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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 implies  <a href="mfold_2.html#K4" title="MFOLD_2:func.4">TUnitSphere</a> <font color="Maroon" title="c1">n</font> is  <a href="topalg_6.html#V1" title="TOPALG_6:attr.1">having_trivial_Fundamental_Group</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:83"/><span class="lab"><font color="Green" title="E55">A1</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 ; <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">  <a href="mfold_2.html#K4" title="MFOLD_2:func.4">TUnitSphere</a> <font color="Maroon" title="c1">n</font> is  <a href="topalg_6.html#V1" title="TOPALG_6:attr.1">having_trivial_Fundamental_Group</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">T</font> =  <a href="mfold_2.html#K4" title="MFOLD_2:func.4">TUnitSphere</a> <font color="Maroon" title="c1">n</font>;<br/>
<a NAME="E2:83"/>
 for <font color="Olive" title="b1">t</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="mfold_2.html#K4" title="MFOLD_2:func.4">TUnitSphere</a> <font color="Maroon" title="c1">n</font></span>)</span><br/>  for <font color="Olive" title="b2">f</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">f</font> is  <a href="topalg_6.html#V3" title="TOPALG_6:attr.3">nullhomotopic</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="83_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">t</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="mfold_2.html#K4" title="MFOLD_2:func.4">TUnitSphere</a> <font color="Maroon" title="c1">n</font></span>)</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">  for <font color="Olive" title="b1">f</font> being   <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c3">t</font> holds  <font color="Olive" title="b1">f</font> is  <a href="topalg_6.html#V3" title="TOPALG_6:attr.3">nullhomotopic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font> be   <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c3">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="c4">f</font> is  <a href="topalg_6.html#V3" title="TOPALG_6:attr.3">nullhomotopic</a> </span><br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:83_1_1"/>
(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="mfold_2.html#K4" title="MFOLD_2:func.4">TUnitSphere</a> <font color="Maroon" title="c1">n</font></span>)</span> or  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="mfold_2.html#K4" title="MFOLD_2:func.4">TUnitSphere</a> <font color="Maroon" title="c1">n</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:83_1_1_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="mfold_2.html#K4" title="MFOLD_2:func.4">TUnitSphere</a> <font color="Maroon" title="c1">n</font></span>)</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"> <font color="Maroon" title="c4">f</font> is  <a href="topalg_6.html#V3" title="TOPALG_6:attr.3">nullhomotopic</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:83_1_1_1"/>
<font color="Maroon" title="c4">f</font> is  <a href="topalg_6.html#V3" title="TOPALG_6:attr.3">nullhomotopic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="topalg_6.html#E62"><span class="lab"><font color="Green" title="E52">Lm3</font></span></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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:83_1_1_2"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="mfold_2.html#K4" title="MFOLD_2:func.4">TUnitSphere</a> <font color="Maroon" title="c1">n</font></span>)</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"> <font color="Maroon" title="c4">f</font> is  <a href="topalg_6.html#V3" title="TOPALG_6:attr.3">nullhomotopic</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">g</font> being   <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c3">t</font><span class="kw"> such that </span><br/><a NAME="E3:83_1_1_2"/><span class="lab"><font color="Green" title="E56">A2</font></span>: 
<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 <span class="kw">and </span><br/><a NAME="E4:83_1_1_2"/><span class="lab"><font color="Green" title="E57">A3</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="mfold_2.html#K4" title="MFOLD_2:func.4">TUnitSphere</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="txt" href="topalg_6.html#E63"><span class="lab"><font color="Green" title="E53">Lm4</font></span></a></span>;<br/><a NAME="E5:83_1_1_2"/>
<font color="Maroon" title="c5">g</font> is  <a href="topalg_6.html#V3" title="TOPALG_6:attr.3">nullhomotopic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_1_1_2"><span class="lab"><font color="Green" title="E57">A3</font></span></a>, <a class="txt" href="topalg_6.html#E62"><span class="lab"><font color="Green" title="E52">Lm3</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">C</font> being   <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c3">t</font><span class="kw"> such that </span><br/><a NAME="E7:83_1_1_2"/><span class="lab"><font color="Green" title="E58">A4</font></span>: 
<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">C</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 ;<br/><a NAME="E8:83_1_1_2"/>
<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">C</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="txt" href="#E3:83_1_1_2"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="txt" href="#E7:83_1_1_2"><span class="lab"><font color="Green" title="E58">A4</font></span></a>, <a class="ref" href="borsuk_6.html#T79" onmouseover="rs('borsuk_6/T79')" onmouseout="rh()">BORSUK_6:79</a></span>;<br/><span class="kw">hence </span><a NAME="E9:83_1_1_2"/>
<font color="Maroon" title="c4">f</font> is  <a href="topalg_6.html#V3" title="TOPALG_6:attr.3">nullhomotopic</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"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:83"/>
 <a href="mfold_2.html#K4" title="MFOLD_2:func.4">TUnitSphere</a> <font color="Maroon" title="c1">n</font> is  <a href="topalg_6.html#V1" title="TOPALG_6:attr.1">having_trivial_Fundamental_Group</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_6.html#T17" target="_self" onmouseover="rs('topalg_6/T17')" onmouseout="rh()">Th17</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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