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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">L</font> being  <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">x1</font> st <br/>( <font color="Maroon">a<sub>1</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" target="_self">EqRel</a> <font color="Maroon">T</font>,<font color="Maroon">x1</font></span>)</span>,<font color="Olive">L</font> &amp; <font color="Maroon">a<sub>2</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" target="_self">EqRel</a> <font color="Maroon">T</font>,<font color="Maroon">x0</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Olive">L</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">P</font></span>)</span></span>)</span> );<br/>
<a NAME="E1:60_1_1"/><i><font color="Green">E46</font></i>: 
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">T</font>,<font color="Maroon">x1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">T</font>,<font color="Maroon">x1</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:60_1_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <font color="Maroon">T</font>,<font color="Maroon">x1</font></span>)</span>  ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <font color="Maroon">T</font>,<font color="Maroon">x0</font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <font color="Maroon">T</font>,<font color="Maroon">x1</font></span>)</span>;<br/>

<b>consider </b><font color="Maroon">Q</font> being   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">x1</font><b> such that </b><br/><a NAME="E2:60_1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">x</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" target="_self">EqRel</a> <font color="Maroon">T</font>,<font color="Maroon">x1</font></span>)</span>,<font color="Maroon">Q</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">y</font> =  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">T</font>,<font color="Maroon">x0</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">P</font></span>)</span></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <font color="Maroon">T</font>,<font color="Maroon">x0</font></span>)</span> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>thus </b><a NAME="E4:60_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>]
 <b>by </b><i><a class="txt" href="topalg_1.html#E1:60"><i><font color="Green">E40</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of 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">T</font>,<font color="Maroon">x1</font></span>)</span>,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">T</font>,<font color="Maroon">x0</font></span>)</span><b> such that </b><br/><a NAME="E4:60_1_1"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <font color="Maroon">T</font>,<font color="Maroon">x1</font></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <font color="Maroon">T</font>,<font color="Maroon">x1</font></span>)</span>,<span class="p1">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <font color="Maroon">T</font>,<font color="Maroon">x0</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>let </b><font color="Maroon">Q</font> be   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">x1</font>;<br/>

<a NAME="E6:60_1_1"/>
<font color="Maroon">Q</font> <a href="hidden.html#R2">in</a>  <a href="topalg_1.html#K1" target="_self">Loops</a> <font color="Maroon">x1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:60_1_1"/><b>then </b>
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">T</font>,<font color="Maroon">x1</font></span>)</span>,<font color="Maroon">Q</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <font color="Maroon">T</font>,<font color="Maroon">x1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">L</font> being   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">x1</font><b> such that </b><br/><a NAME="E9:60_1_1"/><i><font color="Green">E50</font></i>: 
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">T</font>,<font color="Maroon">x1</font></span>)</span>,<font color="Maroon">Q</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" target="_self">EqRel</a> <font color="Maroon">T</font>,<font color="Maroon">x1</font></span>)</span>,<font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E10:60_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">T</font>,<font color="Maroon">x1</font></span>)</span>,<font color="Maroon">Q</font></span>)</span> <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" target="_self">EqRel</a> <font color="Maroon">T</font>,<font color="Maroon">x0</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">L</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">P</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="topalg_1.html#E1:60"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E11:60_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">P</font>,<font color="Maroon">P</font> <a href="borsuk_2.html#R3">are_homotopic</a> 
 
<b>by </b><i>, <a class="ref" href="borsuk_2.html#T15">BORSUK_2:15</a></i>;<br/>
<a NAME="E12:60_1_1"/><i><font color="Green">E81</font></i>: 
 <a href="borsuk_2.html#K2">-</a> <font color="Maroon">P</font>, <a href="borsuk_2.html#K2">-</a> <font color="Maroon">P</font> <a href="borsuk_2.html#R3">are_homotopic</a> 
 
<b>by </b><i>, <a class="ref" href="borsuk_2.html#T15">BORSUK_2:15</a></i>;<br/>
<a NAME="E13:60_1_1"/>
<font color="Maroon">L</font>,<font color="Maroon">Q</font> <a href="borsuk_2.html#R3">are_homotopic</a> 
 
<b>by </b><i><a class="ref" href="topalg_1.html#T2" target="_self">Th2</a>, </i>;<br/>
<a NAME="E14:60_1_1"/><b>then </b>
<font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">L</font>,<font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q</font> <a href="borsuk_2.html#R3">are_homotopic</a> 
 
<b>by </b><i>, , <a class="ref" href="borsuk_6.html#T83">BORSUK_6:83</a></i>;<br/>
<a NAME="E15:60_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">L</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">P</font></span>)</span> <a href="borsuk_2.html#R3">are_homotopic</a> 
 
<b>by </b><i>, , <a class="ref" href="borsuk_6.html#T83">BORSUK_6:83</a></i>;<br/>
<b>hence </b><a NAME="E16:60_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">T</font>,<font color="Maroon">x1</font></span>)</span>,<font color="Maroon">Q</font></span>)</span> <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" target="_self">EqRel</a> <font color="Maroon">T</font>,<font color="Maroon">x0</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">P</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/>


</div>
