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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <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>] means <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">a<sub>1</sub></font>) <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">a<sub>1</sub></font>);<br/>
<b>reconsider </b><font color="Maroon">c<sub>1</sub></font> = <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> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  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> <b>from </b><i><a class="ref" href="jgraph_2.html#S1" target="_self">JGRAPH_2:sch 1</a>();<br/></i>
<a NAME="E2:47_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</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> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 
;<br/>
<a NAME="E3:47_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</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> :  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="jgraph_2.html#S2" target="_self">JGRAPH_2:sch 2</a>(<a class="txt" href="jgraph_2.html#E2:47_1"><i><font color="Green">E47</font></i></a>);<br/></i>
<a NAME="E4:47_1"/>
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>1</sub></font> <a href="subset_1.html#K3">`</a>  holds <br/>ex <font color="Olive">b<sub>2</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</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="E1:47_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</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>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>)</span>)</span> <a href="xcmplx_0.html#K7">/</a> 4;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:47_1_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>5</sub></font>) <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>5</sub></font>) )
 <b>by </b><i><a class="txt" href="jgraph_2.html#E3:47_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jgraph_2.html#E1:47_1_1"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E5:47_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E4:47_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/>
<a NAME="E6:47_1_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>)</span>)</span> <a href="xcmplx_0.html#K7">/</a> 4 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<a NAME="E7:47_1_1"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>)</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E5:47_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<a NAME="E8:47_1_1"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>)</span>)</span> <a href="xcmplx_0.html#K7">/</a> 4</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</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="E1:47_1_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>)</span>)</span> <a href="xcmplx_0.html#K7">/</a> 4</span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</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/>
<a NAME="E3:47_1_1_1"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>)</span>)</span> <a href="xcmplx_0.html#K7">/</a> 4</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> :  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>)</span>)</span> <a href="xcmplx_0.html#K7">/</a> 4 </span>}</span> 
 
<b>by </b><i><a class="ref" href="metric_1.html#T18">METRIC_1:18</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM1">Element</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="E5:47_1_1_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>)</span>)</span> <a href="xcmplx_0.html#K7">/</a> 4 )
 <b>by </b><i><a class="txt" href="jgraph_2.html#E1:47_1_1_1"><i><font color="Green">E54</font></i></a></i>;<br/>
<a NAME="E6:47_1_1_1"/><i><font color="Green">E56</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</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>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E5:47_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="jgraph_1.html#T45">JGRAPH_1:45</a></i>;<br/>
<a NAME="E7:47_1_1_1"/><i><font color="Green">E57</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>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>) <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>) <a href="square_1.html#K3">^2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E2:47"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E8:47_1_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>) <a href="square_1.html#K3">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E9:47_1_1_1"/>
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>) <a href="square_1.html#K3">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E10:47_1_1_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>) <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>) <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>) <a href="square_1.html#K3">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E11:47_1_1_1"/><i><font color="Green">E60</font></i>: 
0 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>) <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>) <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>) <a href="square_1.html#K3">^2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E8:47_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E12:47_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T5">METRIC_1:5</a></i>;<br/>
<a NAME="E13:47_1_1_1"/><b>then </b><i><font color="Green">E61</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>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>)</span>)</span> <a href="xcmplx_0.html#K7">/</a> 4</span>)</span> <a href="square_1.html#K3">^2</a> 
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E5:47_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jgraph_2.html#E6:47_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="square_1.html#T77">SQUARE_1:77</a></i>;<br/>
<a NAME="E14:47_1_1_1"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>) <a href="square_1.html#K3">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>)</span>)</span> <a href="xcmplx_0.html#K7">/</a> 4</span>)</span> <a href="square_1.html#K3">^2</a> 
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E7:47_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jgraph_2.html#E10:47_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E15:47_1_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>) <a href="square_1.html#K3">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>)</span>)</span> <a href="xcmplx_0.html#K7">/</a> 4</span>)</span> <a href="square_1.html#K3">^2</a> 
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E7:47_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jgraph_2.html#E11:47_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jgraph_2.html#E13:47_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E16:47_1_1_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>7</sub></font>) &amp; <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>7</sub></font>) )
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E1:47"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E17:47_1_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
(  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>)</span>)</span> <a href="xcmplx_0.html#K7">/</a> 4</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>7</sub></font>) &amp; <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>7</sub></font>) <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>)</span>)</span> <a href="xcmplx_0.html#K7">/</a> 4 )
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E6:47_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jgraph_2.html#E14:47_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="jgraph_2.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E18:47_1_1_1"/>
(  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>)</span>)</span> <a href="xcmplx_0.html#K7">/</a> 4</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>7</sub></font>) &amp; <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>7</sub></font>) <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>)</span>)</span> <a href="xcmplx_0.html#K7">/</a> 4 )
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E6:47_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jgraph_2.html#E15:47_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_2.html#E16:47_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="jgraph_2.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E19:47_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>7</sub></font>)</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>7</sub></font>)</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>)</span>)</span> <a href="xcmplx_0.html#K7">/</a> 4</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>)</span>)</span> <a href="xcmplx_0.html#K7">/</a> 4</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E17:47_1_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E20:47_1_1_1"/><b>then </b>
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>7</sub></font>) <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>7</sub></font>) <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E7:47_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jgraph_2.html#E50"><i><font color="Green">Lemma44</font></i></a></i>;<br/>
<a NAME="E21:47_1_1_1"/><b>then </b>
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>7</sub></font>) <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>7</sub></font>)
 
<b>by </b><i><a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/>
<b>hence </b><a NAME="E22:47_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="txt" href="jgraph_2.html#E3:47_1"><i><font color="Green">E48</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:47_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a>  )
 <b>by </b><i><a class="txt" href="jgraph_2.html#E6:47_1_1"><i><font color="Green">E52</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:47_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E1"><i><font color="Green">Lemma1</font></i></a>, <a class="ref" href="topmetr.html#T22">TOPMETR:22</a></i>;<br/>
<b>hence </b><a NAME="E6:47_1"/>
<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> : <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) </span>}</span>  is  <a href="pre_topc.html#V4">closed</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="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>


</div>
