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<b>consider </b><font color="Maroon" title="c3">E</font> being   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of  <a href="topalg_1.html#K1" title="TOPALG_1:func.1">Loops</a> <font color="Maroon" title="c2">a</font><b> such that </b><br/><a NAME="E2:50_1_1"/><i><font color="Green" title="E31">A1</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topalg_1.html#K1" title="TOPALG_1:func.1">Loops</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topalg_1.html#K1" title="TOPALG_1:func.1">Loops</a> <font color="Maroon" title="c2">a</font> &amp;  ex <font color="Olive" title="b3">P</font>, <font color="Olive" title="b4">Q</font> being  <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c2">a</font> st <br/>( <font color="Olive" title="b3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b4">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b3">P</font>,<font color="Olive" title="b4">Q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a>  ) ) )
 <b>by </b><i><a class="txt" href="topalg_1.html#E48"><i><font color="Green" title="E30">Lm2</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">E</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">P</font>, <font color="Olive" title="b2">Q</font> being  <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c2">a</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">Q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E</font> iff <font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">Q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a>  )</span><br/>

<b>let </b><font color="Maroon" title="c4">P</font>, <font color="Maroon" title="c5">Q</font> be   <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c2">a</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E</font> iff <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a>  )</span><br/>

<b>thus </b><a NAME="E3:50_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E</font> implies <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a>  implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:50_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> </span><br/>

<a NAME="E2:50_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">P1</font>, <font color="Olive" title="b2">Q1</font> being  <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c2">a</font> st <br/>( <font color="Olive" title="b1">P1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Olive" title="b2">Q1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">Q</font> &amp; <font color="Olive" title="b1">P1</font>,<font color="Olive" title="b2">Q1</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a>  )
 
<b>by </b><i><a class="txt" href="#E2:50_1_1"><i><font color="Green" title="E31">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:50_1_1_1"/>
<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<a NAME="E4:50_1_1"/>
( <font color="Maroon" title="c4">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topalg_1.html#K1" title="TOPALG_1:func.1">Loops</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c5">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topalg_1.html#K1" title="TOPALG_1:func.1">Loops</a> <font color="Maroon" title="c2">a</font> )
 
<b>by </b><i><a class="ref" href="topalg_1.html#D1" target="_self" title="TOPALG_1:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E5:50_1_1"/>
( <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a>  implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E</font> )
 <b>by </b><i><a class="txt" href="#E2:50_1_1"><i><font color="Green" title="E31">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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