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

<b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="xxreal_0.html#V3">negative</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>







<b>assume </b><a NAME="E1:20"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="abian.html#NR4" target="_self">has_no_fixpoint</a> 
 ;<br/>

<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="brouwer.html#K5" target="_self">BR-map</a> <font color="Maroon">c<sub>5</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E2:20"/><i><font color="Green">E15</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="brouwer.html#K5" target="_self">BR-map</a> <font color="Maroon">c<sub>5</sub></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="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:20"/><i><font color="Green">E16</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="brouwer.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E4:20"/>
 <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="topreal9.html#T17">TOPREAL9:17</a></i>;<br/>
<a NAME="E5:20"/><b>then </b><i><font color="Green">E17</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="brouwer.html#K5" target="_self">BR-map</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="brouwer.html#E2:20"><i><font color="Green">E15</font></i></a>, <a class="txt" href="brouwer.html#E3:20"><i><font color="Green">E16</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E6:20"/><i><font color="Green">E18</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <span class="p2">(<span class="default"><a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>;<br/>
<a NAME="E7:20"/><i><font color="Green">E19</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">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="toprealb.html#T9">TOPREALB:9</a></i>;<br/>
<a NAME="E8:20"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="brouwer.html#K5" target="_self">BR-map</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="brouwer.html#K5" target="_self">BR-map</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <span class="p2">(<span class="default"><a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:20_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="brouwer.html#K5" target="_self">BR-map</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="brouwer.html#E1:20_1"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E3:20_1"/><i><font color="Green">E21</font></i>: 
not <font color="Maroon">c<sub>9</sub></font> <a href="abian.html#R1">is_a_fixpoint_of</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="brouwer.html#E1:20"><i><font color="Green">E14</font></i></a>, <a class="ref" href="abian.html#D5">ABIAN:def 5</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="brouwer.html#K5" target="_self">BR-map</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> = 
<span class="p1">(<span class="default"><a href="brouwer.html#K5" target="_self">BR-map</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="brouwer.html#E5:20"><i><font color="Green">E17</font></i></a>, <a class="txt" href="brouwer.html#E1:20_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="brouwer.html#E5:20"><i><font color="Green">E17</font></i></a>, <a class="txt" href="brouwer.html#E1:20_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="brouwer.html#E3:20_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="brouwer.html#E7:20"><i><font color="Green">E19</font></i></a>, <a class="ref" href="brouwer.html#T11" target="_self">Th11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <span class="p2">(<span class="default"><a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="brouwer.html#E5:20"><i><font color="Green">E17</font></i></a>, <a class="txt" href="brouwer.html#E1:20_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="brouwer.html#E7:20"><i><font color="Green">E19</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:20"/>
<span class="p1">(<span class="default"><a href="brouwer.html#K5" target="_self">BR-map</a> <font color="Maroon">c<sub>5</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="grcat_1.html#K7">id</a> <span class="p1">(<span class="default"><a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="brouwer.html#E5:20"><i><font color="Green">E17</font></i></a>, <a class="txt" href="brouwer.html#E7:20"><i><font color="Green">E19</font></i></a>, <a class="txt" href="brouwer.html#E6:20"><i><font color="Green">E18</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div>
