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<b>consider </b><font color="Maroon">F</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span><b> such that </b><br/><a NAME="E2:37"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="relset_1.html#K10">.:</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="relset_1.html#K10">.:</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>and </b><br/><a NAME="E3:37"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">F</font> <a href="pre_topc.html#R1">is_a_cover_of</a>  <a href="toprealb.html#K8">Tunit_circle</a> 2 &amp; <font color="Maroon">F</font> is <a href="tops_2.html#V1">open</a> )
 <b>and </b><br/><a NAME="E4:37"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">U</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> holds <br/> ( ( <font color="Olive">U</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2">.[</a></span> implies (  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="toprealb.html#K3">IntIntervals</a> 0,1</span>)</span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="relset_1.html#K11">"</a> <font color="Olive">U</font> &amp; ( for <font color="Olive">d</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>   st <font color="Olive">d</font> <a href="hidden.html#R2">in</a>  <a href="toprealb.html#K3">IntIntervals</a> 0,1 holds <br/>for <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Olive">d</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Olive">U</font></span>)</span>  st <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Olive">d</font> holds <br/><font color="Olive">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  ) ) ) &amp; ( <font color="Olive">U</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> implies (  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="toprealb.html#K3">IntIntervals</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="relset_1.html#K11">"</a> <font color="Olive">U</font> &amp; ( for <font color="Olive">d</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>   st <font color="Olive">d</font> <a href="hidden.html#R2">in</a>  <a href="toprealb.html#K3">IntIntervals</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> holds <br/>for <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Olive">d</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Olive">U</font></span>)</span>  st <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Olive">d</font> holds <br/><font color="Olive">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  ) ) ) )
 <b>by </b><i><a class="ref" href="toprealb.html#T49">TOPREALB:49</a></i>;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>thus </b><a NAME="E5:37"/>
( <font color="Maroon">F</font> <a href="pre_topc.html#R1">is_a_cover_of</a>  <a href="toprealb.html#K8">Tunit_circle</a> 2 &amp; <font color="Maroon">F</font> is <a href="tops_2.html#V1">open</a> )
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">U</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E6:37"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1"/>
( <font color="Maroon">U</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2">.[</a></span> or <font color="Maroon">U</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> )
 </span><b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1"><b>suppose </b></a><a NAME="E1:37_1_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">U</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">D</font> =  <a href="toprealb.html#K3">IntIntervals</a> 0,1 as  <a href="taxonom2.html#V4">mutually-disjoint</a> <a href="tops_2.html#V1">open</a> <a href="struct_0.html#NM3">Subset-Family</a> of <a href="topalg_2.html#K3">R^1</a>  <b>by </b><i>, <a class="ref" href="toprealb.html#T4">TOPREALB:4</a></i>;<br/><b>take </b>
<font color="Maroon">D</font>
;<br/><b>thus </b><a NAME="E3:37_1_1"/>
(  <a href="setfam_1.html#K5">union</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="relset_1.html#K11">"</a> <font color="Maroon">U</font> &amp; ( for <font color="Olive">d</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>   st <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> holds <br/>for <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Olive">d</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">U</font></span>)</span>  st <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Olive">d</font> holds <br/><font color="Olive">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  ) )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_2"><b>suppose </b></a><a NAME="E1:37_1_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">U</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">D</font> =  <a href="toprealb.html#K3">IntIntervals</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> as  <a href="taxonom2.html#V4">mutually-disjoint</a> <a href="tops_2.html#V1">open</a> <a href="struct_0.html#NM3">Subset-Family</a> of <a href="topalg_2.html#K3">R^1</a>  <b>by </b><i>, <a class="ref" href="toprealb.html#T4">TOPREALB:4</a></i>;<br/><b>take </b>
<font color="Maroon">D</font>
;<br/><b>thus </b><a NAME="E3:37_1_2"/>
(  <a href="setfam_1.html#K5">union</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="relset_1.html#K11">"</a> <font color="Maroon">U</font> &amp; ( for <font color="Olive">d</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>   st <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> holds <br/>for <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Olive">d</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">U</font></span>)</span>  st <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Olive">d</font> holds <br/><font color="Olive">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  ) )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
