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

<b>set </b><font color="Maroon">X</font> =  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <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>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K1">Ball</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Olive">r</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Olive">r</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> where <font color="Olive">r</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <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>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K1">Ball</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Olive">r</font></span>)</span> <a href="subset_1.html#K9">/\</a> <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> </span>)</span> where <font color="Olive">r</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="topgen_5.html#K1" target="_self">y=0-line</a> ;<br/>
<b>consider </b><font color="Maroon">B</font> being    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> <b> such that </b><br/><a NAME="E2:35_1_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">a</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  holds <br/>( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>] implies <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">a</font>) ) &amp; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>] implies <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">a</font>) ) )
 <b>from </b><i><a class="ref" href="pre_circ.html#S2">PRE_CIRC:sch 2</a>();<br/></i>
<a NAME="E3:35_1_1"/>
<font color="Maroon">B</font> is <a href="relat_1.html#V2">non-empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D16">PBOOLE:def 16</a><br/>

<b>assume </b><a NAME="E1:35_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1_1_1_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">z</font>] or  not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">z</font>] )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:35_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">z</font>]
 ;<br/><div class="add"><a NAME="E2:35_1_1_1_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">s</font>)
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">r</font> being  <a href="xxreal_0.html#V2">positive</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>set </b><font color="Maroon">a</font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></span>;<br/><a NAME="E4:35_1_1_1_1_1"/>
<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"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">r</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:35_1_1_1_1_1"/>
not <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:35_1_1_1_1_2"/>
 not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">z</font>]
 ;<br/><div class="add"><a NAME="E2:35_1_1_1_1_2"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">s</font>)
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">r</font> being  <a href="xxreal_0.html#V2">positive</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><a NAME="E4:35_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="topreal9.html#K1">Ball</a> <font color="Maroon">s</font>,<font color="Maroon">r</font></span>)</span> <a href="subset_1.html#K9">/\</a> <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:35_1_1_1_1_2"/>
not <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">B</font> as  <a href="relat_1.html#V2">V5</a>  <a href="pboole.html#M1">ManySortedSet</a> of  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  ;<br/>
<a NAME="E5:35_1_1"/><i><font color="Green">E53</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">w</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="E1:35_1_1_2"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:35_1_1_2"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:35_1_1_2"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a</font>] or  not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a</font>] )
 
;<br/>
<a NAME="E6:35_1_1_2"/>
<font color="Maroon">w</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</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="E1:35_1_1_2_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">w</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1_1_2_1_1"/>
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a</font>) or <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">a</font>) )
 </span><b>by </b><i>, , </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:35_1_1_2_1_1_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a</font>)
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">r</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:35_1_1_2_1_1_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">z</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"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">r</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i/>;<br/><a NAME="E4:35_1_1_2_1_1_1"/>
(  <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a>  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  &amp; <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  )
 <b>by </b><i>, </i>;<br/><a NAME="E5:35_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">z</font> <a href="tarski.html#R1">c=</a>  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> 
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E6:35_1_1_2_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:35_1_1_2_1_1_2"/>
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">a</font>)
 ;<br/><div class="add"><a NAME="E2:35_1_1_2_1_1_2"/><b>then </b>
 ex <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1">Ball</a> <font color="Maroon">a</font>,<font color="Olive">r</font></span>)</span> <a href="subset_1.html#K9">/\</a> <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i/>;<br/><a NAME="E3:35_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">z</font> <a href="tarski.html#R1">c=</a>  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><b>hence </b><a NAME="E4:35_1_1_2_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E7:35_1_1_2"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> </span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:35_1_1"/><i><font color="Green">E63</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">U</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  &amp; <font color="Olive">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">U</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">U</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:35_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  ;<br/>
<a NAME="E3:35_1_1_3"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a</font>] or  not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a</font>] )
 
;<br/>
<b>assume </b><a NAME="E4:35_1_1_3"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1_1_3_1"/>
( <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a</font>) or <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">a</font>) )
 </span><b>by </b><i>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:35_1_1_3_1_1"/>
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a</font>)
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">r</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:35_1_1_3_1_1"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">U</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"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">r</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i/>;<br/><a NAME="E4:35_1_1_3_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:35_1_1_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:35_1_1_3_1_2"/>
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">a</font>)
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">r</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:35_1_1_3_1_2"/><i><font color="Green">E120</font></i>: 
( <font color="Maroon">U</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1">Ball</a> <font color="Maroon">a</font>,<font color="Maroon">r</font></span>)</span> <a href="subset_1.html#K9">/\</a> <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i/>;<br/><a NAME="E4:35_1_1_3_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">a</font>,<font color="Maroon">r</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:35_1_1_3_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E7:35_1_1"/><i><font color="Green">E121</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">U</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">U</font> &amp; <font color="Olive">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  holds <br/> ex <font color="Olive">V</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> &amp; <font color="Olive">V</font> <a href="tarski.html#R1">c=</a> <font color="Olive">U</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">U</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:35_1_1_4"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font>
 ;<br/>

<b>assume </b><a NAME="E2:35_1_1_4"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 ;<br/>

<b>assume </b><a NAME="E3:35_1_1_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1_1_4_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">y</font>] or  not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">y</font>] )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_4_1_1"><b>suppose </b></a><a NAME="E1:35_1_1_4_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">y</font>]
 ;<br/><div class="add"><a NAME="E2:35_1_1_4_1_1"/><b>then </b>
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">y</font>)
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">r</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E4:35_1_1_4_1_1"/><i><font color="Green">E125</font></i>: 
( <font color="Maroon">U</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"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">r</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i/>;<br/><a NAME="E5:35_1_1_4_1_1"/><i><font color="Green">E126</font></i>: 
( <font color="Maroon">x</font> <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"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">r</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i>, , <a class="ref" href="setwiseo.html#T6">SETWISEO:6</a></i>;<br/><b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">x</font> as   <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> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_4_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_1_1_4_1_1_1"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">x</font> <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"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">r</font>
 ;<br/><a NAME="E2:35_1_1_4_1_1_1"/>
(  <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a>  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  &amp;  <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">r</font> <a href="xboole_0.html#R1">misses</a>  <a href="topgen_5.html#K1" target="_self">y=0-line</a>  )
 <b>by </b><i>, , </i>;<br/><a NAME="E3:35_1_1_4_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  &amp;  not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>] )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E4:35_1_1_4_1_1_1"/><b>then </b><i><font color="Green">E128</font></i>: 
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">z</font>)
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">r2</font> = <font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K20">-</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>;<br/><b>take </b><font color="Maroon">V</font> = <span class="p1">(<span class="default"><a href="topreal9.html#K1">Ball</a> <font color="Maroon">z</font>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K20">-</a> <span class="p5"><a href="euclid.html#K23">|[</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> ;<br/><a NAME="E5:35_1_1_4_1_1_1"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K20">-</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="topreal9.html#T7">TOPREAL9:7</a></i>;<br/><b>then reconsider </b><font color="Maroon">r1</font> = <font color="Maroon">r</font>, <font color="Maroon">r2</font> = <font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K20">-</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> as  <a href="xxreal_0.html#V2">positive</a> <a href="real_1.html#NM1">Real</a> <b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><a NAME="E7:35_1_1_4_1_1_1"/>
<font color="Maroon">r2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>hence </b><a NAME="E8:35_1_1_4_1_1_1"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><a NAME="E9:35_1_1_4_1_1_1"/>
<font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r2</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"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <font color="Maroon">z</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i><a class="ref" href="toprns_1.html#T28">TOPRNS_1:28</a></i>;<br/><a NAME="E10:35_1_1_4_1_1_1"/><b>then </b>
( <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">z</font>,<font color="Maroon">r2</font> &amp;  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">z</font>,<font color="Maroon">r2</font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">r1</font> )
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E11:35_1_1_4_1_1_1"/><b>then </b>
( <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">r1</font> &amp;  <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font> )
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E12:35_1_1_4_1_1_1"/>
<font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:35_1_1_4_1_1"/>
 ex <font color="Olive">V</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Olive">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_4_1_2"><b>suppose </b></a><a NAME="E1:35_1_1_4_1_2"/>
 not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">y</font>]
 ;<br/><div class="add"><a NAME="E2:35_1_1_4_1_2"/><b>then </b>
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">y</font>)
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">r</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E4:35_1_1_4_1_2"/><i><font color="Green">E129</font></i>: 
( <font color="Maroon">U</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1">Ball</a> <font color="Maroon">y</font>,<font color="Maroon">r</font></span>)</span> <a href="subset_1.html#K9">/\</a> <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>set </b><font color="Maroon">r2</font> = <font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K20">-</a> <font color="Maroon">y</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>;<br/><a NAME="E6:35_1_1_4_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">y</font>,<font color="Maroon">r</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E7:35_1_1_4_1_2"/><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">z</font> <a href="euclid.html#K20">-</a> <font color="Maroon">y</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T7">TOPREAL9:7</a></i>;<br/><b>then reconsider </b><font color="Maroon">r1</font> = <font color="Maroon">r</font>, <font color="Maroon">r2</font> = <font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K20">-</a> <font color="Maroon">y</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> as  <a href="xxreal_0.html#V2">positive</a> <a href="real_1.html#NM1">Real</a> <b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><a NAME="E9:35_1_1_4_1_2"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_4_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1_1_4_1_2_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">z</font>] or  not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">z</font>] )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_4_1_2_1_1"><b>suppose </b></a><a NAME="E1:35_1_1_4_1_2_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">z</font>]
 ;<br/><div class="add"><a NAME="E2:35_1_1_4_1_2_1_1"/><b>then </b><i><font color="Green">E131</font></i>: 
( <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">z</font>) &amp; <font color="Maroon">z</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, , </i>;<br/><b>set </b><font color="Maroon">V</font> = <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"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">z</font></span><a href="domain_1.html#K6">}</a></span>;<br/><b>take </b>
<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"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">z</font></span><a href="domain_1.html#K6">}</a></span>
;<br/><b>thus </b><a NAME="E3:35_1_1_4_1_2_1_1"/>
<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"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">z</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><a NAME="E4:35_1_1_4_1_2_1_1"/>
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</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">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, , <a class="ref" href="euclid.html#T66">EUCLID:66</a></i>;<br/><b>then </b><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"><span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <font color="Maroon">z</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="topreal6.html#T31">TOPREAL6:31</a></i>
<br/>.= 
<font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> 2
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><a NAME="E6:35_1_1_4_1_2_1_1"/><b>then </b><i><font color="Green">E132</font></i>: 
<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"><span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <font color="Maroon">y</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"><font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> 2</span>)</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">z</font> <a href="euclid.html#K20">-</a> <font color="Maroon">y</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i><a class="ref" href="toprns_1.html#T35">TOPRNS_1:35</a></i>;<br/><a NAME="E7:35_1_1_4_1_2_1_1"/><b>then </b>
( <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"><span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <font color="Maroon">y</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> 2</span>)</span> &amp; <font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T46">XREAL_1:46</a></i>;<br/><a NAME="E8:35_1_1_4_1_2_1_1"/><b>then </b>
<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"><span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <font color="Maroon">y</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E9:35_1_1_4_1_2_1_1"/><b>then </b>
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">y</font>,<font color="Maroon">r</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K20">-</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> 2</span>)</span> )
 <b>by </b><i>, <a class="ref" href="topreal9.html#T7">TOPREAL9:7</a>, <a class="ref" href="toprns_1.html#T28">TOPRNS_1:28</a></i>;<br/><a NAME="E10:35_1_1_4_1_2_1_1"/><b>then </b>
(  <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  &amp;  <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">y</font>,<font color="Maroon">r1</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E11:35_1_1_4_1_2_1_1"/><b>then </b>
(  <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font> &amp; <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">z</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font> )
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><b>hence </b><a NAME="E12:35_1_1_4_1_2_1_1"/>
<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"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">z</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_4_1_2_1_2"><b>suppose </b></a><a NAME="E1:35_1_1_4_1_2_1_2"/>
 not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">z</font>]
 ;<br/><div class="add"><a NAME="E2:35_1_1_4_1_2_1_2"/><b>then </b><i><font color="Green">E133</font></i>: 
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">z</font>)
 <b>by </b><i/>;<br/><b>take </b><font color="Maroon">V</font> = <span class="p1">(<span class="default"><a href="topreal9.html#K1">Ball</a> <font color="Maroon">z</font>,<font color="Maroon">r2</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>thus </b><a NAME="E3:35_1_1_4_1_2_1_2"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><a NAME="E4:35_1_1_4_1_2_1_2"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K20">-</a> <font color="Maroon">z</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font>
 <b>by </b><i><a class="ref" href="toprns_1.html#T28">TOPRNS_1:28</a></i>;<br/><a NAME="E5:35_1_1_4_1_2_1_2"/><b>then </b>
 <a href="topreal9.html#K1">Ball</a> <font color="Maroon">z</font>,<font color="Maroon">r2</font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">y</font>,<font color="Maroon">r1</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:35_1_1_4_1_2_1_2"/>
<font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E8:35_1_1"/><i><font color="Green">E134</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">U1</font>, <font color="Olive">U2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  &amp; <font color="Olive">U1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> &amp; <font color="Olive">U2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> holds <br/> ex <font color="Olive">U</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> &amp; <font color="Olive">U</font> <a href="tarski.html#R1">c=</a> <font color="Olive">U1</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">U2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">U1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">U2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:35_1_1_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  ;<br/>
<b>assume </b><a NAME="E3:35_1_1_5"/><i><font color="Green">E137</font></i>: 
( <font color="Maroon">U1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">U2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1_1_5_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">z</font>] or  not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">z</font>] )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_5_1_1"><b>suppose </b></a><a NAME="E1:35_1_1_5_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">z</font>]
 ;<br/><div class="add"><a NAME="E2:35_1_1_5_1_1"/><b>then </b><i><font color="Green">E138</font></i>: 
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">z</font>)
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:35_1_1_5_1_1"/><i><font color="Green">E139</font></i>: 
( <font color="Maroon">U1</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"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">r1</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">r1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">z</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">r2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:35_1_1_5_1_1"/><i><font color="Green">E140</font></i>: 
( <font color="Maroon">U2</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"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">r2</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">r2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">z</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">r2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i>, </i>;<br/><a NAME="E7:35_1_1_5_1_1"/>
( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> or <font color="Maroon">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r2</font> )
 ;<br/><b>then consider </b><font color="Maroon">U</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:35_1_1_5_1_1"/><i><font color="Green">E141</font></i>: 
( ( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">U</font> <a href="hidden.html#R1">=</a> <font color="Maroon">U1</font> ) or ( <font color="Maroon">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">U</font> <a href="hidden.html#R1">=</a> <font color="Maroon">U2</font> ) )
 ;<br/><b>take </b>
<font color="Maroon">U</font>
;<br/><b>thus </b><a NAME="E10:35_1_1_5_1_1"/>
<font color="Maroon">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/><a NAME="E11:35_1_1_5_1_1"/>
( ( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">U</font> <a href="hidden.html#R1">=</a> <font color="Maroon">U1</font> &amp;  <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">r1</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">r1</font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">r2</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">r2</font> ) or ( <font color="Maroon">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">U</font> <a href="hidden.html#R1">=</a> <font color="Maroon">U2</font> &amp;  <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">r2</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">r2</font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">r1</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">r1</font> ) )
 <b>by </b><i>, , , </i>;<br/><a NAME="E12:35_1_1_5_1_1"/><b>then </b>
( <font color="Maroon">U</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U1</font> &amp; <font color="Maroon">U</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U2</font> )
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T9">XBOOLE_1:9</a></i>;<br/><b>hence </b><a NAME="E13:35_1_1_5_1_1"/>
<font color="Maroon">U</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">U2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_5_1_2"><b>suppose </b></a><a NAME="E1:35_1_1_5_1_2"/>
 not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">z</font>]
 ;<br/><div class="add"><a NAME="E2:35_1_1_5_1_2"/><b>then </b><i><font color="Green">E142</font></i>: 
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">z</font>)
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:35_1_1_5_1_2"/><i><font color="Green">E143</font></i>: 
( <font color="Maroon">U1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1">Ball</a> <font color="Maroon">z</font>,<font color="Maroon">r1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">r2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:35_1_1_5_1_2"/><i><font color="Green">E144</font></i>: 
( <font color="Maroon">U2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1">Ball</a> <font color="Maroon">z</font>,<font color="Maroon">r2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  &amp; <font color="Maroon">r2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i>, </i>;<br/><a NAME="E7:35_1_1_5_1_2"/>
( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> or <font color="Maroon">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r2</font> )
 ;<br/><b>then consider </b><font color="Maroon">U</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:35_1_1_5_1_2"/><i><font color="Green">E145</font></i>: 
( ( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">U</font> <a href="hidden.html#R1">=</a> <font color="Maroon">U1</font> ) or ( <font color="Maroon">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">U</font> <a href="hidden.html#R1">=</a> <font color="Maroon">U2</font> ) )
 ;<br/><b>take </b>
<font color="Maroon">U</font>
;<br/><b>thus </b><a NAME="E10:35_1_1_5_1_2"/>
<font color="Maroon">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/><a NAME="E11:35_1_1_5_1_2"/>
( ( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">U</font> <a href="hidden.html#R1">=</a> <font color="Maroon">U1</font> &amp;  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">z</font>,<font color="Maroon">r1</font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">z</font>,<font color="Maroon">r2</font> ) or ( <font color="Maroon">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">U</font> <a href="hidden.html#R1">=</a> <font color="Maroon">U2</font> &amp;  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">z</font>,<font color="Maroon">r2</font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">z</font>,<font color="Maroon">r1</font> ) )
 <b>by </b><i><a class="ref" href="jordan.html#T18">JORDAN:18</a>, </i>;<br/><a NAME="E12:35_1_1_5_1_2"/><b>then </b>
( <font color="Maroon">U</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U1</font> &amp; <font color="Maroon">U</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U2</font> )
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><b>hence </b><a NAME="E13:35_1_1_5_1_2"/>
<font color="Maroon">U</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">U2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">P</font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> <b> such that </b><br/><a NAME="E10:35_1_1"/><i><font color="Green">E146</font></i>: 
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">B</font> &amp; ( for <font color="Olive">T</font> being   <a href="pre_topc.html#L1">TopStruct</a>   st the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">T</font> <a href="hidden.html#R1">=</a>  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  &amp; the <a href="pre_topc.html#U1">topology</a> of <font color="Olive">T</font> <a href="hidden.html#R1">=</a>  <a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">P</font> holds <br/>( <font color="Olive">T</font> is   <a href="pre_topc.html#NM1">TopSpace</a> &amp; ( for <font color="Olive">T'</font> being non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>  st <font color="Olive">T'</font> <a href="hidden.html#R1">=</a> <font color="Olive">T</font> holds <br/><font color="Maroon">B</font> is    <a href="topgen_2.html#M1">Neighborhood_System</a> of <font color="Olive">T'</font> ) ) ) )
 <b>by </b><i>, , , , <a class="ref" href="topgen_3.html#T3">TOPGEN_3:3</a></i>;<br/>
<b>set </b><font color="Maroon">T</font> =  <a href="pre_topc.html#G1">TopStruct</a>(# <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> ,<span class="p1">(<span class="default"><a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">P</font></span>)</span> #);<br/>
<b>reconsider </b><font color="Maroon">T</font> =  <a href="pre_topc.html#G1">TopStruct</a>(# <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> ,<span class="p1">(<span class="default"><a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">P</font></span>)</span> #) as  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V1">strict</a> <a href="pre_topc.html#NM1">TopSpace</a> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">B</font> as    <a href="topgen_2.html#M1">Neighborhood_System</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">T</font>
;<br/>

<b>thus </b><a NAME="E13:35_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> 
 ;<br/>

<b>take </b>
<font color="Maroon">B</font>
;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="real_1.html#NM1">Real</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</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> = <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">x</font>,<font color="Maroon">a<sub>1</sub></font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,0</span><a href="euclid.html#K23">]|</a></span></span><a href="domain_1.html#K6">}</a></span>;<br/><b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(  <a href="real_1.html#NM1">Real</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</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> = <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"><span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">a<sub>1</sub></font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,0</span><a href="euclid.html#K23">]|</a></span></span><a href="domain_1.html#K6">}</a></span>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="real_1.html#NM1">Real</a>] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0;<br/><a NAME="E1:35_1_1_6"/><i><font color="Green">E151</font></i>: 
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> holds  <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">r</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">r</font>)
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E2:35_1_1_6"/><i><font color="Green">E152</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">r</font>) where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">r</font>] </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">r</font>) where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">r</font>] </span>}</span> 
 <b>from </b><i><a class="ref" href="fraenkel.html#S5">FRAENKEL:sch 5</a>();<br/></i><a NAME="E3:35_1_1_6"/>
( <font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,0</span><a href="euclid.html#K23">]|</a></span>] &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,0</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>  )
 ;<br/><b>hence </b><a NAME="E4:35_1_1_6"/>
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K1">Ball</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Olive">r</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Olive">r</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,0</span><a href="euclid.html#K23">]|</a></span></span><a href="domain_1.html#K6">}</a></span></span>)</span> where <font color="Olive">r</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> 
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">x</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">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E15:35_1_1"/>
<font color="Maroon">y</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E16:35_1_1"/><b>then </b>
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</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>  &amp; not <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topgen_5.html#K1" target="_self">y=0-line</a>  )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E17:35_1_1"/>
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K1">Ball</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Olive">r</font></span>)</span> <a href="subset_1.html#K9">/\</a> <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> </span>)</span> where <font color="Olive">r</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> 
 <b>by </b><i/>;<br/>


</div>
