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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="xxreal_0.html#V2">positive</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>







<b>assume </b><a NAME="E1:99"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1">Ball</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  <a href="xxreal_0.html#V2">positive</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/>
<b>take </b><font color="Maroon">c<sub>6</sub></font> =  <a href="topgen_5.html#K5" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>

<a NAME="E3:99"/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<b>hence </b><a NAME="E4:99"/>
<font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="topgen_5.html#T81" target="_self">Th81</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>thus </b><a NAME="E5:99"/>
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a>  implies <font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:99_1"/><i><font color="Green">E77</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> 
 ;<br/>

<a NAME="E2:99_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> 
 
<b>by </b><i><a class="txt" href="topgen_5.html#E33"><i><font color="Green">Lemma27</font></i></a></i>;<br/>
<a NAME="E3:99_1"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="topgen_5.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E4:99_1"/>
not <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="topgen_5.html#E1:99_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E5:99_1"/><b>then </b>
not <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="topgen_5.html#E1:99"><i><font color="Green">E76</font></i></a>, <a class="txt" href="topgen_5.html#E1:99_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a>, <a class="txt" href="topgen_5.html#E33"><i><font color="Green">Lemma27</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:99_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="topgen_5.html#E3:99_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="topgen_5.html#D6" target="_self">Def6</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E6:99"/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E7:99"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="topgen_5.html#E1:99"><i><font color="Green">E76</font></i></a>, <a class="ref" href="topgen_5.html#T22" target="_self">Th22</a>, <a class="txt" href="topgen_5.html#E33"><i><font color="Green">Lemma27</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E8:99"/>
<font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="euclid.html#K23">]|</a></span> <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"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="topgen_5.html#D6" target="_self">Def6</a></i>;<br/>


</div>
