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

<b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topalg_2.html#K3">R^1</a> ;<br/><b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>;<br/>





<b>assume </b><a NAME="E1:41"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E2:41"/><b>then </b><i><font color="Green">E39</font></i>: 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> =  <a href="compts_1.html#K1">1TopSp</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>;<br/>
<a NAME="E3:41"/><i><font color="Green">E40</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></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="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ,   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> = <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span><b> such that </b><br/><a NAME="E5:41"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  holds <font color="Maroon">c<sub>13</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4">BINOP_1:sch 4</a>();<br/></i>
<a NAME="E6:41"/><i><font color="Green">E42</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = 1 as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span> <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E9:41"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> st <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Olive">b<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>;<br/><b>let </b><font color="Maroon">c<sub>17</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:41_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font>
 <b>and </b><br/><a NAME="E2:41_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>17</sub></font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>, <font color="Maroon">c<sub>19</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E4:41_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T50">BORSUK_1:50</a></i>;<br/>
<a NAME="E5:41_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E5:41"><i><font color="Green">E41</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E7:41_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font>
 <b>and </b><br/><a NAME="E8:41_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>20</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E9:41_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>20</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E1:41_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="topalg_5.html#E2:41_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="topalg_5.html#E4:41_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="jgraph_2.html#T20">JGRAPH_2:20</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>21</sub></font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>20</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<a NAME="E10:41_1"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>20</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>22</sub></font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>20</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>22</sub></font>
;<br/>

<b>thus </b><a NAME="E12:41_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E4:41_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="topalg_5.html#E7:41_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>

<b>thus </b><a NAME="E13:41_1"/>
<font color="Maroon">c<sub>22</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E3:41"><i><font color="Green">E40</font></i></a>, <a class="txt" href="topalg_5.html#E8:41_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="borsuk_1.html#T46">BORSUK_1:46</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>23</sub></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="E14:41_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>24</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E16:41_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>
 <b>and </b><br/><a NAME="E17:41_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font>
 <b>and </b><br/><a NAME="E18:41_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>25</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>, <font color="Maroon">c<sub>26</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E20:41_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E16:41_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="borsuk_1.html#T50">BORSUK_1:50</a></i>;<br/>
<i><font color="Green">E53</font></i>: <font color="Maroon">c<sub>23</sub></font> = 
<font color="Maroon">c<sub>14</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E20:41_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="topalg_5.html#E18:41_1"><i><font color="Green">E51</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E5:41"><i><font color="Green">E41</font></i></a></i>
;<br/>
<a NAME="E22:41_1"/>
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E17:41_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="topalg_5.html#E20:41_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E23:41_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<b>hence </b><a NAME="E24:41_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E9:41_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="topalg_5.html#E21:41_1"><i><font color="Green">E53</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:41"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>14</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="jgraph_2.html#T20">JGRAPH_2:20</a></i>;<br/>
<a NAME="E11:41"/><i><font color="Green">E44</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E12:41"/><i><font color="Green">E45</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E11:41"><i><font color="Green">E44</font></i></a>, <a class="txt" href="topalg_5.html#E6:41"><i><font color="Green">E42</font></i></a>, <a class="txt" href="topalg_5.html#E3"><i><font color="Green">Lemma3</font></i></a>, <a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E13:41"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E14:41"/><i><font color="Green">E47</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>16</sub></font> = <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>10</sub></font>;<br/>
<a NAME="E15:41"/><i><font color="Green">E48</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p4">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p4">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E39"><i><font color="Green">Lemma33</font></i></a>, <a class="txt" href="topalg_5.html#E14:41"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E16:41"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2">Element</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> holds <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p3">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>18</sub></font> be    <a href="subset_1.html#M2">Element</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>;<br/>



<a NAME="E1:41_2"/><i><font color="Green">E50</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#R2">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="borsuk_1.html#K1">--&gt;</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D2">BORSUK_1:def 2</a></i>;<br/>
<a NAME="E2:41_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E39"><i><font color="Green">Lemma33</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:41_2"/>
<span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p3">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E1:41_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="topalg_5.html#E14:41"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:41"/>
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"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p4">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p4">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></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="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p4">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p4">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>10</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="41_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:41_3"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p4">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p4">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:41_3"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>18</sub></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="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>and </b><br/><a NAME="E4:41_3"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E5:41_3"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E1:41_3"><i><font color="Green">E50</font></i></a>, <a class="txt" href="topalg_5.html#E39"><i><font color="Green">Lemma33</font></i></a>, <a class="txt" href="topalg_5.html#E14:41"><i><font color="Green">E47</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E6:41_3"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="topalg_5.html#E4:41_3"><i><font color="Green">E52</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> = 
<font color="Maroon">c<sub>14</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E5:41_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="topalg_5.html#E1:41_3"><i><font color="Green">E50</font></i></a>, <a class="txt" href="topalg_5.html#E39"><i><font color="Green">Lemma33</font></i></a>, <a class="txt" href="topalg_5.html#E14:41"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E5:41"><i><font color="Green">E41</font></i></a>, <a class="txt" href="topalg_5.html#E3:41_3"><i><font color="Green">E51</font></i></a>, <a class="txt" href="topalg_5.html#E6:41_3"><i><font color="Green">E54</font></i></a>, <a class="txt" href="topalg_5.html#E2"><i><font color="Green">Lemma2</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E6:41_3"><i><font color="Green">E54</font></i></a>, <a class="ref" href="borsuk_2.html#D4">BORSUK_2:def 4</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E2:41"><i><font color="Green">E39</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p4">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p4">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E16:41"><i><font color="Green">E49</font></i></a>, <a class="txt" href="topalg_5.html#E3:41_3"><i><font color="Green">E51</font></i></a>, <a class="txt" href="topalg_5.html#E5:41_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="topalg_5.html#E4:41_3"><i><font color="Green">E52</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p4">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p4">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E1:41_3"><i><font color="Green">E50</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E18:41"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p3">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E13:41"><i><font color="Green">E46</font></i></a>, <a class="txt" href="topalg_5.html#E15:41"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a> <b> such that </b><br/><a NAME="E20:41"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>17</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E21:41"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#R2">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>17</sub></font>
 <b>and </b><br/><a NAME="E22:41"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E23:41"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>  st <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#R2">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>10</sub></font> holds <br/><font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#R2">=</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E10:41"><i><font color="Green">E43</font></i></a>, <a class="ref" href="topalg_5.html#T22" target="_self">Th22</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>18</sub></font> =  <a href="topalg_5.html#K4" target="_self">Prj2</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>17</sub></font>;<br/>

<a NAME="E24:41"/><i><font color="Green">E54</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E4"><i><font color="Green">Lemma4</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>18</sub></font> <a href="seq_1.html#K2">.</a> 0 = 
<font color="Maroon">c<sub>17</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="topalg_5.html#D3" target="_self">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p3">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="borsuk_1.html#K7">]</a></span>
<b>by </b><i><a class="txt" href="topalg_5.html#E22:41"><i><font color="Green">E52</font></i></a>, <a class="txt" href="topalg_5.html#E24:41"><i><font color="Green">E54</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="topalg_5.html#E16:41"><i><font color="Green">E49</font></i></a>, <a class="txt" href="topalg_5.html#E4"><i><font color="Green">Lemma4</font></i></a></i>
;<br/>

<a NAME="E26:41"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  holds <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<a NAME="E1:41_5"/><i><font color="Green">E55</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> = 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="topalg_5.html#D3" target="_self">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E1:41_5"><i><font color="Green">E55</font></i></a>, <a class="ref" href="binop_1.html#T30">BINOP_1:30</a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E21:41"><i><font color="Green">E51</font></i></a>, <a class="txt" href="topalg_5.html#E5:41"><i><font color="Green">E41</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E27:41"/>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#R2">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>

<a NAME="E28:41"/>
for <font color="Olive">b<sub>1</sub></font> being <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>  holds  <a href="topalg_5.html#K4" target="_self">Prj2</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
;<br/>
<b>hence </b><a NAME="E29:41"/>
<font color="Maroon">c<sub>18</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E20:41"><i><font color="Green">E50</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>19</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<a href="topalg_2.html#K3">R^1</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E30:41"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>19</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E31:41"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#R2">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>19</sub></font>
 <b>and </b><br/><a NAME="E32:41"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ,   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topalg_2.html#K3">R^1</a>  = <font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <a href="topalg_2.html#K3">R^1</a> <b> such that </b><br/><a NAME="E34:41"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  holds <font color="Maroon">c<sub>20</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4">BINOP_1:sch 4</a>();<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>20</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E6:41"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E36:41"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> holds <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>22</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>;<br/>

<b>consider </b><font color="Maroon">c<sub>23</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>, <font color="Maroon">c<sub>24</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E2:41_6"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T50">BORSUK_1:50</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> = 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a>, <a class="txt" href="topalg_5.html#E2:41_6"><i><font color="Green">E59</font></i></a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E34:41"><i><font color="Green">E58</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">c<sub>14</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E31:41"><i><font color="Green">E56</font></i></a>, <a class="txt" href="topalg_5.html#E5:41"><i><font color="Green">E41</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E2:41_6"><i><font color="Green">E59</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E37:41"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#R2">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>
<a NAME="E38:41"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>  st <font color="Maroon">c<sub>21</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Olive">b<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>22</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>;<br/><b>let </b><font color="Maroon">c<sub>23</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:41_7"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font>
 <b>and </b><br/><a NAME="E2:41_7"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>23</sub></font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>24</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>, <font color="Maroon">c<sub>25</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E4:41_7"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T50">BORSUK_1:50</a></i>;<br/>
<font color="Maroon">c<sub>21</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> = 
<font color="Maroon">c<sub>21</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E4:41_7"><i><font color="Green">E62</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E34:41"><i><font color="Green">E58</font></i></a></i>
;<br/>
<b>then consider </b><font color="Maroon">c<sub>26</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E7:41_7"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>26</sub></font>
 <b>and </b><br/><a NAME="E8:41_7"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>26</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E9:41_7"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>26</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E1:41_7"><i><font color="Green">E60</font></i></a>, <a class="txt" href="topalg_5.html#E2:41_7"><i><font color="Green">E61</font></i></a>, <a class="txt" href="topalg_5.html#E30:41"><i><font color="Green">E55</font></i></a>, <a class="ref" href="jgraph_2.html#T20">JGRAPH_2:20</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>27</sub></font> = <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>;<br/>

<b>thus </b><a NAME="E10:41_7"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E4:41_7"><i><font color="Green">E62</font></i></a>, <a class="txt" href="topalg_5.html#E7:41_7"><i><font color="Green">E63</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>

<b>thus </b><a NAME="E11:41_7"/>
<font color="Maroon">c<sub>27</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E8:41_7"><i><font color="Green">E64</font></i></a>, <a class="ref" href="borsuk_1.html#T46">BORSUK_1:46</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>28</sub></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="E12:41_7"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>27</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>29</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E14:41_7"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>27</sub></font>
 <b>and </b><br/><a NAME="E15:41_7"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>29</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>30</sub></font>, <font color="Maroon">c<sub>31</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E17:41_7"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>and </b><br/><a NAME="E18:41_7"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>26</sub></font>
 <b>and </b><br/><a NAME="E19:41_7"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>30</sub></font>,<font color="Maroon">c<sub>31</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E14:41_7"><i><font color="Green">E66</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<i><font color="Green">E71</font></i>: <font color="Maroon">c<sub>21</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>29</sub></font> = 
<font color="Maroon">c<sub>21</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font>,<font color="Maroon">c<sub>31</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E19:41_7"><i><font color="Green">E70</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>19</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>31</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E34:41"><i><font color="Green">E58</font></i></a>, <a class="txt" href="topalg_5.html#E17:41_7"><i><font color="Green">E68</font></i></a>, <a class="txt" href="topalg_5.html#E18:41_7"><i><font color="Green">E69</font></i></a></i>
;<br/>
<a NAME="E21:41_7"/>
<font color="Maroon">c<sub>19</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>26</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E18:41_7"><i><font color="Green">E69</font></i></a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<b>hence </b><a NAME="E22:41_7"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E9:41_7"><i><font color="Green">E65</font></i></a>, <a class="txt" href="topalg_5.html#E15:41_7"><i><font color="Green">E67</font></i></a>, <a class="txt" href="topalg_5.html#E20:41_7"><i><font color="Green">E71</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E39:41"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>21</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="jgraph_2.html#T20">JGRAPH_2:20</a></i>;<br/>
<a NAME="E40:41"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  holds <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>22</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<a NAME="E1:41_8"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:41_8"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E11:41"><i><font color="Green">E44</font></i></a>, <a class="txt" href="topalg_5.html#E12:41"><i><font color="Green">E45</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E3:41_8"/><i><font color="Green">E62</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p3">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E39"><i><font color="Green">Lemma33</font></i></a>, <a class="txt" href="topalg_5.html#E14:41"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:41_8"/>
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"><font color="Maroon">c<sub>21</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p3">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_8_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>23</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:41_8_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>24</sub></font>, <font color="Maroon">c<sub>25</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:41_8_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>24</sub></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="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>and </b><br/><a NAME="E4:41_8_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E5:41_8_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E1:41_8_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="topalg_5.html#E2:41_8"><i><font color="Green">E61</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E6:41_8_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E4:41_8_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font> = 
<font color="Maroon">c<sub>21</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E5:41_8_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="topalg_5.html#E1:41_8_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon">c<sub>19</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E34:41"><i><font color="Green">E58</font></i></a>, <a class="txt" href="topalg_5.html#E3:41_8_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="topalg_5.html#E6:41_8_1"><i><font color="Green">E67</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p3">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E3:41_8_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="topalg_5.html#E4:41_8_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="topalg_5.html#E5:41_8_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="topalg_5.html#E16:41"><i><font color="Green">E49</font></i></a>, <a class="txt" href="topalg_5.html#E6:41_8_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="topalg_5.html#E32:41"><i><font color="Green">E57</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E5:41_8"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E2:41_8"><i><font color="Green">E61</font></i></a>, <a class="txt" href="topalg_5.html#E3:41_8"><i><font color="Green">E62</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> = 
<font color="Maroon">c<sub>17</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="ref" href="topalg_5.html#D3" target="_self">Def3</a></i>
<br/>.= 
<font color="Maroon">c<sub>21</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E39:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="topalg_5.html#E37:41"><i><font color="Green">E59</font></i></a>, <a class="txt" href="topalg_5.html#E23:41"><i><font color="Green">E53</font></i></a>, <a class="txt" href="topalg_5.html#E5:41_8"><i><font color="Green">E63</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E34:41"><i><font color="Green">E58</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E41:41"/>
<font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


</div>
