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

<b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="xxreal_0.html#V2">positive</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</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> 2</span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of  <a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>





<b>assume </b><a NAME="E1:17"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<b>given </b><font color="Maroon">c<sub>6</sub></font> being  <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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font><b> such that </b><a NAME="E2:17"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="borsuk_1.html#NR1" target="_self">is_a_retraction</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_1.html#D20">BORSUK_1:def 20</a><br/>

<a NAME="E3:17"/><i><font color="Green">E14</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>3</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>3</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="E4:17"/><i><font color="Green">E15</font></i>: 
 <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">c<sub>3</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>3</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/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E6:17"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="brouwer.html#E1:17"><i><font color="Green">E12</font></i></a>, <a class="txt" href="brouwer.html#E3:17"><i><font color="Green">E14</font></i></a>, <a class="txt" href="brouwer.html#E4:17"><i><font color="Green">E15</font></i></a>, <a class="ref" href="brouwer.html#T3" target="_self">Th3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being  <a href="funct_1.html#V3">constant</a> <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">c<sub>8</sub></font>;<br/>
<i><font color="Green">E16</font></i>: <font color="Maroon">c<sub>9</sub></font> = 
<a href="borsuk_1.html#K20">I[01]</a>  <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="borsuk_2.html#T6">BORSUK_2:6</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a>  <a href="borsuk_1.html#K1">--&gt;</a> <font color="Maroon">c<sub>7</sub></font>

;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <a href="borsuk_1.html#K20">I[01]</a> ,<font color="Maroon">c<sub>4</sub></font> ;<br/>
<a NAME="E11:17"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#R2">=</a> <a href="borsuk_1.html#K20">I[01]</a>  <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="brouwer.html#E9:17"><i><font color="Green">E16</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = 0, <font color="Maroon">c<sub>12</sub></font> = 1 as   <a href="pre_topc.html#NM2">Point</a> of <a href="borsuk_1.html#K20">I[01]</a>  <b>by </b><i><a class="ref" href="borsuk_1.html#D17">BORSUK_1:def 17</a>, <a class="ref" href="borsuk_1.html#D18">BORSUK_1:def 18</a></i>;<br/>
<a NAME="E13:17"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="borsuk_2.html#R2">are_connected</a> 
 
;<br/>
<a NAME="E14:17"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="brouwer.html#E9:17"><i><font color="Green">E16</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>10</sub></font> as  <a href="funct_1.html#V3">constant</a> <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="brouwer.html#E11:17"><i><font color="Green">E17</font></i></a>, <a class="txt" href="brouwer.html#E13:17"><i><font color="Green">E18</font></i></a>, <a class="ref" href="borsuk_2.html#D2">BORSUK_2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="brouwer.html#E1:17"><i><font color="Green">E12</font></i></a></i>;<br/>
<a NAME="E17:17"/><i><font color="Green">E19</font></i>: 
not the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <span class="p2">(<span class="default"><a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>14</sub></font></span>)</span> is <a href="finset_1.html#V1">finite</a>
 
;<br/>
<a NAME="E18:17"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p3">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>15</sub></font> =  <a href="topalg_1.html#K2">EqRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font>;<br/>
<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">c<sub>16</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:17_2_1"/>
<font color="Maroon">c<sub>16</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="topalg_1.html#NK4" target="_self">pi_1</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">c<sub>7</sub></font><b> such that </b><br/><a NAME="E3:17_2_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="ref" href="topalg_1.html#T48">TOPALG_1:48</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font> as   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">c<sub>8</sub></font> <b>by </b><i><a class="ref" href="topalg_2.html#T1">TOPALG_2:1</a></i>;<br/><a NAME="E5:17_2_1"/>
<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="borsuk_2.html#R4">are_homotopic</a> 
 <b>by </b><i><a class="ref" href="topalg_2.html#T2">TOPALG_2:2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>19</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"><a href="borsuk_1.html#K20">I[01]</a> ,<a href="borsuk_1.html#K20">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E7:17_2_1"/><i><font color="Green">E21</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="E8:17_2_1"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="borsuk_1.html#K20">I[01]</a>  holds <br/>( <font color="Maroon">c<sub>19</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="borsuk_1.html#K20">I[01]</a>  holds <br/>( <font color="Maroon">c<sub>19</sub></font> <a href="binop_1.html#K1">.</a> 0,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="binop_1.html#K1">.</a> 1,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> ) ) )
 <b>by </b><i><a class="ref" href="borsuk_2.html#D7">BORSUK_2:def 7</a></i>;<br/><a NAME="E9:17_2_1"/>
<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="borsuk_2.html#R3">are_homotopic</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>19</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_2.html#D7">BORSUK_2:def 7</a><br/>

<b>thus </b><a NAME="E1:17_2_1_1"/>
<font color="Maroon">c<sub>20</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="brouwer.html#E7:17_2_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>21</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="borsuk_1.html#K20">I[01]</a> ;<br/>

<b>thus </b><font color="Maroon">c<sub>20</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>,0 = 
<font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="borsuk_1.html#K7">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,0</span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>,0</span>)</span>

<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <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>21</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="brouwer.html#E8:17_2_1"><i><font color="Green">E22</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="brouwer.html#E2:17"><i><font color="Green">E13</font></i></a>, <a class="ref" href="borsuk_1.html#D19">BORSUK_1:def 19</a></i>
;<br/>

<b>thus </b><font color="Maroon">c<sub>20</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>,1 = 
<font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="borsuk_1.html#K7">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,1</span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>,1</span>)</span>

<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="brouwer.html#E8:17_2_1"><i><font color="Green">E22</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="brouwer.html#E2:17"><i><font color="Green">E13</font></i></a>, <a class="ref" href="borsuk_1.html#D19">BORSUK_1:def 19</a></i>
;<br/>

<b>let </b><font color="Maroon">c<sub>22</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="borsuk_1.html#K20">I[01]</a> ;<br/>

<b>thus </b><font color="Maroon">c<sub>20</sub></font> <a href="binop_1.html#K1">.</a> 0,<font color="Maroon">c<sub>22</sub></font> = 
<font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="borsuk_1.html#K7">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>22</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="binop_1.html#K1">.</a> 0,<font color="Maroon">c<sub>22</sub></font></span>)</span>

<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="brouwer.html#E8:17_2_1"><i><font color="Green">E22</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="brouwer.html#E2:17"><i><font color="Green">E13</font></i></a>, <a class="ref" href="borsuk_1.html#D19">BORSUK_1:def 19</a></i>
;<br/>

<b>thus </b><font color="Maroon">c<sub>20</sub></font> <a href="binop_1.html#K1">.</a> 1,<font color="Maroon">c<sub>22</sub></font> = 
<font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="borsuk_1.html#K7">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>22</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="binop_1.html#K1">.</a> 1,<font color="Maroon">c<sub>22</sub></font></span>)</span>

<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="brouwer.html#E8:17_2_1"><i><font color="Green">E22</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="brouwer.html#E2:17"><i><font color="Green">E13</font></i></a>, <a class="ref" href="borsuk_1.html#D19">BORSUK_1:def 19</a></i>
;<br/>


</div><b>end;</b></div><a NAME="E10:17_2_1"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="brouwer.html#E3:17_2_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="topalg_1.html#T47">TOPALG_1:47</a></i>;<br/><b>hence </b><a NAME="E11:17_2_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p3">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>16</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="E2:17_2"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p3">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E3:17_2"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:17_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="topalg_1.html#K1">Loops</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="topalg_1.html#D1">TOPALG_1:def 1</a></i>;<br/>
<a NAME="E5:17_2"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="brouwer.html#E3:17_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:17_2"/>
<font color="Maroon">c<sub>16</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="topalg_1.html#NK4" target="_self">pi_1</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="topalg_1.html#D3">TOPALG_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E19:17"/>
contradiction
 <b>by </b><i><a class="txt" href="brouwer.html#E1:17"><i><font color="Green">E12</font></i></a>, <a class="txt" href="brouwer.html#E17:17"><i><font color="Green">E19</font></i></a></i>;<br/>


</div>
