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<b>let </b><font color="Maroon" title="c2">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c3">T</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="topalg_2.html#V1" title="TOPALG_2:attr.1">convex</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font>;<br/><b>let </b><font color="Maroon" title="c4">a</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">T</font>;<br/><b>let </b><font color="Maroon" title="c5">C</font> be   <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c4">a</font>;<br/>







<b>set </b><font color="Maroon" title="c6">E</font> =  <a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">a</font>;<br/>
<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:12_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" title="TOPALG_1:NK.4">pi_1</a> <font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">a</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c8">P</font> being   <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c4">a</font><b> such that </b><br/><a NAME="E3:12_1"/><i><font color="Green" title="E12">A1</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">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="c3">T</font>,<font color="Maroon" title="c4">a</font></span>)</span>,<font color="Maroon" title="c8">P</font>
 <b>by </b><i><a class="ref" href="topalg_1.html#T48" title="TOPALG_1:th.48">TOPALG_1:48</a></i>;<br/><a NAME="E4:12_1"/>
<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c5">C</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a> 
 <b>by </b><i><a class="ref" href="topalg_2.html#T2" target="_self" title="TOPALG_2:th.2">Th2</a></i>;<br/><a NAME="E5:12_1"/><b>then </b>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">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="c3">T</font>,<font color="Maroon" title="c4">a</font></span>)</span>,<font color="Maroon" title="c5">C</font>
 <b>by </b><i><a class="txt" href="topalg_2.html#E3:12_1"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="topalg_1.html#T47" title="TOPALG_1:th.47">TOPALG_1:47</a></i>;<br/><b>hence </b><a NAME="E6:12_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p3">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">a</font></span>)</span>,<font color="Maroon" title="c5">C</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:12"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p3">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">a</font></span>)</span>,<font color="Maroon" title="c5">C</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<a NAME="E3:12"/><b>then </b><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">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="c3">T</font>,<font color="Maroon" title="c4">a</font></span>)</span>,<font color="Maroon" title="c5">C</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:12"/>
<font color="Maroon" title="c5">C</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="c4">a</font>
 
<b>by </b><i><a class="ref" href="topalg_1.html#D1" title="TOPALG_1:def.1">TOPALG_1:def 1</a></i>;<br/>
<a NAME="E5:12"/><b>then </b>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">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="c3">T</font>,<font color="Maroon" title="c4">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="topalg_2.html#E3:12"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:12"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" title="TOPALG_1:NK.4">pi_1</a> <font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="topalg_1.html#D3" title="TOPALG_1:def.3">TOPALG_1:def 3</a></i>;<br/>


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