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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="funct_1.html#V3">constant</a> <a href="goboard5.html#V2">standard</a> <a href="goboard5.html#NM1">special_circular_sequence</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>3</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><b>that </b><br/><a NAME="E1:1"/><i><font color="Green">E2</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>and </b><br/><a NAME="E2:1"/><i><font color="Green">E3</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="jordan2c.html#R2">is_outside_component_of</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font> as  <a href="tbsp_1.html#V6">bounded</a> <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="jordan2c.html#D2">JORDAN2C:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">c<sub>6</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span><b> such that </b><br/><a NAME="E5:1"/><i><font color="Green">E4</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E6:1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E7:1"/><i><font color="Green">E5</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#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>4</sub></font> holds <br/> <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="tbsp_1.html#T15">TBSP_1:15</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1;<br/>
<a NAME="E8:1"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</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="E1:1_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</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> ;<br/>
<a NAME="E3:1_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="gobrd14.html#E7:1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="gobrd14.html#E1:1_1"><i><font color="Green">E7</font></i></a></i>;<br/>
<a NAME="E4:1_1"/><b>then </b><i><font color="Green">E8</font></i>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E5:1_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="metric_1.html#T4">METRIC_1:4</a></i>;<br/>
<a NAME="E6:1_1"/><b>then </b><i><font color="Green">E9</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="gobrd14.html#E4:1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E8:1_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="txt" href="gobrd14.html#E6:1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E9:1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1
;<br/>

<a NAME="E9:1"/><i><font color="Green">E7</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T5">METRIC_1:5</a></i>;<br/>
<a NAME="E10:1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E11:1"/><i><font color="Green">E8</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="txt" href="gobrd14.html#E5:1"><i><font color="Green">E4</font></i></a>, <a class="txt" href="gobrd14.html#E9:1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>8</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="E12:1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</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> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</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="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <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><b> such that </b><br/><a NAME="E16:1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="jordan2c.html#R2">is_outside_component_of</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E17:1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan2c.html#K2">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan2c.html#T76">JORDAN2C:76</a></i>;<br/>
<a NAME="E18:1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="gobrd14.html#E2:1"><i><font color="Green">E3</font></i></a>, <a class="txt" href="gobrd14.html#E16:1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="gobrd14.html#T6" target="_self">Th6</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> 2</span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <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> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></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="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1 </span>}</span> ;<br/>
<a NAME="E19:1"/><i><font color="Green">E13</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>  <a href="euclid.html#K1">REAL</a> 2
 
<b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<a NAME="E20:1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="gobrd14.html#E11:1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E21:1"/><b>then </b>
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> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> or <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1 )
 
<b>by </b><i><a class="ref" href="topreal6.html#T31">TOPREAL6:31</a></i>;<br/>
<a NAME="E22:1"/><b>then </b>
not <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <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> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></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="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1 </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> 2</span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <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> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></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="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1 </span>}</span>  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> <b>by </b><i><a class="txt" href="gobrd14.html#E19:1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E24:1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="euclid.html#K1">REAL</a> 2
 
;<br/>
<a NAME="E25:1"/><i><font color="Green">E15</font></i>: 
not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="gobrd14.html#E12:1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E26:1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i><a class="txt" href="gobrd14.html#E1:1"><i><font color="Green">E2</font></i></a>, <a class="ref" href="topreal6.html#T33">TOPREAL6:33</a></i>;<br/>
<a NAME="E27:1"/><b>then </b>
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> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font> or <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1 )
 
<b>by </b><i><a class="txt" href="gobrd14.html#E25:1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<a NAME="E28:1"/><b>then </b>
not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <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> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></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="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1 </span>}</span> 
 
;<br/>
<a NAME="E29:1"/><b>then </b><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="gobrd14.html#E24:1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E30:1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>13</sub></font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T61">JORDAN2C:61</a></i>;<br/>
<a NAME="E31:1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</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="E1:1_2"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<a NAME="E2:1_2"/><b>then </b><i><font color="Green">E19</font></i>: 
not <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <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> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></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="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1 </span>}</span> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</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="gobrd14.html#E1:1_2"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</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="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<a NAME="E5:1_2"/><i><font color="Green">E20</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="txt" href="gobrd14.html#E2:1_2"><i><font color="Green">E19</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_2_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:1_2_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="txt" href="gobrd14.html#E8:1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/><b>hence </b><a NAME="E3:1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd14.html#E1:1"><i><font color="Green">E2</font></i></a>, <a class="txt" href="gobrd14.html#E5:1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="topreal6.html#T33">TOPREAL6:33</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:1_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="txt" href="gobrd14.html#E1:1_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="subset_1.html#T50">SUBSET_1:50</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being   <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><b> such that </b><br/><a NAME="E33:1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>and </b><br/><a NAME="E34:1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="gobrd14.html#E30:1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="goboard9.html#T5">GOBOARD9:5</a></i>;<br/>
<a NAME="E35:1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="jordan2c.html#R2">is_outside_component_of</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:1_3"/>
<font color="Maroon">c<sub>14</sub></font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="txt" href="gobrd14.html#E33:1"><i><font color="Green">E18</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="jordan2c.html#D4">JORDAN2C:def 4</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>13</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> ;<br/>
<a NAME="E3:1_3"/>
not <font color="Maroon">c<sub>15</sub></font> is <a href="tbsp_1.html#V6">bounded</a>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T70">JORDAN2C:70</a></i>;<br/>
<a NAME="E4:1_3"/><b>then </b>
not <font color="Maroon">c<sub>13</sub></font> is <a href="jordan2c.html#V1">Bounded</a>
 
<b>by </b><i><a class="ref" href="jordan2c.html#D2">JORDAN2C:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:1_3"/>
not <font color="Maroon">c<sub>14</sub></font> is <a href="jordan2c.html#V1">Bounded</a>
 <b>by </b><i><a class="txt" href="gobrd14.html#E34:1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="jordan2c.html#T16">JORDAN2C:16</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E36:1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="jordan2c.html#R2">is_outside_component_of</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E37:1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="gobrd14.html#E17:1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="gobrd14.html#E18:1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="gobrd14.html#E29:1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="gobrd14.html#E34:1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div>
