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

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:96"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:96"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">p</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">p</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</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/>
<b>let </b><font color="Maroon">x</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">a</font> be  non <a href="xxreal_0.html#V3">negative</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">y</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> , <font color="Maroon">r</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>set </b><font color="Maroon">f</font> =  <a href="topgen_5.html#K5" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">r</font>;<br/>
<b>assume </b><a NAME="E3:96"/><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><a href="topgen_5.html#K5" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E4:96"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:96"/><b>then </b>
<span class="p1">(<span class="default"><a href="topgen_5.html#K5" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/>
<a NAME="E6:96"/><b>then </b><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><a href="topgen_5.html#K5" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E7:96"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:96"/><i><font color="Green">E56</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"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</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"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p3"><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></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/>
<b>thus </b><a NAME="E9:96"/>
<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">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:96_1_1"/>
( <span class="p1">(<span class="default"><a href="topgen_5.html#K5" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 or <span class="p1">(<span class="default"><a href="topgen_5.html#K5" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1 )
 </span><b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_1"><b>suppose </b></a><a NAME="E1:96_1_1_1"/>
<span class="p1">(<span class="default"><a href="topgen_5.html#K5" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><div class="add"><a NAME="E2:96_1_1_1"/><b>then </b>
<font color="Maroon">p</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"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">r</font>
 <b>by </b><i>, , <a class="ref" href="topgen_5.html#T10" target="_self">Th10</a></i>;<br/><a NAME="E3:96_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topgen_5.html#K5" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</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"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">r</font>
 <b>by </b><i>, , <a class="ref" href="topgen_5.html#T10" target="_self">Th10</a></i>;<br/><b>hence </b><a NAME="E4:96_1_1_1"/>
<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">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T81">XREAL_1:81</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_2"><b>suppose </b></a><a NAME="E1:96_1_1_2"/><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><a href="topgen_5.html#K5" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:96_1_1_2_1"/>
<font color="Maroon">p</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"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">r</font>
 ;<br/><a NAME="E2:96_1_1_2_1"/><b>then </b>
( 1 <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">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xcmplx_0.html#K7">/</a> <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">p</font> <a href="euclid.html#K20">-</a> <span class="p3"><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></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> &amp; <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i>, , <a class="txt" href="topgen_5.html#E17:96"><i><font color="Green">E80</font></i></a>, <a class="ref" href="topgen_5.html#T10" target="_self">Th10</a>, <a class="ref" href="topreal9.html#T7">TOPREAL9:7</a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/><b>hence </b><a NAME="E3:96_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="topgen_5.html#E16:96"><i><font color="Green">E79</font></i></a>, <a class="ref" href="xreal_1.html#T76">XREAL_1:76</a></i>;<br/></div><b>end;</b></div><a NAME="E3:96_1_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">p</font> <a href="euclid.html#K20">-</a> <span class="p3"><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></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font> )
 <b>by </b><i>, <a class="ref" href="topreal9.html#T7">TOPREAL9:7</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><b>hence </b><a NAME="E4:96_1_1_2"/>
<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">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="topgen_5.html#E16:96"><i><font color="Green">E79</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>then reconsider </b><font color="Maroon">r1</font> = <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">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> 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>  <b>by </b><i><a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/>
<b>let </b><font color="Maroon">u</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="E11:96"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1">Ball</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5"><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="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</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="E12:96"/><b>then </b><i><font color="Green">E62</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><span class="p2"><a href="toprns_1.html#K5">|.</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>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> &amp; <font color="Maroon">u</font> <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="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">u</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a>  <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">z</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> <b>by </b><i><a class="txt" href="topgen_5.html#E17:96"><i><font color="Green">E80</font></i></a></i>;<br/>
<a NAME="E15:96"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">q</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">q</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q</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/>
<a NAME="E16:96"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, </i>;<br/>
<a NAME="E17:96"/><i><font color="Green">E80</font></i>: 
( not <font color="Maroon">q</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"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">r</font> implies <span class="p1">(<span class="default"><a href="topgen_5.html#K5" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i>, , <a class="ref" href="topgen_5.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E18:96"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p</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">r1</font>
 
<b>by </b><i><a class="txt" href="topgen_5.html#E17:96"><i><font color="Green">E80</font></i></a>, <a class="ref" href="topreal9.html#T7">TOPREAL9:7</a></i>;<br/>
<a NAME="E19:96"/><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"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <font color="Maroon">q</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><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">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <font color="Maroon">q</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r1</font> &amp; <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">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><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="euclid.html#K20">-</a> <font color="Maroon">q</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="toprns_1.html#T35">TOPRNS_1:35</a></i>;<br/>
<a NAME="E20:96"/><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"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><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">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <font color="Maroon">q</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r1</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E21:96"/><b>then </b><i><font color="Green">E92</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"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2"><a href="toprns_1.html#K5">|.</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>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <font color="Maroon">q</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r1</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<div><i><font color="Green">E120</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:96_2"/>
<font color="Maroon">q</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"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">r</font>
 ;<br/><a NAME="E2:96_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topgen_5.html#K5" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">q</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"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <font color="Maroon">q</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">r</font>
 <b>by </b><i>, , <a class="ref" href="topgen_5.html#T10" target="_self">Th10</a></i>;<br/><b>hence </b><a NAME="E3:96_2"/>
<span class="p1">(<span class="default"><a href="topgen_5.html#K5" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T83">XREAL_1:83</a></i>;<br/></div><b>end;</b></div>
<a NAME="E23:96"/>
<span class="p1">(<span class="default"><a href="topgen_5.html#K5" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/>
<a NAME="E24:96"/><b>then </b>
<span class="p1">(<span class="default"><a href="topgen_5.html#K5" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E25:96"/><b>then </b>
<span class="p1">(<span class="default"><a href="topgen_5.html#K5" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">a</font>,1</span><a href="rcomp_2.html#K2">.]</a></span>
 
<b>by </b><i>, , , <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/>
<b>hence </b><a NAME="E26:96"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K5" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">r</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">a</font>,1</span><a href="rcomp_2.html#K2">.]</a></span>
 <b>by </b><i><a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>


</div>
