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<span class="kw">let </span><font color="Maroon" title="c1">T</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">T</font> is  <a href="topalg_6.html#V1" title="TOPALG_6:attr.1">having_trivial_Fundamental_Group</a>  implies  for <font color="Olive" title="b1">t</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</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">t</font> holds  <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/>

<span class="kw">assume </span><a NAME="E1:17"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
<font color="Maroon" title="c1">T</font> is  <a href="topalg_6.html#V1" title="TOPALG_6:attr.1">having_trivial_Fundamental_Group</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">t</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</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">t</font> holds  <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/>

<span class="kw">let </span><font color="Maroon" title="c2">t</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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">t</font> holds  <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/><span class="kw">let </span><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">t</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>



<span class="kw">set </span><font color="Maroon" title="c5">E</font> =  <a href="topalg_1.html#K4" title="TOPALG_1:func.4">EqRel</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">t</font>);<br/>
<a NAME="E2:17"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
 <a href="topalg_1.html#NK6" title="TOPALG_1:NK.6">pi_1</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">t</font>) is  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E12">A1</font></span></a></span>;<br/>
<a NAME="E3:17"/>
(  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="topalg_1.html#K4" title="TOPALG_1:func.4">EqRel</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">t</font>)</span>)</span>,<font color="Maroon" title="c3">P</font>) <a href="tarski.html#R2" title="TARSKI: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#NK6" title="TOPALG_1:NK.6">pi_1</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">t</font>)</span>)</span> &amp;  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="topalg_1.html#K4" title="TOPALG_1:func.4">EqRel</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">t</font>)</span>)</span>,<font color="Maroon" title="c4">Q</font>) <a href="tarski.html#R2" title="TARSKI: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#NK6" title="TOPALG_1:NK.6">pi_1</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">t</font>)</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_1.html#T47" onmouseover="rs('topalg_1/T47')" onmouseout="rh()">TOPALG_1:47</a></span>;<br/>
<a NAME="E4:17"/><span class="kw">then </span>
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="topalg_1.html#K4" title="TOPALG_1:func.4">EqRel</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">t</font>)</span>)</span>,<font color="Maroon" title="c3">P</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K4" title="TOPALG_1:func.4">EqRel</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">t</font>)</span>)</span>,<font color="Maroon" title="c4">Q</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E13">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:17"/>
<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 class="kw">by</span> <span class="lab"><a class="ref" href="topalg_1.html#T46" onmouseover="rs('topalg_1/T46')" onmouseout="rh()">TOPALG_1:46</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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