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<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of  <a href="topalg_1.html#K1" target="_self">Loops</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E2:51_1_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="topalg_1.html#K1" target="_self">Loops</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topalg_1.html#K1" target="_self">Loops</a> <font color="Maroon">c<sub>2</sub></font> &amp; ex <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font> <a href="borsuk_2.html#R3">are_homotopic</a>  ) ) )
 <b>by </b><i><a class="txt" href="topalg_1.html#E48"><i><font color="Green">Lemma30</font></i></a></i>;<br/>
<a NAME="E3:51_1_1"/>
 <a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D2">RELAT_1:def 2</a><br/>

<b>thus </b><a NAME="E1:51_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:51_1_1_1_1"/><i><font color="Green">E33</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:51_1_1_1_1"/><b>then </b><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="topalg_1.html#K1" target="_self">Loops</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="topalg_1.html#K1" target="_self">Loops</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="topalg_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E4:51_1_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="borsuk_2.html#R3">are_homotopic</a> 
 
<b>by </b><i><a class="txt" href="topalg_1.html#E1:51_1_1_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="topalg_1.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E5:51_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="topalg_1.html#E2:51_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="topalg_1.html#E2:51_1_1_1_1"><i><font color="Green">E34</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:51_1_1_1"/><i><font color="Green">E33</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E3:51_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="topalg_1.html#K1" target="_self">Loops</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="topalg_1.html#K1" target="_self">Loops</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="topalg_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E5:51_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="borsuk_2.html#R3">are_homotopic</a>  )
 
<b>by </b><i><a class="txt" href="topalg_1.html#E2:51_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="topalg_1.html#E2:51_1_1_1"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:51_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="topalg_1.html#D2" target="_self">Def2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:51_1_1"/>
( not  <a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> is <a href="partfun1.html#V1">total</a> &amp;  <a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> is <a href="relat_2.html#V3">symmetric</a> &amp;  <a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> is <a href="relat_2.html#V8">transitive</a> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#T16">EQREL_1:16</a>, <a class="ref" href="relat_1.html#T63">RELAT_1:63</a></i>;<br/>


</div>
