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

<b>let </b><font color="Maroon">C</font> be   <a href="topreal2.html#NM1">Simple_closed_curve</a>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">U</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:108"/><i><font color="Green">E69</font></i>: 
not  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">C</font> is <a href="xboole_0.html#V1">empty</a>
 <b>and </b><br/><a NAME="E2:108"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">U</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E3:108"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">U</font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>and </b><br/><a NAME="E4:108"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="tops_1.html#K2">Fr</a> <font color="Maroon">P</font>
 ;<br/>

<a NAME="E5:108"/><i><font color="Green">E79</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<b>reconsider </b><font color="Maroon">T2C</font> = <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span> as  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of  <a href="euclid.html#K15">TOP-REAL</a> 2 ;<br/>
<a NAME="E7:108"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">T2C</font> is <a href="connsp_2.html#V1">locally_connected</a>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T89">JORDAN2C:89</a></i>;<br/>
<a NAME="E8:108"/><b>then </b>
<font color="Maroon">U</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="connsp_2.html#T21">CONNSP_2:21</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P</font> as  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i>, <a class="ref" href="tsep_1.html#T17">TSEP_1:17</a></i>;<br/>
<a NAME="E10:108"/><i><font color="Green">E81</font></i>: 
 <a href="tops_1.html#K2">Fr</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/>
<b>set </b><font color="Maroon">Z</font> = <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> : ( <font color="Olive">X</font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">U</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">V</font> =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> : ( <font color="Olive">X</font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">U</font> ) </span>}</span> ;<br/>
<a NAME="E11:108"/><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> : ( <font color="Olive">X</font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">U</font> ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">U</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1"><b>proof </b></a><div class="add">

<a NAME="E1:108_1"/><i><font color="Green">E83</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> : ( <font color="Olive">X</font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">U</font> ) </span>}</span>  <a href="tarski.html#R1">c=</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:108_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> : ( <font color="Olive">X</font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">U</font> ) </span>}</span> 
 ;<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="E3:108_1_1"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E4:108_1_1"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> : ( <font color="Olive">X</font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">U</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E5:108_1_1"/>
 ex <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; <font color="Olive">X</font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">U</font> )
 
<b>by </b><i><a class="ref" href="jordan.html#T14" target="_self">Th14</a></i>;<br/>
<b>hence </b><a NAME="E6:108_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</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>
 <b>by </b><i>, <a class="ref" href="jordan.html#T13" target="_self">Th13</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:108_1"/>
<font color="Maroon">U</font> <a href="tarski.html#R1">c=</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>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E3:108_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> : ( <font color="Olive">X</font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">U</font> ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">U</font> <a href="tarski.html#R1">c=</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>
 
<b>by </b><i><a class="ref" href="jordan.html#T12" target="_self">Th12</a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E4:108_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> : ( <font color="Olive">X</font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">U</font> ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">U</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">C</font> <a href="tarski.html#R1">c=</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>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">a</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="E5:108_1"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</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>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:108_1_2"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> or  not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_1"><b>suppose </b></a><a NAME="E1:108_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:108_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> : ( <font color="Olive">X</font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">U</font> ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">U</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">C</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="108_1_2_2"><b>suppose </b></a><a NAME="E1:108_1_2_2"/>
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <b>by </b><i>, <a class="ref" href="jordan.html#T13" target="_self">Th13</a>, <a class="ref" href="subset_1.html#T50">SUBSET_1:50</a></i>;<br/><a NAME="E3:108_1_2_2"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="connsp_1.html#T40">CONNSP_1:40</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:108_1_2_2_1"/>
(  <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">U</font> or  <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">U</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:108_1_2_2_1_1"/>
 <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">U</font>
 ;<br/><div class="add"><a NAME="E2:108_1_2_2_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> : ( <font color="Olive">X</font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">U</font> ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">U</font>
 <b>by </b><i><a class="ref" href="jordan.html#T14" target="_self">Th14</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E3:108_1_2_2_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> : ( <font color="Olive">X</font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">U</font> ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">U</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">C</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="108_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:108_1_2_2_1_2"/><i><font color="Green">E118</font></i>: 
 <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">U</font>
 ;<br/><div class="add"><a NAME="E2:108_1_2_2_1_2"/>
 <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">a</font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>by </b><i><a class="ref" href="connsp_1.html#T43">CONNSP_1:43</a></i>;<br/><a NAME="E3:108_1_2_2_1_2"/><b>then </b>
 <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> : ( <font color="Olive">X</font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">U</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jordan.html#E21"><i><font color="Green">Lemma93</font></i></a></i>;<br/><a NAME="E4:108_1_2_2_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> : ( <font color="Olive">X</font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">U</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="jordan.html#T14" target="_self">Th14</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E5:108_1_2_2_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> : ( <font color="Olive">X</font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">U</font> ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</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/><b>hence </b><a NAME="E6:108_1_2_2_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> : ( <font color="Olive">X</font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">U</font> ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">U</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">C</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><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E12:108"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">P</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E13:108"/>
 <a href="tops_1.html#K2">Fr</a> <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:108_2"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K2">Fr</a> <font color="Maroon">P</font>
 ;<br/>

<a NAME="E2:108_2"/><b>then </b><i><font color="Green">E121</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  )
 
<b>by </b><i><a class="ref" href="jordan.html#T11" target="_self">Th11</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>assume </b><a NAME="E3:108_2"/>
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>

<a NAME="E4:108_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> : ( <font color="Olive">X</font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">U</font> ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">U</font>
 
<b>by </b><i><a class="ref" href="jordan.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan.html#E16"><i><font color="Green">Lemma89</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:108_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> : ( <font color="Olive">X</font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">U</font> ) </span>}</span>  or <font color="Maroon">a</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/>
<b>then consider </b><font color="Maroon">O</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:108_2"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 <b>and </b><br/><a NAME="E8:108_2"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">O</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> : ( <font color="Olive">X</font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">U</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="jordan.html#T12" target="_self">Th12</a>, <a class="ref" href="jordan.html#T14" target="_self">Th14</a>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">X</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span><b> such that </b><br/><a NAME="E10:108_2"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">O</font>
 <b>and </b><br/><a NAME="E11:108_2"/><i><font color="Green">E178</font></i>: 
<font color="Maroon">X</font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>and </b><br/><a NAME="E12:108_2"/><i><font color="Green">E180</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">U</font>
 <b>by </b><i/>;<br/>
<a NAME="E13:108_2"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, , , </i>;<br/>
<b>hence </b><a NAME="E14:108_2"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan.html#T14" target="_self">Th14</a>, <a class="txt" href="jordan.html#E21"><i><font color="Green">Lemma93</font></i></a>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">p1</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">p2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">A</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E15:108"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">A</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>and </b><br/><a NAME="E16:108"/><i><font color="Green">E182</font></i>: 
 <a href="tops_1.html#K2">Fr</a> <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E17:108"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan.html#T10" target="_self">Th10</a>, <a class="ref" href="borsuk_4.html#T84">BORSUK_4:84</a></i>;<br/>
<a NAME="E18:108"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">U</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="connsp_1.html#T34">CONNSP_1:34</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:108_3"/>
( <font color="Maroon">P</font> is <a href="jordan2c.html#V1">Bounded</a> or  not <font color="Maroon">P</font> is <a href="jordan2c.html#V1">Bounded</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_3_1"><b>suppose </b></a><a NAME="E1:108_3_1"/>
<font color="Maroon">P</font> is <a href="jordan2c.html#V1">Bounded</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P</font> as  <a href="jordan2c.html#V1">Bounded</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/><b>consider </b><font color="Maroon">p</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:108_3_1"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i>, </i>;<br/><a NAME="E6:108_3_1"/><i><font color="Green">E186</font></i>: 
<font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font> is <a href="jordan2c.html#V1">Bounded</a>
 <b>by </b><i><a class="ref" href="topreal6.html#T76">TOPREAL6:76</a></i>;<br/><b>then reconsider </b><font color="Maroon">PC</font> = <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font> as  <a href="tbsp_1.html#V6">bounded</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="jordan2c.html#D2">JORDAN2C:def 2</a></i>;<br/><b>consider </b><font color="Maroon">r</font> being  <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> such that </b><br/><a NAME="E9:108_3_1"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">PC</font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 <b>by </b><i>, </i>;<br/><a NAME="E10:108_3_1"/>
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">PC</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E11:108_3_1"/><b>then </b><i><font color="Green">E189</font></i>: 
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>set </b><font color="Maroon">D</font> =  <a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>;<br/><b>set </b><font color="Maroon">S</font> =  <a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>;<br/><b>set </b><font color="Maroon">B</font> = <span class="p1">(<span class="default"><a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>;<br/><a NAME="E12:108_3_1"/><i><font color="Green">E190</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="toprealb.html#T9">TOPREALB:9</a></i>;<br/><a NAME="E13:108_3_1"/><i><font color="Green">E192</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="brouwer.html#T3">BROUWER:3</a></i>;<br/><a NAME="E14:108_3_1"/><i><font color="Green">E193</font></i>: 
 <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T17">TOPREAL9:17</a></i>;<br/><a NAME="E15:108_3_1"/><i><font color="Green">E194</font></i>: 
 <a href="topreal9.html#K1">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> <a href="subset_1.html#R1">misses</a>  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T19">TOPREAL9:19</a></i>;<br/><a NAME="E16:108_3_1"/><i><font color="Green">E195</font></i>: 
 <a href="topreal9.html#K1">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T16">TOPREAL9:16</a></i>;<br/><a NAME="E17:108_3_1"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="jordan.html#E21"><i><font color="Green">Lemma93</font></i></a>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E18:108_3_1"/><b>then </b>
<font color="Maroon">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan.html#E23"><i><font color="Green">Lemma100</font></i></a>, <a class="ref" href="jordan.html#T16" target="_self">Th16</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then consider </b><font color="Maroon">R</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span><b> such that </b><br/><a NAME="E20:108_3_1"/><i><font color="Green">E196</font></i>: 
<font color="Maroon">R</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E21:108_3_1"/><i><font color="Green">E198</font></i>: 
<font color="Maroon">R</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="jordan.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan.html#E4:108"><i><font color="Green">E78</font></i></a></i>;<br/><a NAME="E22:108_3_1"/>
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">PC</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E23:108_3_1"/><b>then </b><i><font color="Green">E199</font></i>: 
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E25:108_3_1"/><i><font color="Green">E252</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E26:108_3_1"/><i><font color="Green">E253</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span> holds <br/> ( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font> implies <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> ) &amp; ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  implies <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> ) )
 <b>by </b><i><a class="txt" href="jordan.html#E21"><i><font color="Green">Lemma93</font></i></a>, <a class="ref" href="jordan.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan.html#T14" target="_self">Th14</a>, <a class="ref" href="jordan.html#T11" target="_self">Th11</a>, , , <a class="ref" href="jordan.html#T17" target="_self">Th17</a>, <a class="ref" href="jordan.html#T18" target="_self">Th18</a>, , <a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, <a class="txt" href="jordan.html#E5:108"><i><font color="Green">E79</font></i></a></i>;<br/><b>set </b><font color="Maroon">g</font> =  <a href="jordan.html#K9" target="_self">DiskProj</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font>;<br/><b>set </b><font color="Maroon">h</font> =  <a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font>;<br/><a NAME="E27:108_3_1"/><i><font color="Green">E254</font></i>: 
 <a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> is    <a href="pre_topc.html#M1">SubSpace</a> of  <a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="jordan.html#E22"><i><font color="Green">Lemma99</font></i></a>, <a class="txt" href="jordan.html#E23"><i><font color="Green">Lemma100</font></i></a>, , <a class="ref" href="tsep_1.html#T4">TSEP_1:4</a></i>;<br/><b>reconsider </b><font color="Maroon">F</font> = <span class="p1">(<span class="default"><a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K9" target="_self">DiskProj</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span>,<span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span> <b>by </b><i><a class="txt" href="jordan.html#E22"><i><font color="Green">Lemma99</font></i></a>, <a class="txt" href="jordan.html#E23"><i><font color="Green">Lemma100</font></i></a>, , <a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/><a NAME="E29:108_3_1"/>
<font color="Maroon">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E30:108_3_1"/><b>then </b>
 <a href="jordan.html#K9" target="_self">DiskProj</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="jordan.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E31:108_3_1"/><b>then </b><i><font color="Green">E257</font></i>: 
<span class="p1">(<span class="default"><a href="jordan.html#K9" target="_self">DiskProj</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="jordan.html#T19" target="_self">Th19</a>, <a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/><a NAME="E32:108_3_1"/>
<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> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="toprns_1.html#T29">TOPRNS_1:29</a></i>;<br/><a NAME="E33:108_3_1"/><b>then </b><i><font color="Green">E258</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T7">TOPREAL9:7</a></i>;<br/><a NAME="E34:108_3_1"/><b>then </b>
 <a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="jordan.html#E3:108"><i><font color="Green">E74</font></i></a></i>;<br/><a NAME="E35:108_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K9" target="_self">DiskProj</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i>, <a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/><a NAME="E36:108_3_1"/><b>then </b><i><font color="Green">E259</font></i>: 
<font color="Maroon">F</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i>, <a class="ref" href="topmetr.html#T7">TOPMETR:7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_3_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:108_3_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> or  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:108_3_1_1_1_1"/><i><font color="Green">E260</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 ;<br/><div class="add"><a NAME="E2:108_3_1_1_1_1"/><i><font color="Green">E261</font></i>: 
<span class="p1">(<span class="default"><a href="topreal9.html#K1">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T18">TOPREAL9:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_3_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_3_1_1_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="E1:108_3_1_1_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> )
 </span><b>by </b><i><a class="txt" href="jordan.html#E23"><i><font color="Green">Lemma100</font></i></a>, <a class="ref" href="jordan.html#T23" target="_self">Th23</a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_3_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:108_3_1_1_1_1_1_1_1"/><i><font color="Green">E262</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 ;<br/><div class="add"><a NAME="E2:108_3_1_1_1_1_1_1_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan.html#T23" target="_self">Th23</a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><b>hence </b><a NAME="E3:108_3_1_1_1_1_1_1_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="jordan.html#E22"><i><font color="Green">Lemma99</font></i></a>, , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_3_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:108_3_1_1_1_1_1_1_2"/><i><font color="Green">E263</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 ;<br/><div class="add"><a NAME="E2:108_3_1_1_1_1_1_1_2"/><i><font color="Green">E264</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E3:108_3_1_1_1_1_1_1_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E4:108_3_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E265</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i>, <a class="ref" href="subset_1.html#T50">SUBSET_1:50</a></i>;<br/><a NAME="E5:108_3_1_1_1_1_1_1_2"/><i><font color="Green">E271</font></i>: 
<span class="p1">(<span class="default"><a href="jordan.html#K9" target="_self">DiskProj</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan.html#T21" target="_self">Th21</a>, <a class="ref" href="jordan.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E6:108_3_1_1_1_1_1_1_2"/>
 <a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font> <a href="abian.html#NR4" target="_self">has_no_fixpoint</a> 
 <b>by </b><i><a class="ref" href="jordan.html#T21" target="_self">Th21</a>, <a class="ref" href="jordan.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E7:108_3_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E272</font></i>: 
not <font color="Maroon">x</font> <a href="abian.html#R1">is_a_fixpoint_of</a>  <a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="abian.html#D5">ABIAN:def 5</a></i>;<br/><a NAME="E8:108_3_1_1_1_1_1_1_2"/><i><font color="Green">E273</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K9" target="_self">DiskProj</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="jordan.html#T23" target="_self">Th23</a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K9" target="_self">DiskProj</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan.html#E23"><i><font color="Green">Lemma100</font></i></a>, , , , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K9" target="_self">DiskProj</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="jordan.html#T20" target="_self">Th20</a>, , , <a class="txt" href="jordan.html#E23"><i><font color="Green">Lemma100</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/><b>hence </b><a NAME="E10:108_3_1_1_1_1_1_1_2"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="txt" href="jordan.html#E22"><i><font color="Green">Lemma99</font></i></a>, <a class="txt" href="jordan.html#E12:108"><i><font color="Green">E119</font></i></a>, , <a class="ref" href="abian.html#D3">ABIAN:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:108_3_1_1_1_1"/>
not <font color="Maroon">x</font> <a href="abian.html#R1">is_a_fixpoint_of</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="abian.html#D3">ABIAN:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:108_3_1_1_1_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:108_3_1_1_1_2"/>
not <font color="Maroon">x</font> <a href="abian.html#R1">is_a_fixpoint_of</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="abian.html#D3">ABIAN:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E38:108_3_1"/><b>then </b>
not <font color="Maroon">F</font> <a href="abian.html#R3">has_a_fixpoint</a> 
 <b>by </b><i><a class="ref" href="abian.html#D5">ABIAN:def 5</a></i>;<br/><b>hence </b><a NAME="E39:108_3_1"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan.html#T22" target="_self">Th22</a>, <a class="ref" href="brouwer.html#T14">BROUWER:14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_3_2"><b>suppose </b></a><a NAME="E1:108_3_2"/><i><font color="Green">E274</font></i>: 
not <font color="Maroon">P</font> is <a href="jordan2c.html#V1">Bounded</a>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">p</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:108_3_2"/><i><font color="Green">E275</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="txt" href="jordan.html#E11:108"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>consider </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:108_3_2"/><i><font color="Green">E276</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E6:108_3_2"/><i><font color="Green">E277</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : <font color="Olive">B</font> <a href="jordan2c.html#R1">is_inside_component_of</a> <font color="Maroon">C</font> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">P1</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E8:108_3_2"/><i><font color="Green">E278</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font>
 <b>and </b><br/><a NAME="E9:108_3_2"/><i><font color="Green">E279</font></i>: 
<font color="Maroon">P1</font> <a href="jordan2c.html#R1">is_inside_component_of</a> <font color="Maroon">C</font>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">U1</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span><b> such that </b><br/><a NAME="E11:108_3_2"/><i><font color="Green">E280</font></i>: 
( <font color="Maroon">U1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> &amp; <font color="Maroon">U1</font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span> )
 <b>and </b><br/><a NAME="E12:108_3_2"/>
<font color="Maroon">U1</font> is  <a href="tbsp_1.html#V6">bounded</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="jordan.html#E23"><i><font color="Green">Lemma100</font></i></a>, <a class="ref" href="jordan2c.html#T17">JORDAN2C:17</a></i>;<br/><a NAME="E13:108_3_2"/>
<font color="Maroon">U1</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, , <a class="ref" href="connsp_2.html#T21">CONNSP_2:21</a></i>;<br/><b>then reconsider </b><font color="Maroon">P1</font> = <font color="Maroon">P1</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="jordan2c.html#V1">Bounded</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i>, <a class="txt" href="jordan.html#E22"><i><font color="Green">Lemma99</font></i></a>, <a class="txt" href="jordan.html#E23"><i><font color="Green">Lemma100</font></i></a>, , <a class="ref" href="tsep_1.html#T17">TSEP_1:17</a>, <a class="ref" href="jordan2c.html#D3">JORDAN2C:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i>, <a class="txt" href="jordan.html#E22"><i><font color="Green">Lemma99</font></i></a></i>;<br/><a NAME="E16:108_3_2"/><i><font color="Green">E281</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font>
 <b>by </b><i>, <a class="txt" href="jordan.html#E22"><i><font color="Green">Lemma99</font></i></a></i>;<br/><a NAME="E17:108_3_2"/><i><font color="Green">E282</font></i>: 
<font color="Maroon">P1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font> is <a href="jordan2c.html#V1">Bounded</a>
 <b>by </b><i><a class="ref" href="topreal6.html#T76">TOPREAL6:76</a></i>;<br/><b>then reconsider </b><font color="Maroon">PC</font> = <font color="Maroon">P1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font> as  <a href="tbsp_1.html#V6">bounded</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="jordan2c.html#D2">JORDAN2C:def 2</a></i>;<br/><b>consider </b><font color="Maroon">rv</font> being  <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> such that </b><br/><a NAME="E20:108_3_2"/><i><font color="Green">E283</font></i>: 
<font color="Maroon">PC</font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">rv</font>
 <b>by </b><i><a class="ref" href="jordan.html#T16" target="_self">Th16</a>, </i>;<br/><a NAME="E21:108_3_2"/>
not <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">rv</font>
 <b>by </b><i>, <a class="ref" href="jordan2c.html#T16">JORDAN2C:16</a></i>;<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="E23:108_3_2"/><i><font color="Green">E284</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E24:108_3_2"/><i><font color="Green">E285</font></i>: 
not <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">rv</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">u</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="jordan.html#T18" target="_self">Th18</a></i>;<br/><b>set </b><font color="Maroon">r</font> = <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>;<br/><a NAME="E26:108_3_2"/>
<font color="Maroon">P</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">P1</font>
 <b>by </b><i>, , <a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, , <a class="ref" href="connsp_1.html#T37">CONNSP_1:37</a></i>;<br/><a NAME="E27:108_3_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">u</font>
 <b>by </b><i>, <a class="txt" href="jordan.html#E22"><i><font color="Green">Lemma99</font></i></a>, <a class="ref" href="jordan.html#T18" target="_self">Th18</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>then reconsider </b><font color="Maroon">r</font> = <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> as  non <a href="xboole_0.html#V1">empty</a> 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>  <b>by </b><i><a class="ref" href="toprns_1.html#T29">TOPRNS_1:29</a></i>;<br/><a NAME="E29:108_3_2"/><i><font color="Green">E286</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">rv</font>
 <b>by </b><i>, <a class="ref" href="topreal9.html#T7">TOPREAL9:7</a></i>;<br/><a NAME="E30:108_3_2"/><b>then </b>
 <a href="topreal9.html#K1">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">rv</font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 <b>by </b><i/>;<br/><a NAME="E31:108_3_2"/><b>then </b><i><font color="Green">E287</font></i>: 
<font color="Maroon">PC</font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="jordan.html#T17" target="_self">Th17</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E32:108_3_2"/><i><font color="Green">E288</font></i>: 
 <a href="tops_1.html#K2">Fr</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 <b>by </b><i/>;<br/><a NAME="E33:108_3_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T9">TOPREAL9:9</a></i>;<br/><a NAME="E34:108_3_2"/><b>then </b><i><font color="Green">E289</font></i>: 
<font color="Maroon">P</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="jordan.html#T18" target="_self">Th18</a>, , <a class="ref" href="tops_1.html#T61">TOPS_1:61</a></i>;<br/><a NAME="E35:108_3_2"/><i><font color="Green">E290</font></i>: 
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">PC</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E36:108_3_2"/><b>then </b><i><font color="Green">E291</font></i>: 
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="jordan.html#T20" target="_self">Th20</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>set </b><font color="Maroon">D</font> =  <a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>;<br/><b>set </b><font color="Maroon">S</font> =  <a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>;<br/><b>set </b><font color="Maroon">B</font> = <span class="p1">(<span class="default"><a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>;<br/><a NAME="E37:108_3_2"/><i><font color="Green">E292</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="toprealb.html#T9">TOPREALB:9</a></i>;<br/><a NAME="E38:108_3_2"/><i><font color="Green">E293</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="brouwer.html#T3">BROUWER:3</a></i>;<br/><a NAME="E39:108_3_2"/><i><font color="Green">E294</font></i>: 
 <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T17">TOPREAL9:17</a></i>;<br/><a NAME="E40:108_3_2"/><i><font color="Green">E295</font></i>: 
 <a href="topreal9.html#K1">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> <a href="subset_1.html#R1">misses</a>  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T19">TOPREAL9:19</a></i>;<br/><a NAME="E41:108_3_2"/><i><font color="Green">E296</font></i>: 
 <a href="topreal9.html#K1">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T16">TOPREAL9:16</a></i>;<br/><a NAME="E42:108_3_2"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="jordan.html#E21"><i><font color="Green">Lemma93</font></i></a>, <a class="ref" href="jordan.html#T22" target="_self">Th22</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E43:108_3_2"/><b>then </b>
<font color="Maroon">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then consider </b><font color="Maroon">R</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span><b> such that </b><br/><a NAME="E45:108_3_2"/><i><font color="Green">E297</font></i>: 
<font color="Maroon">R</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E46:108_3_2"/><i><font color="Green">E298</font></i>: 
<font color="Maroon">R</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="jordan.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan.html#E4:108"><i><font color="Green">E78</font></i></a></i>;<br/><a NAME="E47:108_3_2"/>
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="jordan.html#T13" target="_self">Th13</a>, <a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/><b>then consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E49:108_3_2"/><i><font color="Green">E299</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E50:108_3_2"/><i><font color="Green">E300</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span> holds <br/> ( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font> implies <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> ) &amp; ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  implies <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> ) )
 <b>by </b><i>, <a class="txt" href="jordan.html#E21"><i><font color="Green">Lemma93</font></i></a>, <a class="ref" href="jordan.html#T14" target="_self">Th14</a>, <a class="ref" href="jordan.html#T11" target="_self">Th11</a>, , <a class="ref" href="jordan.html#T22" target="_self">Th22</a>, , <a class="txt" href="jordan.html#E12:108"><i><font color="Green">E119</font></i></a>, , <a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, , , <a class="txt" href="jordan.html#E7:108"><i><font color="Green">E80</font></i></a></i>;<br/><b>set </b><font color="Maroon">g</font> =  <a href="jordan.html#K9" target="_self">DiskProj</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font>;<br/><b>set </b><font color="Maroon">h</font> =  <a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font>;<br/><a NAME="E51:108_3_2"/><i><font color="Green">E301</font></i>: 
 <a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> is    <a href="pre_topc.html#M1">SubSpace</a> of  <a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="jordan.html#T23" target="_self">Th23</a>, , , <a class="ref" href="tsep_1.html#T4">TSEP_1:4</a></i>;<br/><b>reconsider </b><font color="Maroon">F</font> = <span class="p1">(<span class="default"><a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K9" target="_self">DiskProj</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span>,<span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span> <b>by </b><i><a class="ref" href="jordan.html#T23" target="_self">Th23</a>, , , <a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/><a NAME="E53:108_3_2"/>
<font color="Maroon">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E54:108_3_2"/><b>then </b>
 <a href="jordan.html#K9" target="_self">DiskProj</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="jordan.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E55:108_3_2"/><b>then </b><i><font color="Green">E302</font></i>: 
<span class="p1">(<span class="default"><a href="jordan.html#K9" target="_self">DiskProj</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i>, <a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/><a NAME="E56:108_3_2"/>
<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> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="toprns_1.html#T29">TOPRNS_1:29</a></i>;<br/><a NAME="E57:108_3_2"/><b>then </b><i><font color="Green">E303</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T7">TOPREAL9:7</a></i>;<br/><a NAME="E58:108_3_2"/><b>then </b>
 <a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="jordan.html#E3:108"><i><font color="Green">E74</font></i></a></i>;<br/><a NAME="E59:108_3_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K9" target="_self">DiskProj</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i>, <a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/><a NAME="E60:108_3_2"/><b>then </b><i><font color="Green">E304</font></i>: 
<font color="Maroon">F</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i>, <a class="ref" href="topmetr.html#T7">TOPMETR:7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_3_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_3_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:108_3_2_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> or  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_3_2_1_1_1"><b>suppose </b></a><a NAME="E1:108_3_2_1_1_1"/><i><font color="Green">E305</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 ;<br/><div class="add"><a NAME="E2:108_3_2_1_1_1"/><i><font color="Green">E306</font></i>: 
<span class="p1">(<span class="default"><a href="topreal9.html#K1">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T18">TOPREAL9:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_3_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_3_2_1_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="E1:108_3_2_1_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> )
 </span><b>by </b><i>, , <a class="ref" href="jordan.html#T26" target="_self">Th26</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_3_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:108_3_2_1_1_1_1_1_1"/><i><font color="Green">E307</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 ;<br/><div class="add"><a NAME="E2:108_3_2_1_1_1_1_1_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><b>hence </b><a NAME="E3:108_3_2_1_1_1_1_1_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="jordan.html#T23" target="_self">Th23</a>, , <a class="ref" href="jordan.html#T27" target="_self">Th27</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_3_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:108_3_2_1_1_1_1_1_2"/><i><font color="Green">E308</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 ;<br/><div class="add"><a NAME="E2:108_3_2_1_1_1_1_1_2"/><i><font color="Green">E309</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E3:108_3_2_1_1_1_1_1_2"/><i><font color="Green">E310</font></i>: 
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/><b>set </b><font color="Maroon">SS</font> =  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>;<br/><a NAME="E4:108_3_2_1_1_1_1_1_2"/>
 <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_3_2_1_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:108_3_2_1_1_1_1_1_2_1"/><i><font color="Green">E311</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 ;<br/>

<b>assume </b><a NAME="E2:108_3_2_1_1_1_1_1_2_1"/>
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> 
 ;<br/>

<a NAME="E3:108_3_2_1_1_1_1_1_2_1"/><b>then </b><i><font color="Green">E312</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="jordan.html#T29" target="_self">Th29</a>, <a class="ref" href="subset_1.html#T50">SUBSET_1:50</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="jordan.html#T29" target="_self">Th29</a></i>;<br/>
<a NAME="E5:108_3_2_1_1_1_1_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PC</font>
 
<b>by </b><i><a class="ref" href="jordan.html#T21" target="_self">Th21</a>, </i>;<br/>
<a NAME="E6:108_3_2_1_1_1_1_1_2_1"/><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">a</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#R1">&lt;</a> <font color="Maroon">rv</font>
 
<b>by </b><i><a class="ref" href="jordan.html#T17" target="_self">Th17</a>, <a class="ref" href="topreal9.html#T7">TOPREAL9:7</a></i>;<br/>
<b>hence </b><a NAME="E7:108_3_2_1_1_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan.html#T19" target="_self">Th19</a>, <a class="ref" href="jordan.html#T29" target="_self">Th29</a>, <a class="ref" href="topreal9.html#T9">TOPREAL9:9</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">SS</font> =  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E6:108_3_2_1_1_1_1_1_2"/><i><font color="Green">E313</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SS</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T9">TOPREAL9:9</a></i>;<br/><a NAME="E7:108_3_2_1_1_1_1_1_2"/>
<font color="Maroon">SS</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="ref" href="connsp_1.html#T24">CONNSP_1:24</a></i>;<br/><a NAME="E8:108_3_2_1_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">SS</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">U</font> or <font color="Maroon">SS</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font> )
 <b>by </b><i><a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="connsp_1.html#T38">CONNSP_1:38</a></i>;<br/><a NAME="E9:108_3_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E314</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jordan.html#T27" target="_self">Th27</a>, <a class="ref" href="jordan.html#T29" target="_self">Th29</a>, , <a class="ref" href="jordan.html#T18" target="_self">Th18</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E10:108_3_2_1_1_1_1_1_2"/><i><font color="Green">E316</font></i>: 
<span class="p1">(<span class="default"><a href="jordan.html#K9" target="_self">DiskProj</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan.html#T24" target="_self">Th24</a>, <a class="ref" href="jordan.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E11:108_3_2_1_1_1_1_1_2"/>
 <a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font> <a href="abian.html#NR4" target="_self">has_no_fixpoint</a> 
 <b>by </b><i><a class="ref" href="jordan.html#T24" target="_self">Th24</a>, <a class="ref" href="jordan.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E12:108_3_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E317</font></i>: 
not <font color="Maroon">x</font> <a href="abian.html#R1">is_a_fixpoint_of</a>  <a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="abian.html#D5">ABIAN:def 5</a></i>;<br/><a NAME="E13:108_3_2_1_1_1_1_1_2"/><i><font color="Green">E318</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K9" target="_self">DiskProj</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K9" target="_self">DiskProj</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="jordan.html#T27" target="_self">Th27</a>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K9" target="_self">DiskProj</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="jordan.html#T27" target="_self">Th27</a>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="jordan.html#T27" target="_self">Th27</a>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="jordan.html#T27" target="_self">Th27</a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/><b>hence </b><a NAME="E15:108_3_2_1_1_1_1_1_2"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="jordan.html#T27" target="_self">Th27</a>, <a class="ref" href="jordan.html#T23" target="_self">Th23</a>, , , <a class="ref" href="abian.html#D3">ABIAN:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:108_3_2_1_1_1"/>
not <font color="Maroon">x</font> <a href="abian.html#R1">is_a_fixpoint_of</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="abian.html#D3">ABIAN:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_3_2_1_1_2"><b>suppose </b></a><a NAME="E1:108_3_2_1_1_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:108_3_2_1_1_2"/>
not <font color="Maroon">x</font> <a href="abian.html#R1">is_a_fixpoint_of</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="abian.html#D3">ABIAN:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E62:108_3_2"/><b>then </b>
not <font color="Maroon">F</font> <a href="abian.html#R3">has_a_fixpoint</a> 
 <b>by </b><i><a class="ref" href="abian.html#D5">ABIAN:def 5</a></i>;<br/><b>hence </b><a NAME="E63:108_3_2"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan.html#T25" target="_self">Th25</a>, <a class="ref" href="brouwer.html#T14">BROUWER:14</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
