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<span class="kw">let </span><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">q</font> be   <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> 2</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">C</font> being   <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Maroon" title="c1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  holds <br/> <a href="jordan18.html#K1" title="JORDAN18:func.1">North-Bound</a> (<font color="Maroon" title="c1">p</font>,<font color="Olive" title="b1">C</font>), <a href="jordan18.html#K2" title="JORDAN18:func.2">South-Bound</a> (<font color="Maroon" title="c2">q</font>,<font color="Olive" title="b1">C</font>), <a href="jordan18.html#K1" title="JORDAN18:func.1">North-Bound</a> (<font color="Maroon" title="c2">q</font>,<font color="Olive" title="b1">C</font>), <a href="jordan18.html#K2" title="JORDAN18:func.2">South-Bound</a> (<font color="Maroon" title="c1">p</font>,<font color="Olive" title="b1">C</font>) <a href="zfmisc_1.html#R2" title="ZFMISC_1:pred.2">are_mutually_distinct</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">C</font> be   <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</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="c1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Maroon" title="c2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Maroon" title="c1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  implies  <a href="jordan18.html#K1" title="JORDAN18:func.1">North-Bound</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">C</font>), <a href="jordan18.html#K2" title="JORDAN18:func.2">South-Bound</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">C</font>), <a href="jordan18.html#K1" title="JORDAN18:func.1">North-Bound</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">C</font>), <a href="jordan18.html#K2" title="JORDAN18:func.2">South-Bound</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">C</font>) <a href="zfmisc_1.html#R2" title="ZFMISC_1:pred.2">are_mutually_distinct</a>  )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">np</font> =  <a href="jordan18.html#K1" title="JORDAN18:func.1">North-Bound</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">C</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c5">sq</font> =  <a href="jordan18.html#K2" title="JORDAN18:func.2">South-Bound</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">C</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c6">nq</font> =  <a href="jordan18.html#K1" title="JORDAN18:func.1">North-Bound</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">C</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c7">sp</font> =  <a href="jordan18.html#K2" title="JORDAN18:func.2">South-Bound</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">C</font>);<br/>
<a NAME="E1:24"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
( <span class="p1">(<span class="default"><a href="jordan18.html#K1" title="JORDAN18:func.1">North-Bound</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">C</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><a href="jordan18.html#K2" title="JORDAN18:func.2">South-Bound</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">C</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" onmouseover="rs('euclid/T52')" onmouseout="rh()">EUCLID:52</a></span>;<br/>
<a NAME="E2:24"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
( <span class="p1">(<span class="default"><a href="jordan18.html#K1" title="JORDAN18:func.1">North-Bound</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">C</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">north_halfline</a> <font color="Maroon" title="c2">q</font></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan18.html#K2" title="JORDAN18:func.2">South-Bound</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">C</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="topreal1.html#K6" title="TOPREAL1:func.6">south_halfline</a> <font color="Maroon" title="c2">q</font></span>)</span></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" onmouseover="rs('euclid/T52')" onmouseout="rh()">EUCLID:52</a></span>;<br/>
<a NAME="E3:24"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
( <span class="p1">(<span class="default"><a href="jordan18.html#K1" title="JORDAN18:func.1">North-Bound</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">C</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">north_halfline</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan18.html#K2" title="JORDAN18:func.2">South-Bound</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">C</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="topreal1.html#K6" title="TOPREAL1:func.6">south_halfline</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" onmouseover="rs('euclid/T52')" onmouseout="rh()">EUCLID:52</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:24"/>
( <font color="Maroon" title="c1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Maroon" title="c2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Maroon" title="c1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</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">  <a href="jordan18.html#K1" title="JORDAN18:func.1">North-Bound</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">C</font>), <a href="jordan18.html#K2" title="JORDAN18:func.2">South-Bound</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">C</font>), <a href="jordan18.html#K1" title="JORDAN18:func.1">North-Bound</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">C</font>), <a href="jordan18.html#K2" title="JORDAN18:func.2">South-Bound</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">C</font>) <a href="zfmisc_1.html#R2" title="ZFMISC_1:pred.2">are_mutually_distinct</a> </span><br/>

<span class="kw">hence </span><a NAME="E5:24"/>
(  <a href="jordan18.html#K1" title="JORDAN18:func.1">North-Bound</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">C</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="jordan18.html#K2" title="JORDAN18:func.2">South-Bound</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">C</font>) &amp;  <a href="jordan18.html#K1" title="JORDAN18:func.1">North-Bound</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">C</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="jordan18.html#K1" title="JORDAN18:func.1">North-Bound</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">C</font>) &amp;  <a href="jordan18.html#K1" title="JORDAN18:func.1">North-Bound</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">C</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="jordan18.html#K2" title="JORDAN18:func.2">South-Bound</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">C</font>) &amp;  <a href="jordan18.html#K2" title="JORDAN18:func.2">South-Bound</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">C</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="jordan18.html#K1" title="JORDAN18:func.1">North-Bound</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">C</font>) &amp;  <a href="jordan18.html#K2" title="JORDAN18:func.2">South-Bound</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">C</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="jordan18.html#K2" title="JORDAN18:func.2">South-Bound</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">C</font>) &amp;  <a href="jordan18.html#K1" title="JORDAN18:func.1">North-Bound</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">C</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="jordan18.html#K2" title="JORDAN18:func.2">South-Bound</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">C</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E3:24"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="jordan18.html#T19" target="_self" onmouseover="rs('jordan18/T19')" onmouseout="rh()">Th19</a>, <a class="ref" href="euclid.html#T52" onmouseover="rs('euclid/T52')" onmouseout="rh()">EUCLID:52</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="zfmisc_1.html#D6" onmouseover="rs('zfmisc_1/D6')" onmouseout="rh()">ZFMISC_1:def 6</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/>


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