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<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="topreal2.html#NM1" title="TOPREAL2:NM.1">Simple_closed_curve</a>; <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">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <span class="p1">(<span class="default"><a href="jordan19.html#K2" title="JORDAN19:func.2">Lower_Appr</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> &amp; <span class="p1">(<span class="default"><a href="jordan19.html#K2" title="JORDAN19:func.2">Lower_Appr</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> )</span><br/><b>let </b><font color="Maroon" title="c2">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="jordan19.html#K2" title="JORDAN19:func.2">Lower_Appr</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">i</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> &amp; <span class="p1">(<span class="default"><a href="jordan19.html#K2" title="JORDAN19:func.2">Lower_Appr</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">i</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> )</span><br/>



<a NAME="E1:6"/>
 <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 
;<br/>
<b>hence </b><a NAME="E2:6"/>
<span class="p1">(<span class="default"><a href="jordan19.html#K2" title="JORDAN19:func.2">Lower_Appr</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">i</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 <b>by </b><i><a class="ref" href="jordan19.html#D2" title="JORDAN19:def.2">JORDAN19:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="jordan19.html#K2" title="JORDAN19:func.2">Lower_Appr</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">i</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a></span><br/>

<a NAME="E3:6"/>
 <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
;<br/>
<b>hence </b><a NAME="E4:6"/>
<span class="p1">(<span class="default"><a href="jordan19.html#K2" title="JORDAN19:func.2">Lower_Appr</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">i</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 <b>by </b><i><a class="ref" href="jordan19.html#D2" title="JORDAN19:def.2">JORDAN19:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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