52_8"/>
( <font color="Maroon">c<sub>5</sub></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">c<sub>16</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E49:52"><i><font color="Green">E78</font></i></a>, <a class="txt" href="jgraph_1.html#E4:52_8"><i><font color="Green">E79</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E51:52"/><b>then </b><i><font color="Green">E79</font></i>: 
(  <a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c<sub>16</sub></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">c<sub>16</sub></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">c<sub>16</sub></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">c<sub>16</sub></font></span>)</span> &amp;  <a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c<sub>16</sub></font> <a href="goboard4.html#R1">lies_between</a> <font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K22">`2</a> ,<font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K22">`2</a>  )
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E48:52"><i><font color="Green">E77</font></i></a>, <a class="ref" href="goboard4.html#D1">GOBOARD4:def 1</a></i>;<br/>
<a NAME="E52:52"/><i><font color="Green">E80</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></font> holds <br/>( <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></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">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></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">c<sub>19</sub></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">c<sub>19</sub></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">c<sub>20</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:52_9"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/>

<a NAME="E2:52_9"/><b>then </b>
<font color="Maroon">c<sub>20</sub></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">c<sub>19</sub></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">E81</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E41:52"><i><font color="Green">E72</font></i></a>, <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">E82</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E45:52"><i><font color="Green">E74</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">c<sub>17</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E41:52"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_1.html#E46:52"><i><font color="Green">E75</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">c<sub>17</sub></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">E83</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="goboard1.html#K1">.</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E45:52"><i><font color="Green">E74</font></i></a></i>;<br/>
<a NAME="E8:52_9"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></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">c<sub>17</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E41:52"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_1.html#E46:52"><i><font color="Green">E75</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">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></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">E84</font></i>: 
<font color="Maroon">c<sub>19</sub></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">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E41:52"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_1.html#E45:52"><i><font color="Green">E74</font></i></a></i>;<br/>
<a NAME="E11:52_9"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E3:52_9"><i><font color="Green">E81</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">c<sub>10</sub></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#E11:52"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jgraph_1.html#E15:52"><i><font color="Green">E57</font></i></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">E86</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E15:52"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jgraph_1.html#E35:52"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jgraph_1.html#E4:52_9"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jgraph_1.html#E11:52_9"><i><font color="Green">E85</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">c<sub>8</sub></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">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_1.html#E11:52"><i><font color="Green">E55</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></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">c<sub>19</sub></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">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>20</sub></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">c<sub>19</sub></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">c<sub>19</sub></font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E5:52"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jgraph_1.html#E11:52"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jgraph_1.html#E15:52"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jgraph_1.html#E35:52"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jgraph_1.html#E7:52_9"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jgraph_1.html#E10:52_9"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jgraph_1.html#E13:52_9"><i><font color="Green">E86</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E53:52"/><i><font color="Green">E81</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></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">c<sub>19</sub></font></span>)</span> holds <br/>( <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c<sub>19</sub></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">c<sub>19</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>19</sub></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">c<sub>19</sub></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">c<sub>20</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:52_10"/>
<font color="Maroon">c<sub>20</sub></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">c<sub>19</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:52_10"/><b>then </b>
<font color="Maroon">c<sub>20</sub></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">c<sub>19</sub></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">c<sub>20</sub></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">c<sub>19</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E46:52"><i><font color="Green">E75</font></i></a>, <a class="ref" href="jgraph_1.html#T60" target="_self">Th60</a></i>;<br/>
<a NAME="E4:52_10"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></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">E83</font></i>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T61" target="_self">Th61</a></i>;<br/>
<a NAME="E6:52_10"/><i><font color="Green">E84</font></i>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c<sub>19</sub></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">c<sub>19</sub></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#E46:52"><i><font color="Green">E75</font></i></a>, <a class="ref" href="jgraph_1.html#T60" target="_self">Th60</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">c<sub>19</sub></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">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></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">c<sub>19</sub></font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E46:52"><i><font color="Green">E75</font></i></a>, <a class="ref" href="jgraph_1.html#T60" target="_self">Th60</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">c<sub>19</sub></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">c<sub>19</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></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">c<sub>19</sub></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">c<sub>19</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E52:52"><i><font color="Green">E80</font></i></a>, <a class="txt" href="jgraph_1.html#E4:52_10"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jgraph_1.html#E5:52_10"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jgraph_1.html#E6:52_10"><i><font color="Green">E84</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E54:52"/><i><font color="Green">E82</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></font> holds <br/>( <font color="Maroon">c<sub>3</sub></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">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></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">c<sub>20</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:52_11"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/>

<a NAME="E2:52_11"/><b>then </b>
<font color="Maroon">c<sub>20</sub></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">c<sub>19</sub></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">E83</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E41:52"><i><font color="Green">E72</font></i></a>, <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">E84</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E45:52"><i><font color="Green">E74</font></i></a></i>;<br/>
<a NAME="E5:52_11"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E3:52_11"><i><font color="Green">E83</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">c<sub>10</sub></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#E11:52"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jgraph_1.html#E15:52"><i><font color="Green">E57</font></i></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">E86</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E15:52"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jgraph_1.html#E35:52"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jgraph_1.html#E4:52_11"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jgraph_1.html#E5:52_11"><i><font color="Green">E85</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">c<sub>8</sub></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">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_1.html#E11:52"><i><font color="Green">E55</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></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">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E3:52"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jgraph_1.html#E7:52_11"><i><font color="Green">E86</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E55:52"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></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">c<sub>19</sub></font></span>)</span> holds <br/>( <font color="Maroon">c<sub>3</sub></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">c<sub>19</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></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">c<sub>20</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:52_12"/>
<font color="Maroon">c<sub>20</sub></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">c<sub>19</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:52_12"/><b>then </b>
<font color="Maroon">c<sub>20</sub></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">c<sub>19</sub></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">c<sub>20</sub></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">c<sub>19</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E46:52"><i><font color="Green">E75</font></i></a>, <a class="ref" href="jgraph_1.html#T59" target="_self">Th59</a></i>;<br/>
<a NAME="E4:52_12"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></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">c<sub>19</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T61" target="_self">Th61</a></i>;<br/>
<b>hence </b><a NAME="E6:52_12"/>
( <font color="Maroon">c<sub>3</sub></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">c<sub>19</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E54:52"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jgraph_1.html#E4:52_12"><i><font color="Green">E83</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E56:52"/><b>then </b><i><font color="Green">E83</font></i>: 
(  <a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c<sub>19</sub></font> <a href="goboard4.html#R1">lies_between</a> <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K21">`1</a> ,<font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c<sub>19</sub></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">c<sub>19</sub></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">c<sub>19</sub></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">c<sub>19</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E53:52"><i><font color="Green">E81</font></i></a>, <a class="ref" href="goboard4.html#D1">GOBOARD4:def 1</a></i>;<br/>
<a NAME="E57:52"/><i><font color="Green">E84</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>19</sub></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">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Olive">b<sub>1</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></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">c<sub>20</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:52_13"/><i><font color="Green">E85</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>20</sub></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">c<sub>19</sub></font> )
 ;<br/>

<a NAME="E2:52_13"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>19</sub></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">E87</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:52_13"><i><font color="Green">E85</font></i></a>, <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">E88</font></i>: 
<font color="Maroon">c<sub>20</sub></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">c<sub>17</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E41:52"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_1.html#E1:52_13"><i><font color="Green">E85</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:52_13"/>
<font color="Maroon">c<sub>20</sub></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">c<sub>17</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E41:52"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_1.html#E1:52_13"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jgraph_1.html#E2:52_13"><i><font color="Green">E86</font></i></a>, <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">E89</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E41:52"><i><font color="Green">E72</font></i></a></i>;<br/>
<a NAME="E7:52_13"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:52_13"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jgraph_1.html#E2:52_13"><i><font color="Green">E86</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E8:52_13"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E41:52"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_1.html#E4:52_13"><i><font color="Green">E88</font></i></a></i>;<br/>
<a NAME="E9:52_13"/>
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:52_13"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jgraph_1.html#E3:52_13"><i><font color="Green">E87</font></i></a>, <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">E92</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E41:52"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_1.html#E4:52_13"><i><font color="Green">E88</font></i></a></i>;<br/>
<a NAME="E11:52_13"/>
<font color="Maroon">c<sub>20</sub></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">c<sub>17</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E41:52"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_1.html#E1:52_13"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jgraph_1.html#E2:52_13"><i><font color="Green">E86</font></i></a>, <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">E93</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></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">E94</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<i><font color="Green">E95</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></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#E11:52"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jgraph_1.html#E15:52"><i><font color="Green">E57</font></i></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">E96</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E15:52"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jgraph_1.html#E35:52"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jgraph_1.html#E6:52_13"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jgraph_1.html#E13:52_13"><i><font color="Green">E94</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<i><font color="Green">E97</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>8</sub></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">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_1.html#E11:52"><i><font color="Green">E55</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></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">c<sub>20</sub></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">c<sub>17</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E41:52"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_1.html#E1:52_13"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jgraph_1.html#E3:52_13"><i><font color="Green">E87</font></i></a>, <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">c<sub>20</sub></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">c<sub>17</sub></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">c<sub>17</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></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">c<sub>17</sub></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">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></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">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E15:52"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jgraph_1.html#E35:52"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jgraph_1.html#E8:52_13"><i><font color="Green">E91</font></i></a>, <a class="txt" href="jgraph_1.html#E14:52_13"><i><font color="Green">E95</font></i></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">E98</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></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#E41:52"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_1.html#E4:52_13"><i><font color="Green">E88</font></i></a>, <a class="txt" href="jgraph_1.html#E16:52_13"><i><font color="Green">E97</font></i></a></i>;<br/>
<a NAME="E22:52_13"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E41:52"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_1.html#E1:52_13"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jgraph_1.html#E2:52_13"><i><font color="Green">E86</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E23:52_13"/>
<font color="Maroon">c<sub>20</sub></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">c<sub>17</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E41:52"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_1.html#E1:52_13"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jgraph_1.html#E3:52_13"><i><font color="Green">E87</font></i></a>, <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">E100</font></i>: 
<font color="Maroon">c<sub>20</sub></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">c<sub>17</sub></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">E101</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></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">c<sub>17</sub></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">E102</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E41:52"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_1.html#E1:52_13"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jgraph_1.html#E3:52_13"><i><font color="Green">E87</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E27:52_13"/>
<font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>20</sub></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">E103</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>17</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>17</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></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#E35:52"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jgraph_1.html#E12:52_13"><i><font color="Green">E93</font></i></a>, <a class="txt" href="jgraph_1.html#E13:52_13"><i><font color="Green">E94</font></i></a>, <a class="txt" href="jgraph_1.html#E22:52_13"><i><font color="Green">E99</font></i></a>, <a class="txt" href="jgraph_1.html#E24:52_13"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jgraph_1.html#E25:52_13"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jgraph_1.html#E26:52_13"><i><font color="Green">E102</font></i></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">c<sub>21</sub></font> = <font color="Maroon">c<sub>2</sub></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#E31:52"><i><font color="Green">E65</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>2</sub></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">E104</font></i>: 
<font color="Maroon">c<sub>22</sub></font> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:52"><i><font color="Green">E47</font></i></a>, <a class="ref" href="jordan5a.html#T1">JORDAN5A:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>22</sub></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">c<sub>24</sub></font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></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#E35:52"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jgraph_1.html#E13:52_13"><i><font color="Green">E94</font></i></a>, <a class="txt" href="jgraph_1.html#E22:52_13"><i><font color="Green">E99</font></i></a>, <a class="txt" href="jgraph_1.html#E25:52_13"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jgraph_1.html#E26:52_13"><i><font color="Green">E102</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">c<sub>25</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>, <font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></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="txt" href="jgraph_1.html#E6:52_13"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jgraph_1.html#E15:52_13"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jgraph_1.html#E16:52_13"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jgraph_1.html#E21:52_13"><i><font color="Green">E98</font></i></a>, <a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<a NAME="E35:52_13"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>24</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E22:52_13"><i><font color="Green">E99</font></i></a>, <a class="txt" href="jgraph_1.html#E28:52_13"><i><font color="Green">E103</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E36:52_13"/>
<font color="Maroon">c<sub>17</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>24</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E26:52_13"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jgraph_1.html#E28:52_13"><i><font color="Green">E103</font></i></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">E106</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>24</sub></font> )
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E14:52_13"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jgraph_1.html#E35:52_13"><i><font color="Green">E105</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>24</sub></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">E107</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></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">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E11:52"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jgraph_1.html#E15:52"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jgraph_1.html#E16:52_13"><i><font color="Green">E97</font></i></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">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></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="txt" href="jgraph_1.html#E39:52_13"><i><font color="Green">E107</font></i></a>, <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">E108</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>21</sub></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">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:52"><i><font color="Green">E47</font></i></a>, <a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/>
<a NAME="E44:52_13"/><b>then </b><i><font color="Green">E109</font></i>: 
(  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c<sub>5</sub></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">c<sub>23</sub></font> &amp;  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>5</sub></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">c<sub>23</sub></font> )
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E31:52_13"><i><font color="Green">E104</font></i></a>, <a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>28</sub></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">c<sub>23</sub></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">c<sub>23</sub></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">c<sub>23</sub></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">c<sub>23</sub></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">c<sub>23</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>23</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E44:52_13"><i><font color="Green">E109</font></i></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">E110</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">c<sub>23</sub></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">c<sub>23</sub></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">c<sub>23</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">c<sub>23</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E44:52_13"><i><font color="Green">E109</font></i></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">c<sub>23</sub></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">c<sub>23</sub></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">E111</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">c<sub>23</sub></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">c<sub>23</sub></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">c<sub>23</sub></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">c<sub>23</sub></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="txt" href="jgraph_1.html#E46:52_13"><i><font color="Green">E110</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E50:52_13"/><i><font color="Green">E112</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">c<sub>23</sub></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">c<sub>23</sub></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">c<sub>23</sub></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">c<sub>23</sub></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">c<sub>23</sub></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">c<sub>23</sub></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">c<sub>23</sub></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">c<sub>23</sub></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">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font> holds <br/> <a href="metric_1.html#K4">dist</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">c<sub>23</sub></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">c<sub>23</sub></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">c<sub>23</sub></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">c<sub>23</sub></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">c<sub>29</sub></font>, <font color="Maroon">c<sub>30</sub></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">E113</font></i>: 
( <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>31</sub></font> = <font color="Maroon">c<sub>29</sub></font>, <font color="Maroon">c<sub>32</sub></font> = <font color="Maroon">c<sub>30</sub></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">E114</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>30</sub></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">c<sub>31</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>32</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T45" target="_self">Th45</a></i>;<br/>
<a NAME="E4:52_13_3"/><i><font color="Green">E115</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>32</sub></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">c<sub>31</sub></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">c<sub>32</sub></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">c<sub>31</sub></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">c<sub>32</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T49" target="_self">Th49</a></i>;<br/>
<a NAME="E5:52_13_3"/><i><font color="Green">E116</font></i>: 
(  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>31</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c<sub>31</sub></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">c<sub>23</sub></font> &amp;  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>31</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>31</sub></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">c<sub>23</sub></font> )
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E31:52_13"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jgraph_1.html#E1:52_13_3"><i><font color="Green">E113</font></i></a>, <a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/>
<a NAME="E6:52_13_3"/><i><font color="Green">E117</font></i>: 
(  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>32</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c<sub>32</sub></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">c<sub>23</sub></font> &amp;  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>32</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>32</sub></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">c<sub>23</sub></font> )
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E31:52_13"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jgraph_1.html#E1:52_13_3"><i><font color="Green">E113</font></i></a>, <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">E118</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>31</sub></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">c<sub>32</sub></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">c<sub>23</sub></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">c<sub>23</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E5:52_13_3"><i><font color="Green">E116</font></i></a>, <a class="ref" href="jgraph_1.html#T31" target="_self">Th31</a></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">c<sub>31</sub></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">c<sub>32</sub></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">c<sub>23</sub></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">c<sub>23</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E5:52_13_3"><i><font color="Green">E116</font></i></a>, <a class="txt" href="jgraph_1.html#E6:52_13_3"><i><font color="Green">E117</font></i></a>, <a class="ref" href="jgraph_1.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E9:52_13_3"/><b>then </b><i><font color="Green">E119</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">c<sub>31</sub></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">c<sub>32</sub></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">c<sub>31</sub></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">c<sub>32</sub></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">c<sub>23</sub></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">c<sub>23</sub></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">c<sub>23</sub></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">c<sub>23</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E7:52_13_3"><i><font color="Green">E118</font></i></a>, <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">c<sub>23</sub></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">c<sub>23</sub></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">c<sub>23</sub></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">c<sub>23</sub></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">c<sub>23</sub></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">c<sub>23</sub></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">c<sub>23</sub></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">c<sub>23</sub></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">c<sub>31</sub></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">c<sub>32</sub></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">c<sub>31</sub></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">c<sub>32</sub></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">c<sub>23</sub></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">c<sub>23</sub></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">c<sub>23</sub></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">c<sub>23</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E9:52_13_3"><i><font color="Green">E119</font></i></a>, <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">c<sub>29</sub></font>,<font color="Maroon">c<sub>30</sub></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">c<sub>23</sub></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">c<sub>23</sub></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">c<sub>23</sub></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">c<sub>23</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="txt" href="jgraph_1.html#E3:52_13_3"><i><font color="Green">E114</font></i></a>, <a class="txt" href="jgraph_1.html#E4:52_13_3"><i><font color="Green">E115</font></i></a>, <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">c<sub>21</sub></font> is <a href="tbsp_1.html#V6">bounded</a>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E49:52_13"><i><font color="Green">E111</font></i></a>, <a class="txt" href="jgraph_1.html#E50:52_13"><i><font color="Green">E112</font></i></a>, <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">E113</font></i>: 
<font color="Maroon">c<sub>27</sub></font> is <a href="tbsp_1.html#V6">bounded</a>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E42:52_13"><i><font color="Green">E108</font></i></a>, <a class="ref" href="tbsp_1.html#T21">TBSP_1:21</a></i>;<br/>
<a NAME="E54:52_13"/><i><font color="Green">E114</font></i>: 
 <a href="tbsp_1.html#K2">diameter</a> <font color="Maroon">c<sub>27</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 5
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E35:52"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jgraph_1.html#E41:52"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_1.html#E1:52_13"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jgraph_1.html#E2:52_13"><i><font color="Green">E86</font></i></a></i>;<br/>
<a NAME="E55:52_13"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>26</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="tbsp_1.html#K2">diameter</a> <font color="Maroon">c<sub>27</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E37:52_13"><i><font color="Green">E106</font></i></a>, <a class="txt" href="jgraph_1.html#E53:52_13"><i><font color="Green">E113</font></i></a>, <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">c<sub>25</sub></font>,<font color="Maroon">c<sub>26</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 5
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E54:52_13"><i><font color="Green">E114</font></i></a>, <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">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>20</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 5
 <b>by </b><i><a class="txt" href="jgraph_1.html#E7:52_13"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jgraph_1.html#E10:52_13"><i><font color="Green">E92</font></i></a>, <a class="ref" href="jgraph_1.html#T45" target="_self">Th45</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="struct_0.html#NM8">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="E59:52"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> is <a href="topreal1.html#V1">special</a> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">c<sub>20</sub></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">c<sub>20</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></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">c<sub>16</sub></font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>16</sub></font> &amp;  <a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c<sub>20</sub></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">c<sub>16</sub></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">c<sub>16</sub></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">c<sub>16</sub></font></span>)</span> &amp;  <a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c<sub>20</sub></font> <a href="goboard4.html#R1">lies_between</a> <font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K22">`2</a> ,<font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K22">`2</a>  &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>20</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>16</sub></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">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>2</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 5 ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>20</sub></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">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Olive">b<sub>1</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 5 ) )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E22:52"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jgraph_1.html#E32:52"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jgraph_1.html#E33:52"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jgraph_1.html#E51:52"><i><font color="Green">E79</font></i></a>, <a class="ref" href="jgraph_1.html#T56" target="_self">Th56</a></i>;<br/>
<a NAME="E60:52"/><i><font color="Green">E86</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E33:52"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jgraph_1.html#E59:52"><i><font color="Green">E85</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="struct_0.html#NM8">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="E62:52"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> is <a href="topreal1.html#V1">special</a> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">c<sub>21</sub></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">c<sub>21</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></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">c<sub>19</sub></font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>19</sub></font> &amp;  <a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c<sub>21</sub></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">c<sub>19</sub></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">c<sub>19</sub></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">c<sub>19</sub></font></span>)</span> &amp;  <a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c<sub>21</sub></font> <a href="goboard4.html#R1">lies_between</a> <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K21">`1</a> ,<font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></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">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>2</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 5 ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>21</sub></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">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Olive">b<sub>1</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 5 ) )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E22:52"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jgraph_1.html#E46:52"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jgraph_1.html#E56:52"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jgraph_1.html#E57:52"><i><font color="Green">E84</font></i></a>, <a class="ref" href="jgraph_1.html#T57" target="_self">Th57</a></i>;<br/>
<a NAME="E63:52"/><i><font color="Green">E88</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E46:52"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jgraph_1.html#E62:52"><i><font color="Green">E87</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E64:52"/><i><font color="Green">E89</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E27:52"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_1.html#E33:52"><i><font color="Green">E67</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">E90</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E27:52"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_1.html#E33:52"><i><font color="Green">E67</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E66:52"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="goboard1.html#K1">.</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E27:52"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_1.html#E30:52"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jgraph_1.html#E64:52"><i><font color="Green">E89</font></i></a></i>;<br/>
<a NAME="E67:52"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E33:52"><i><font color="Green">E67</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">E93</font></i>: 
<font color="Maroon">c<sub>16</sub></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">c<sub>16</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></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">c<sub>16</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E33:52"><i><font color="Green">E67</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E69:52"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E60:52"><i><font color="Green">E86</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E70:52"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">c<sub>20</sub></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">c<sub>20</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></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">c<sub>20</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E60:52"><i><font color="Green">E86</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E71:52"/><i><font color="Green">E96</font></i>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E9:52"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jgraph_1.html#E13:52"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jgraph_1.html#E24:52"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jgraph_1.html#E59:52"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jgraph_1.html#E60:52"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_1.html#E66:52"><i><font color="Green">E91</font></i></a>, <a class="txt" href="jgraph_1.html#E69:52"><i><font color="Green">E94</font></i></a>, <a class="ref" href="jgraph_1.html#T59" target="_self">Th59</a></i>;<br/>
<a NAME="E72:52"/>
<font color="Maroon">c<sub>16</sub></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">c<sub>16</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E9:52"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jgraph_1.html#E13:52"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jgraph_1.html#E24:52"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jgraph_1.html#E27:52"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_1.html#E30:52"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jgraph_1.html#E65:52"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jgraph_1.html#E68:52"><i><font color="Green">E93</font></i></a></i>;<br/>
<a NAME="E73:52"/><b>then </b><i><font color="Green">E97</font></i>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c<sub>20</sub></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">c<sub>20</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E59:52"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jgraph_1.html#E60:52"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_1.html#E68:52"><i><font color="Green">E93</font></i></a>, <a class="txt" href="jgraph_1.html#E70:52"><i><font color="Green">E95</font></i></a>, <a class="ref" href="jgraph_1.html#T59" target="_self">Th59</a></i>;<br/>
<a NAME="E74:52"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E11:52"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jgraph_1.html#E15:52"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jgraph_1.html#E35:52"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jgraph_1.html#E37:52"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jgraph_1.html#E45:52"><i><font color="Green">E74</font></i></a></i>;<br/>
<a NAME="E75:52"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E36:52"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jgraph_1.html#E41:52"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E76:52"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">c<sub>19</sub></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">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></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">c<sub>19</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E36:52"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jgraph_1.html#E41:52"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E77:52"/>
<font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E63:52"><i><font color="Green">E88</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E78:52"/><b>then </b><i><font color="Green">E101</font></i>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E62:52"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jgraph_1.html#E63:52"><i><font color="Green">E88</font></i></a>, <a class="txt" href="jgraph_1.html#E74:52"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jgraph_1.html#E75:52"><i><font color="Green">E99</font></i></a>, <a class="ref" href="jgraph_1.html#T60" target="_self">Th60</a></i>;<br/>
<a NAME="E79:52"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">c<sub>21</sub></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">c<sub>21</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></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">c<sub>21</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E63:52"><i><font color="Green">E88</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E80:52"/>
<font color="Maroon">c<sub>21</sub></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">c<sub>21</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E11:52"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jgraph_1.html#E15:52"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jgraph_1.html#E35:52"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jgraph_1.html#E38:52"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jgraph_1.html#E41:52"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_1.html#E45:52"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jgraph_1.html#E62:52"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jgraph_1.html#E76:52"><i><font color="Green">E100</font></i></a></i>;<br/>
<a NAME="E81:52"/><b>then </b><i><font color="Green">E103</font></i>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c<sub>21</sub></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">c<sub>21</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E63:52"><i><font color="Green">E88</font></i></a>, <a class="txt" href="jgraph_1.html#E79:52"><i><font color="Green">E102</font></i></a>, <a class="ref" href="jgraph_1.html#T60" target="_self">Th60</a></i>;<br/>
<i><font color="Green">E104</font></i>: <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></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#E59:52"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jgraph_1.html#E60:52"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_1.html#E67:52"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jgraph_1.html#E69:52"><i><font color="Green">E94</font></i></a>, <a class="ref" href="jgraph_1.html#T59" target="_self">Th59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="jgraph_1.html#E33:52"><i><font color="Green">E67</font></i></a>, <a class="ref" href="jgraph_1.html#T59" target="_self">Th59</a></i>
;<br/>
<i><font color="Green">E105</font></i>: <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c<sub>20</sub></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">c<sub>20</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></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">c<sub>16</sub></font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="txt" href="jgraph_1.html#E59:52"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jgraph_1.html#E60:52"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_1.html#E68:52"><i><font color="Green">E93</font></i></a>, <a class="txt" href="jgraph_1.html#E70:52"><i><font color="Green">E95</font></i></a>, <a class="ref" href="jgraph_1.html#T59" target="_self">Th59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c<sub>16</sub></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">c<sub>16</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_1.html#E33:52"><i><font color="Green">E67</font></i></a>, <a class="ref" href="jgraph_1.html#T59" target="_self">Th59</a></i>
;<br/>
<a NAME="E84:52"/>
<font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E63:52"><i><font color="Green">E88</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<b>then </b><i><font color="Green">E106</font></i>: <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></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#E62:52"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jgraph_1.html#E63:52"><i><font color="Green">E88</font></i></a>, <a class="txt" href="jgraph_1.html#E75:52"><i><font color="Green">E99</font></i></a>, <a class="ref" href="jgraph_1.html#T60" target="_self">Th60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="jgraph_1.html#E46:52"><i><font color="Green">E75</font></i></a>, <a class="ref" href="jgraph_1.html#T60" target="_self">Th60</a></i>
;<br/>
<a NAME="E86:52"/>
<font color="Maroon">c<sub>21</sub></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">c<sub>21</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></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">c<sub>21</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E63:52"><i><font color="Green">E88</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<b>then </b><i><font color="Green">E107</font></i>: <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c<sub>21</sub></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">c<sub>21</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></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">c<sub>19</sub></font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="txt" href="jgraph_1.html#E62:52"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jgraph_1.html#E63:52"><i><font color="Green">E88</font></i></a>, <a class="txt" href="jgraph_1.html#E76:52"><i><font color="Green">E100</font></i></a>, <a class="ref" href="jgraph_1.html#T60" target="_self">Th60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c<sub>19</sub></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">c<sub>19</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_1.html#E46:52"><i><font color="Green">E75</font></i></a>, <a class="ref" href="jgraph_1.html#T60" target="_self">Th60</a></i>
;<br/>
<a NAME="E88:52"/>
5 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E24:52"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jgraph_1.html#E27:52"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_1.html#E59:52"><i><font color="Green">E85</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E89:52"/><b>then </b><i><font color="Green">E108</font></i>: 
2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E90:52"/>
5 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E35:52"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jgraph_1.html#E41:52"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_1.html#E62:52"><i><font color="Green">E87</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E91:52"/><b>then </b><i><font color="Green">E109</font></i>: 
2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E92:52"/><i><font color="Green">E110</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E24:52"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E93:52"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E94:52"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="goboard1.html#K1">.</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E27:52"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_1.html#E30:52"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jgraph_1.html#E67:52"><i><font color="Green">E92</font></i></a></i>;<br/>
<a NAME="E95:52"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E92:52"><i><font color="Green">E110</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E96:52"/><b>then </b><i><font color="Green">E112</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>20</sub></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">c<sub>20</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E6:52"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jgraph_1.html#E9:52"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jgraph_1.html#E13:52"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jgraph_1.html#E24:52"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jgraph_1.html#E27:52"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_1.html#E30:52"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jgraph_1.html#E59:52"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jgraph_1.html#E67:52"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jgraph_1.html#E69:52"><i><font color="Green">E94</font></i></a>, <a class="txt" href="jgraph_1.html#E70:52"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jgraph_1.html#E94:52"><i><font color="Green">E111</font></i></a></i>;<br/>
<a NAME="E97:52"/><i><font color="Green">E113</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E35:52"><i><font color="Green">E68</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E98:52"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E99:52"/><b>then </b><i><font color="Green">E114</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="goboard1.html#K1">.</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E45:52"><i><font color="Green">E74</font></i></a></i>;<br/>
<a NAME="E100:52"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E97:52"><i><font color="Green">E113</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E101:52"/><b>then </b>
<font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>21</sub></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">c<sub>21</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E7:52"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jgraph_1.html#E11:52"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jgraph_1.html#E15:52"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jgraph_1.html#E35:52"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jgraph_1.html#E41:52"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_1.html#E45:52"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jgraph_1.html#E62:52"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jgraph_1.html#E75:52"><i><font color="Green">E99</font></i></a>, <a class="txt" href="jgraph_1.html#E76:52"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jgraph_1.html#E99:52"><i><font color="Green">E114</font></i></a></i>;<br/>
<a NAME="E102:52"/><b>then </b>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>20</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E59:52"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jgraph_1.html#E62:52"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jgraph_1.html#E69:52"><i><font color="Green">E94</font></i></a>, <a class="txt" href="jgraph_1.html#E70:52"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jgraph_1.html#E71:52"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jgraph_1.html#E73:52"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jgraph_1.html#E78:52"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jgraph_1.html#E81:52"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jgraph_1.html#E82:52"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jgraph_1.html#E83:52"><i><font color="Green">E105</font></i></a>, <a class="txt" href="jgraph_1.html#E85:52"><i><font color="Green">E106</font></i></a>, <a class="txt" href="jgraph_1.html#E87:52"><i><font color="Green">E107</font></i></a>, <a class="txt" href="jgraph_1.html#E89:52"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jgraph_1.html#E91:52"><i><font color="Green">E109</font></i></a>, <a class="txt" href="jgraph_1.html#E96:52"><i><font color="Green">E112</font></i></a>, <a class="ref" href="jgraph_1.html#T30" target="_self">Th30</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>22</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E104:52"/><i><font color="Green">E115</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>21</sub></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">c<sub>23</sub></font> = <font color="Maroon">c<sub>22</sub></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><a class="txt" href="jgraph_1.html#E104:52"><i><font color="Green">E115</font></i></a></i>;<br/>
<a NAME="E106:52"/>
<font color="Maroon">c<sub>23</sub></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">c<sub>20</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>20</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E104:52"><i><font color="Green">E115</font></i></a>, <a class="ref" href="topreal1.html#D6">TOPREAL1:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>24</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E108:52"/><i><font color="Green">E116</font></i>: 
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>24</sub></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">c<sub>20</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>20</sub></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">c<sub>25</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E110:52"/><i><font color="Green">E117</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>25</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>25</sub></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">c<sub>20</sub></font> )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E108:52"><i><font color="Green">E116</font></i></a></i>;<br/>
<a NAME="E111:52"/><i><font color="Green">E118</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E110:52"><i><font color="Green">E117</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E112:52"/>
<font color="Maroon">c<sub>25</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E110:52"><i><font color="Green">E117</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E113:52"/><b>then </b>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E110:52"><i><font color="Green">E117</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>26</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E115:52"/><i><font color="Green">E119</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>16</sub></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">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>26</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 5 )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E59:52"><i><font color="Green">E85</font></i></a></i>;<br/>
<a NAME="E116:52"/><i><font color="Green">E120</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>25</sub></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">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>25</sub></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#E108:52"><i><font color="Green">E116</font></i></a>, <a class="txt" href="jgraph_1.html#E110:52"><i><font color="Green">E117</font></i></a>, <a class="txt" href="jgraph_1.html#E111:52"><i><font color="Green">E118</font></i></a>, <a class="ref" href="jgraph_1.html#T53" target="_self">Th53</a></i>;<br/>
<a NAME="E117:52"/>
<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">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>25</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 5
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E59:52"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jgraph_1.html#E110:52"><i><font color="Green">E117</font></i></a></i>;<br/>
<a NAME="E118:52"/><b>then </b><i><font color="Green">E121</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>25</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 5
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E116:52"><i><font color="Green">E120</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E119:52"/><i><font color="Green">E122</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>26</sub></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">c<sub>23</sub></font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>25</sub></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">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>26</sub></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="E120:52"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>25</sub></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">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>26</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 5</span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E115:52"><i><font color="Green">E119</font></i></a>, <a class="txt" href="jgraph_1.html#E118:52"><i><font color="Green">E121</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E121:52"/><b>then </b><i><font color="Green">E123</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>26</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 5</span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E119:52"><i><font color="Green">E122</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E122:52"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">c<sub>26</sub></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">c<sub>16</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E115:52"><i><font color="Green">E119</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E123:52"/><b>then </b><i><font color="Green">E125</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E27:52"><i><font color="Green">E63</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E124:52"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>16</sub></font> )
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E122:52"><i><font color="Green">E124</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E125:52"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>26</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E126:52"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E27:52"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_1.html#E122:52"><i><font color="Green">E124</font></i></a></i>;<br/>
<a NAME="E127:52"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E123:52"><i><font color="Green">E125</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<i><font color="Green">E128</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></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#E9:52"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jgraph_1.html#E13:52"><i><font color="Green">E56</font></i></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">c<sub>7</sub></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">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_1.html#E9:52"><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">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<a NAME="E130:52"/><b>then </b><i><font color="Green">E129</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E13:52"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jgraph_1.html#E24:52"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jgraph_1.html#E126:52"><i><font color="Green">E126</font></i></a>, <a class="txt" href="jgraph_1.html#E127:52"><i><font color="Green">E127</font></i></a>, <a class="txt" href="jgraph_1.html#E128:52"><i><font color="Green">E128</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>26</sub></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="E132:52"/><i><font color="Green">E130</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>23</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 5</span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E121:52"><i><font color="Green">E123</font></i></a>, <a class="ref" href="toprns_1.html#T28">TOPRNS_1:28</a></i>;<br/>
<a NAME="E133:52"/>
<font color="Maroon">c<sub>23</sub></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">c<sub>21</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>21</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E104:52"><i><font color="Green">E115</font></i></a>, <a class="ref" href="topreal1.html#D6">TOPREAL1:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>28</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E135:52"/><i><font color="Green">E131</font></i>: 
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>28</sub></font> &amp; <font color="Maroon">c<sub>28</sub></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">c<sub>21</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>21</sub></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">c<sub>29</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E137:52"/><i><font color="Green">E132</font></i>: 
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>29</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>29</sub></font> &amp; <font color="Maroon">c<sub>29</sub></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">c<sub>21</sub></font> )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E135:52"><i><font color="Green">E131</font></i></a></i>;<br/>
<a NAME="E138:52"/><i><font color="Green">E133</font></i>: 
(  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>29</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>29</sub></font> &amp; <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>29</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>29</sub></font> &amp; <font color="Maroon">c<sub>29</sub></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">c<sub>21</sub></font> )
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E135:52"><i><font color="Green">E131</font></i></a>, <a class="txt" href="jgraph_1.html#E137:52"><i><font color="Green">E132</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E139:52"/>
( <font color="Maroon">c<sub>29</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font> &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>29</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>29</sub></font> &amp; <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>29</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>29</sub></font> &amp; <font color="Maroon">c<sub>29</sub></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">c<sub>21</sub></font> )
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E135:52"><i><font color="Green">E131</font></i></a>, <a class="txt" href="jgraph_1.html#E137:52"><i><font color="Green">E132</font></i></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="E140:52"/><b>then </b>
<font color="Maroon">c<sub>29</sub></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">c<sub>21</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E141:52"/><b>then </b>
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></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">c<sub>30</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E143:52"/><i><font color="Green">E134</font></i>: 
( <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></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">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>30</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 5 )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E62:52"><i><font color="Green">E87</font></i></a></i>;<br/>
<a NAME="E144:52"/><i><font color="Green">E135</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>29</sub></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">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>29</sub></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#E138:52"><i><font color="Green">E133</font></i></a>, <a class="ref" href="jgraph_1.html#T53" target="_self">Th53</a></i>;<br/>
<a NAME="E145:52"/>
<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">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>29</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 5
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E62:52"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jgraph_1.html#E137:52"><i><font color="Green">E132</font></i></a></i>;<br/>
<a NAME="E146:52"/><b>then </b><i><font color="Green">E136</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>29</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 5
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E144:52"><i><font color="Green">E135</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E147:52"/><i><font color="Green">E137</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>30</sub></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">c<sub>23</sub></font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>29</sub></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">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>30</sub></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="E148:52"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>29</sub></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">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>30</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 5</span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E143:52"><i><font color="Green">E134</font></i></a>, <a class="txt" href="jgraph_1.html#E146:52"><i><font color="Green">E136</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E149:52"/><b>then </b><i><font color="Green">E138</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>30</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 5</span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E147:52"><i><font color="Green">E137</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E150:52"/>
<font color="Maroon">c<sub>30</sub></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">c<sub>19</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E143:52"><i><font color="Green">E134</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E151:52"/><b>then </b>
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E41:52"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E152:52"/><b>then </b><i><font color="Green">E139</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font> )
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E45:52"><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#K4">dom</a> <font color="Maroon">c<sub>10</sub></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#E11:52"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jgraph_1.html#E15:52"><i><font color="Green">E57</font></i></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="E154:52"/><b>then </b><i><font color="Green">E140</font></i>: 
( <font color="Maroon">c<sub>2</sub></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">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E15:52"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jgraph_1.html#E35:52"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jgraph_1.html#E152:52"><i><font color="Green">E139</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<i><font color="Green">E141</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>8</sub></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">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_1.html#E11:52"><i><font color="Green">E55</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></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">c<sub>31</sub></font> = <font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>30</sub></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">c<sub>32</sub></font> = <font color="Maroon">c<sub>27</sub></font>, <font color="Maroon">c<sub>33</sub></font> = <font color="Maroon">c<sub>31</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<a NAME="E158:52"/>
 <a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>33</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E18:52"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jgraph_1.html#E130:52"><i><font color="Green">E129</font></i></a>, <a class="txt" href="jgraph_1.html#E154:52"><i><font color="Green">E140</font></i></a>, <a class="txt" href="jgraph_1.html#E155:52"><i><font color="Green">E141</font></i></a>, <a class="ref" href="weierstr.html#T40">WEIERSTR:40</a></i>;<br/>
<a NAME="E159:52"/><b>then </b><i><font color="Green">E142</font></i>: 
( <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>31</sub></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">c<sub>27</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>23</sub></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">c<sub>23</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>31</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></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">c<sub>27</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>31</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> )
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T45" target="_self">Th45</a>, <a class="ref" href="toprns_1.html#T35">TOPRNS_1:35</a></i>;<br/>
<a NAME="E160:52"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>23</sub></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">c<sub>23</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>31</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 5</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E132:52"><i><font color="Green">E130</font></i></a>, <a class="txt" href="jgraph_1.html#E149:52"><i><font color="Green">E138</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E161:52"/><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">c<sub>27</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>31</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 5</span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E159:52"><i><font color="Green">E142</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E162:52"/><b>then </b>
 <a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 5</span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E159:52"><i><font color="Green">E142</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E163:52"/><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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></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="E164:52"/><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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></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#E22:52"><i><font color="Green">E61</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<a NAME="E165:52"/><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#E22:52"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/>
<b>hence </b><a NAME="E166:52"/>
contradiction
 ;<br/>


</div>
