<?xml version="1.0"?>
<div class="add">

<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">  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>  st  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b2">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> holds <br/><font color="Olive" title="b2">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="c2">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="c2">t</font>  st  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">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> holds <br/><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="c3">f</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"> (  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">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> implies <font color="Maroon" title="c3">f</font> is  <a href="topalg_6.html#V3" title="TOPALG_6:attr.3">nullhomotopic</a>  )</span><br/>





<span class="kw">assume </span><a NAME="E1:81"/><span class="lab"><font color="Green" title="E53">A1</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">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="c3">f</font> is  <a href="topalg_6.html#V3" title="TOPALG_6:attr.3">nullhomotopic</a> </span><br/>

<a NAME="E2:81"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">f</font> holds <br/><font color="Olive" title="b1">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="mfold_2.html#K4" title="MFOLD_2:func.4">TUnitSphere</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E4:81"/><span class="lab"><font color="Green" title="E54">A2</font></span>: 
<font color="Maroon" title="c4">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="mfold_2.html#K4" title="MFOLD_2:func.4">TUnitSphere</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:81"/><span class="lab"><font color="Green" title="E55">A3</font></span>: 
 not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="ref" href="tarski.html#T2" onmouseover="rs('tarski/T2')" onmouseout="rh()">TARSKI:2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">n1</font> = <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/>
<a NAME="E7:81"/><span class="lab"><font color="Green" title="E56">A4</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> <font color="Maroon" title="c5">n1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">n1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D4" onmouseover="rs('pre_topc/D4')" onmouseout="rh()">PRE_TOPC:def 4</a></span>;<br/>
<a NAME="E8:81"/><span class="lab"><font color="Green" title="E57">A5</font></span>: 
<font color="Maroon" title="c4">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="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> <font color="Maroon" title="c5">n1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:81"><span class="lab"><font color="Green" title="E54">A2</font></span></a>, <a class="ref" href="mfold_2.html#D4" onmouseover="rs('mfold_2/D4')" onmouseout="rh()">MFOLD_2:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">p</font> = <font color="Maroon" title="c4">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">n1</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:81"><span class="lab"><font color="Green" title="E56">A4</font></span></a></span>;<br/>
<a NAME="E10:81"/>
<font color="Maroon" title="c6">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="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">n1</font></span>)</span></span>)</span>,1)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:81"><span class="lab"><font color="Green" title="E57">A5</font></span></a>, <a class="ref" href="toprealb.html#D7" onmouseover="rs('toprealb/D7')" onmouseout="rh()">TOPREALB:def 7</a></span>;<br/>
<a NAME="E11:81"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A6</font></span>: 
<font color="Maroon" title="c6">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">n1</font></span>)</span></span>)</span>,1)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="toprealb.html#T9" onmouseover="rs('toprealb/T9')" onmouseout="rh()">TOPREALB:9</a></span>;<br/>
<a NAME="E12:81"/><span class="kw">then </span>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">n1</font></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_6.html#T3" target="_self" onmouseover="rs('topalg_6/T3')" onmouseout="rh()">Th3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">S</font> = <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">n1</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">n1</font></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> as   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</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="c5">n1</font> ;<br/>
<a NAME="E14:81"/><span class="lab"><font color="Green" title="E59">A7</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c7">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">n1</font></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E15:81"/><span class="kw">then </span>
 <a href="mfold_2.html#K5" title="MFOLD_2:func.5">stereographic_projection</a> (<font color="Maroon" title="c7">S</font>,<font color="Maroon" title="c6">p</font>) is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:81"><span class="lab"><font color="Green" title="E58">A6</font></span></a>, <a class="ref" href="mfold_2.html#T31" onmouseover="rs('mfold_2/T31')" onmouseout="rh()">MFOLD_2:31</a></span>;<br/>
<a NAME="E16:81"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A8</font></span>: 
 <a href="mfold_2.html#K3" title="MFOLD_2:func.3">TPlane</a> (<font color="Maroon" title="c6">p</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">n1</font></span>)</span></span>)</span>),<font color="Maroon" title="c7">S</font> <a href="mfold_2.html#R1" title="MFOLD_2:pred.1">are_homeomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="t_0topsp.html#D1" onmouseover="rs('t_0topsp/D1')" onmouseout="rh()">T_0TOPSP:def 1</a></span>;<br/>
<a NAME="E17:81"/><span class="lab"><font color="Green" title="E61">A9</font></span>: 
<font color="Maroon" title="c7">S</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="txt" href="#E16:81"><span class="lab"><font color="Green" title="E60">A8</font></span></a>, <a class="ref" href="topalg_6.html#T13" target="_self" onmouseover="rs('topalg_6/T13')" onmouseout="rh()">Th13</a></span>;<br/>
<a NAME="E18:81"/>
 <a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> <font color="Maroon" title="c5">n1</font> is    <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="c5">n1</font>
 
;<br/>
<a NAME="E19:81"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A10</font></span>: 
 <a href="mfold_2.html#K4" title="MFOLD_2:func.4">TUnitSphere</a> <font color="Maroon" title="c1">n</font> is    <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="c5">n1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mfold_2.html#D4" onmouseover="rs('mfold_2/D4')" onmouseout="rh()">MFOLD_2:def 4</a></span>;<br/>
<a NAME="E20:81"/>
<span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">n1</font></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">n1</font></span>)</span></span>)</span>,1)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T36" onmouseover="rs('xboole_1/T36')" onmouseout="rh()">XBOOLE_1:36</a></span>;<br/>
<a NAME="E21:81"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">n1</font></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">n1</font></span>)</span></span>)</span>,1)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="toprealb.html#T9" onmouseover="rs('toprealb/T9')" onmouseout="rh()">TOPREALB:9</a></span>;<br/>
<a NAME="E22:81"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">n1</font></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> <font color="Maroon" title="c5">n1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="toprealb.html#D7" onmouseover="rs('toprealb/D7')" onmouseout="rh()">TOPREALB:def 7</a></span>;<br/>
<a NAME="E23:81"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">n1</font></span>)</span></span>)</span>,1)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="ref" href="mfold_2.html#D4" onmouseover="rs('mfold_2/D4')" onmouseout="rh()">MFOLD_2:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">S0</font> = <font color="Maroon" title="c7">S</font> as   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of  <a href="mfold_2.html#K4" title="MFOLD_2:func.4">TUnitSphere</a> <font color="Maroon" title="c1">n</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:81"><span class="lab"><font color="Green" title="E59">A7</font></span></a>, <a class="txt" href="#E19:81"><span class="lab"><font color="Green" title="E62">A10</font></span></a>, <a class="ref" href="topmetr.html#T3" onmouseover="rs('topmetr/T3')" onmouseout="rh()">TOPMETR:3</a></span>;<br/>
<a NAME="E25:81"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T43" onmouseover="rs('borsuk_1/T43')" onmouseout="rh()">BORSUK_1:43</a></span>;<br/>
<a NAME="E26:81"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A11</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E27:81"/>
<font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c2">t</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 
;<br/>
<a NAME="E28:81"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A12</font></span>: 
( <font color="Maroon" title="c3">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t</font> &amp; <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_2.html#D2" onmouseover="rs('borsuk_2/D2')" onmouseout="rh()">BORSUK_2:def 2</a></span>;<br/>
<a NAME="E29:81"/><span class="kw">then </span>
<font color="Maroon" title="c2">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:81"><span class="lab"><font color="Green" title="E63">A11</font></span></a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/>
<a NAME="E30:81"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A13</font></span>: 
 not <font color="Maroon" title="c2">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:81"><span class="lab"><font color="Green" title="E55">A3</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="lab"><font color="Green" title="E66">A14</font></span>:  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> = 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> <font color="Maroon" title="c5">n1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mfold_2.html#D4" onmouseover="rs('mfold_2/D4')" onmouseout="rh()">MFOLD_2:def 4</a></span>
<br/>.= 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">n1</font></span>)</span></span>)</span>,1)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="toprealb.html#D7" onmouseover="rs('toprealb/D7')" onmouseout="rh()">TOPREALB:def 7</a></span>
<br/>.= 
 <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">n1</font></span>)</span></span>)</span>,1)
<span class="kw">by</span> <span class="lab"><a class="ref" href="toprealb.html#T9" onmouseover="rs('toprealb/T9')" onmouseout="rh()">TOPREALB:9</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">t0</font> = <font color="Maroon" title="c2">t</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c8">S0</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:81"><span class="lab"><font color="Green" title="E59">A7</font></span></a>, <a class="txt" href="#E31:81"><span class="lab"><font color="Green" title="E66">A14</font></span></a>, <a class="txt" href="#E30:81"><span class="lab"><font color="Green" title="E65">A13</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E33:81"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">f0</font> = <font color="Maroon" title="c3">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>,<font color="Maroon" title="c8">S0</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:81"><span class="lab"><font color="Green" title="E59">A7</font></span></a>, <a class="txt" href="#E5:81"><span class="lab"><font color="Green" title="E55">A3</font></span></a>, <a class="txt" href="#E31:81"><span class="lab"><font color="Green" title="E66">A14</font></span></a>, <a class="ref" href="funct_2.html#T2" onmouseover="rs('funct_2/T2')" onmouseout="rh()">FUNCT_2:2</a>, <a class="ref" href="zfmisc_1.html#T34" onmouseover="rs('zfmisc_1/T34')" onmouseout="rh()">ZFMISC_1:34</a></span>;<br/>
<a NAME="E35:81"/><span class="lab"><font color="Green" title="E67">A15</font></span>: 
<font color="Maroon" title="c9">t0</font>,<font color="Maroon" title="c9">t0</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 
;<br/>
<a NAME="E36:81"/>
<font color="Maroon" title="c10">f0</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan16.html#T8" onmouseover="rs('jordan16/T8')" onmouseout="rh()">JORDAN16:8</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">f0</font> = <font color="Maroon" title="c3">f</font> as   <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c9">t0</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:81"><span class="lab"><font color="Green" title="E64">A12</font></span></a>, <a class="txt" href="#E35:81"><span class="lab"><font color="Green" title="E67">A15</font></span></a>, <a class="ref" href="borsuk_2.html#D2" onmouseover="rs('borsuk_2/D2')" onmouseout="rh()">BORSUK_2:def 2</a></span>;<br/>
<a NAME="E38:81"/>
<font color="Maroon" title="c11">f0</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="#E17:81"><span class="lab"><font color="Green" title="E61">A9</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E39:81"/>
<font color="Maroon" title="c3">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="ref" href="topalg_6.html#T18" target="_self" onmouseover="rs('topalg_6/T18')" onmouseout="rh()">Th18</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>
