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<b>consider </b><font color="Maroon" title="c8">F</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span><b> such that </b><br/><a NAME="E2:34"/><i><font color="Green" title="E33">A1</font></i>: 
<font color="Maroon" title="c8">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <b>and </b><br/><a NAME="E3:34"/><i><font color="Green" title="E34">A2</font></i>: 
( <font color="Maroon" title="c8">F</font> is   <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">Cover</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> &amp; <font color="Maroon" title="c8">F</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> )
 <b>and </b><br/><a NAME="E4:34"/><i><font color="Green" title="E35">A3</font></i>: 
for <font color="Olive" title="b1">U</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> holds <br/> ( ( <font color="Olive" title="b1">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> implies (  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">U</font> &amp; ( for <font color="Olive" title="b2">d</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 holds <br/>for <font color="Olive" title="b3">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b2">d</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">U</font></span>)</span>  st <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b2">d</font> holds <br/><font color="Olive" title="b3">f</font> is <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> ) ) ) &amp; ( <font color="Olive" title="b1">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> implies (  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">U</font> &amp; ( for <font color="Olive" title="b2">d</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> holds <br/>for <font color="Olive" title="b3">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b2">d</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">U</font></span>)</span>  st <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b2">d</font> holds <br/><font color="Olive" title="b3">f</font> is <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> ) ) ) )
 <b>by </b><i><a class="ref" href="toprealb.html#T49" title="TOPREALB:th.49">TOPREALB:49</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">F</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">F</font> is   <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">Cover</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> &amp; <font color="Maroon" title="c8">F</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> &amp; ( for <font color="Olive" title="b1">U</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span>  st <font color="Olive" title="b1">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">F</font> holds <br/> ex <font color="Olive" title="b2">D</font> being <a href="taxonom2.html#V4" title="TAXONOM2:attr.4">mutually-disjoint</a> <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  st <br/>(  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b2">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">U</font> &amp; ( for <font color="Olive" title="b3">d</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b3">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">D</font> holds <br/>for <font color="Olive" title="b4">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b3">d</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">U</font></span>)</span>  st <font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b3">d</font> holds <br/><font color="Olive" title="b4">f</font> is <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> ) ) ) )</span><br/>

<b>thus </b><a NAME="E5:34"/>
( <font color="Maroon" title="c8">F</font> is   <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">Cover</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> &amp; <font color="Maroon" title="c8">F</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> )
 <b>by </b><i><a class="txt" href="#E3:34"><i><font color="Green" title="E34">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">U</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span>  st <font color="Olive" title="b1">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">F</font> holds <br/> ex <font color="Olive" title="b2">D</font> being <a href="taxonom2.html#V4" title="TAXONOM2:attr.4">mutually-disjoint</a> <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  st <br/>(  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b2">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">U</font> &amp; ( for <font color="Olive" title="b3">d</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b3">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">D</font> holds <br/>for <font color="Olive" title="b4">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b3">d</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">U</font></span>)</span>  st <font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b3">d</font> holds <br/><font color="Olive" title="b4">f</font> is <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> ) )</span><br/>

<b>let </b><font color="Maroon" title="c9">U</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">F</font> implies  ex <font color="Olive" title="b1">D</font> being <a href="taxonom2.html#V4" title="TAXONOM2:attr.4">mutually-disjoint</a> <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  st <br/>(  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c9">U</font> &amp; ( for <font color="Olive" title="b2">d</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">D</font> holds <br/>for <font color="Olive" title="b3">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b2">d</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">U</font></span>)</span>  st <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b2">d</font> holds <br/><font color="Olive" title="b3">f</font> is <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> ) ) )</span><br/>

<b>assume </b><a NAME="E6:34"/><i><font color="Green" title="E36">A4</font></i>: 
<font color="Maroon" title="c9">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">D</font> being <a href="taxonom2.html#V4" title="TAXONOM2:attr.4">mutually-disjoint</a> <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  st <br/>(  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c9">U</font> &amp; ( for <font color="Olive" title="b2">d</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">D</font> holds <br/>for <font color="Olive" title="b3">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b2">d</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">U</font></span>)</span>  st <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b2">d</font> holds <br/><font color="Olive" title="b3">f</font> is <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> ) )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_1"/>
( <font color="Maroon" title="c9">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> or <font color="Maroon" title="c9">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )
 </span><b>by </b><i><a class="txt" href="#E2:34"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="#E6:34"><i><font color="Green" title="E36">A4</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1"><b>suppose </b></a><a NAME="E1:34_1_1"/><i><font color="Green" title="E37">A5</font></i>: 
<font color="Maroon" title="c9">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">D</font> being <a href="taxonom2.html#V4" title="TAXONOM2:attr.4">mutually-disjoint</a> <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  st <br/>(  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c9">U</font> &amp; ( for <font color="Olive" title="b2">d</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">D</font> holds <br/>for <font color="Olive" title="b3">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b2">d</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">U</font></span>)</span>  st <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b2">d</font> holds <br/><font color="Olive" title="b3">f</font> is <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> ) )</span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c10">D</font> =  <a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 as  <a href="taxonom2.html#V4" title="TAXONOM2:attr.4">mutually-disjoint</a> <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E9"><i><font color="Green" title="E8">Lm8</font></i></a>, <a class="ref" href="toprealb.html#T4" title="TOPREALB:th.4">TOPREALB:4</a></i>;<br/><b>take </b>
<font color="Maroon" title="c10">D</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c10">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c9">U</font> &amp; ( for <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">D</font> holds <br/>for <font color="Olive" title="b2">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">d</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">U</font></span>)</span>  st <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">d</font> holds <br/><font color="Olive" title="b2">f</font> is <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> ) )</span><br/><b>thus </b><a NAME="E3:34_1_1"/>
(  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c10">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c9">U</font> &amp; ( for <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">D</font> holds <br/>for <font color="Olive" title="b2">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">d</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">U</font></span>)</span>  st <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">d</font> holds <br/><font color="Olive" title="b2">f</font> is <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> ) )
 <b>by </b><i><a class="txt" href="#E4:34"><i><font color="Green" title="E35">A3</font></i></a>, <a class="txt" href="#E1:34_1_1"><i><font color="Green" title="E37">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2"><b>suppose </b></a><a NAME="E1:34_1_2"/><i><font color="Green" title="E37">A6</font></i>: 
<font color="Maroon" title="c9">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">D</font> being <a href="taxonom2.html#V4" title="TAXONOM2:attr.4">mutually-disjoint</a> <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  st <br/>(  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c9">U</font> &amp; ( for <font color="Olive" title="b2">d</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">D</font> holds <br/>for <font color="Olive" title="b3">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b2">d</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">U</font></span>)</span>  st <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b2">d</font> holds <br/><font color="Olive" title="b3">f</font> is <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> ) )</span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c10">D</font> =  <a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> as  <a href="taxonom2.html#V4" title="TAXONOM2:attr.4">mutually-disjoint</a> <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E10"><i><font color="Green" title="E9">Lm9</font></i></a>, <a class="ref" href="toprealb.html#T4" title="TOPREALB:th.4">TOPREALB:4</a></i>;<br/><b>take </b>
<font color="Maroon" title="c10">D</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c10">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c9">U</font> &amp; ( for <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">D</font> holds <br/>for <font color="Olive" title="b2">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">d</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">U</font></span>)</span>  st <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">d</font> holds <br/><font color="Olive" title="b2">f</font> is <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> ) )</span><br/><b>thus </b><a NAME="E3:34_1_2"/>
(  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c10">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c9">U</font> &amp; ( for <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">D</font> holds <br/>for <font color="Olive" title="b2">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">d</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">U</font></span>)</span>  st <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">d</font> holds <br/><font color="Olive" title="b2">f</font> is <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> ) )
 <b>by </b><i><a class="txt" href="#E4:34"><i><font color="Green" title="E35">A3</font></i></a>, <a class="txt" href="#E1:34_1_2"><i><font color="Green" title="E37">A6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
