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

<b>let </b><font color="Maroon" title="c7">n</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c8">o</font>, <font color="Maroon" title="c9">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c7">n</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c10">r</font> be  <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a> <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>





<b>assume </b><a NAME="E1:98"/><i><font color="Green" title="E73">A1</font></i>: 
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font>
 ;<br/>

<a NAME="E2:98"/><i><font color="Green" title="E74">A2</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font>
 
<b>by </b><i><a class="ref" href="brouwer.html#T3" title="BROUWER:th.3">BROUWER:3</a></i>;<br/>
<a NAME="E3:98"/><i><font color="Green" title="E75">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E4:98"/><i><font color="Green" title="E76">A4</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="jordan.html#K9" title="JORDAN:func.9">DiskProj</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c9">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:98"/><i><font color="Green" title="E77">A5</font></i>: 
 <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font>
 
<b>by </b><i><a class="ref" href="topreal9.html#T19" title="TOPREAL9:th.19">TOPREAL9:19</a></i>;<br/>
<a NAME="E6:98"/><i><font color="Green" title="E78">A6</font></i>: 
 <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font>
 
<b>by </b><i><a class="ref" href="topreal9.html#T17" title="TOPREAL9:th.17">TOPREAL9:17</a></i>;<br/>
<a NAME="E7:98"/><i><font color="Green" title="E79">A7</font></i>: 
 <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font>
 
<b>by </b><i><a class="ref" href="topreal9.html#T16" title="TOPREAL9:th.16">TOPREAL9:16</a></i>;<br/>
<a NAME="E8:98"/>
 <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:98_1"/><i><font color="Green" title="E80">A8</font></i>: 
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font>
 ;<br/>

<a NAME="E2:98_1"/><b>then </b>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">p</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E1:98"><i><font color="Green" title="E73">A1</font></i></a>, <a class="txt" href="jordan.html#E5:98"><i><font color="Green" title="E77">A5</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E3:98_1"/>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="jordan.html#E6:98"><i><font color="Green" title="E78">A6</font></i></a>, <a class="txt" href="jordan.html#E1:98_1"><i><font color="Green" title="E80">A8</font></i></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:98"/><b>then </b><i><font color="Green" title="E80">A9</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K9" title="JORDAN:func.9">DiskProj</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c9">p</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E3:98"><i><font color="Green" title="E75">A3</font></i></a>, <a class="txt" href="jordan.html#E4:98"><i><font color="Green" title="E76">A4</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<a NAME="E10:98"/><i><font color="Green" title="E81">A10</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:98_2"/><i><font color="Green" title="E82">A11</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K9" title="JORDAN:func.9">DiskProj</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c9">p</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font></span>)</span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c12">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c7">n</font></span>)</span><b> such that </b><br/><a NAME="E3:98_2"/><i><font color="Green" title="E83">A12</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y</font>
 <b>and </b><br/><a NAME="E4:98_2"/><i><font color="Green" title="E84">A13</font></i>: 
<span class="p1">(<span class="default"><a href="jordan.html#K9" title="JORDAN:func.9">DiskProj</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c9">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="brouwer.html#K3" title="BROUWER:func.3">HC</a> <font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font>
 <b>by </b><i><a class="txt" href="jordan.html#E1:98"><i><font color="Green" title="E73">A1</font></i></a>, <a class="txt" href="jordan.html#E2:98"><i><font color="Green" title="E74">A2</font></i></a>, <a class="txt" href="jordan.html#E7:98"><i><font color="Green" title="E79">A7</font></i></a>, <a class="ref" href="jordan.html#D7" target="_self" title="JORDAN:def.7">Def7</a></i>;<br/><a NAME="E5:98_2"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> <font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c12">y</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T28" title="TOPREAL9:th.28">TOPREAL9:28</a></i>;<br/><a NAME="E6:98_2"/><b>then </b><i><font color="Green" title="E85">A14</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> <font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c12">y</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan.html#E9:98"><i><font color="Green" title="E80">A9</font></i></a>, <a class="txt" href="jordan.html#E1:98_2"><i><font color="Green" title="E82">A11</font></i></a>, <a class="txt" href="jordan.html#E3:98_2"><i><font color="Green" title="E83">A12</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E7:98_2"/><i><font color="Green" title="E86">A15</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">p</font>
 <b>by </b><i><a class="txt" href="jordan.html#E1:98"><i><font color="Green" title="E73">A1</font></i></a>, <a class="txt" href="jordan.html#E5:98"><i><font color="Green" title="E77">A5</font></i></a>, <a class="txt" href="jordan.html#E9:98"><i><font color="Green" title="E80">A9</font></i></a>, <a class="txt" href="jordan.html#E1:98_2"><i><font color="Green" title="E82">A11</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K9" title="JORDAN:func.9">DiskProj</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c9">p</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> = 
<span class="p1">(<span class="default"><a href="jordan.html#K9" title="JORDAN:func.9">DiskProj</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c9">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font>
<b>by </b><i><a class="txt" href="jordan.html#E1:98_2"><i><font color="Green" title="E82">A11</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon" title="c11">x</font>
<b>by </b><i><a class="txt" href="jordan.html#E1:98"><i><font color="Green" title="E73">A1</font></i></a>, <a class="txt" href="jordan.html#E2:98"><i><font color="Green" title="E74">A2</font></i></a>, <a class="txt" href="jordan.html#E6:98"><i><font color="Green" title="E78">A6</font></i></a>, <a class="txt" href="jordan.html#E7:98"><i><font color="Green" title="E79">A7</font></i></a>, <a class="txt" href="jordan.html#E9:98"><i><font color="Green" title="E80">A9</font></i></a>, <a class="txt" href="jordan.html#E1:98_2"><i><font color="Green" title="E82">A11</font></i></a>, <a class="txt" href="jordan.html#E3:98_2"><i><font color="Green" title="E83">A12</font></i></a>, <a class="txt" href="jordan.html#E4:98_2"><i><font color="Green" title="E84">A13</font></i></a>, <a class="txt" href="jordan.html#E6:98_2"><i><font color="Green" title="E85">A14</font></i></a>, <a class="txt" href="jordan.html#E7:98_2"><i><font color="Green" title="E86">A15</font></i></a>, <a class="ref" href="brouwer.html#D3" title="BROUWER:def.3">BROUWER:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font>
<b>by </b><i><a class="txt" href="jordan.html#E9:98"><i><font color="Green" title="E80">A9</font></i></a>, <a class="txt" href="jordan.html#E1:98_2"><i><font color="Green" title="E82">A11</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:98"/>
<span class="p1">(<span class="default"><a href="jordan.html#K9" title="JORDAN:func.9">DiskProj</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c9">p</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c10">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan.html#E9:98"><i><font color="Green" title="E80">A9</font></i></a>, <a class="txt" href="jordan.html#E10:98"><i><font color="Green" title="E81">A10</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


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