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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <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">a</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">P</font>, <font color="Olive" title="b3">Q</font> being  <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Olive" title="b1">a</font> holds <br/> ( <font color="Olive" title="b3">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">a</font></span>)</span>,<font color="Olive" title="b2">P</font> iff <font color="Olive" title="b2">P</font>,<font color="Olive" title="b3">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="c2">a</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</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/> ( <font color="Olive" title="b2">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>,<font color="Olive" title="b1">P</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="c3">P</font>, <font color="Maroon" title="c4">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"> ( <font color="Maroon" title="c4">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">P</font> iff <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a>  )</span><br/>





<b>set </b><font color="Maroon" title="c5">E</font> =  <a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font>;<br/>
<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a>  implies <font color="Maroon" title="c4">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">P</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:52_1"/>
<font color="Maroon" title="c4">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> </span><br/><a NAME="E2:52_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">Q</font>,<font color="Maroon" title="c3">P</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></i>;<br/><b>hence </b><a NAME="E3:52_1"/>
<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 <b>by </b><i><a class="ref" href="topalg_1.html#D2" target="_self" title="TOPALG_1:def.2">Def2</a></i>; <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>

<b>assume </b><a NAME="E2:52"/>
<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">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">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">P</font></span><br/>

<a NAME="E3:52"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">Q</font>,<font color="Maroon" title="c3">P</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="ref" href="topalg_1.html#D2" target="_self" title="TOPALG_1:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E4:52"/>
<font color="Maroon" title="c4">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">P</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</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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