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

<span class="kw">let </span><font color="Maroon" title="c1">p</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" 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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Olive" title="b1">C</font> 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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp;  <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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">north_halfline</a> <font color="Maroon" title="c1">p</font> &amp;  <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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp;  <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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K6" title="TOPREAL1:func.6">south_halfline</a> <font color="Maroon" title="c1">p</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">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" 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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">C</font> implies (  <a href="jordan18.html#K1" title="JORDAN18:func.1">North-Bound</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">C</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">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="c2">C</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">north_halfline</a> <font color="Maroon" title="c1">p</font> &amp;  <a href="jordan18.html#K2" title="JORDAN18:func.2">South-Bound</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">C</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> &amp;  <a href="jordan18.html#K2" title="JORDAN18:func.2">South-Bound</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">C</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K6" title="TOPREAL1:func.6">south_halfline</a> <font color="Maroon" title="c1">p</font> ) )</span><br/>



<a NAME="E1:18"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
<font color="Maroon" title="c2">C</font> is  <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#T65" title="JORDAN2C:th.65">JORDAN2C:65</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">X</font> = <font color="Maroon" title="c2">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">north_halfline</a> <font color="Maroon" title="c1">p</font></span>)</span>;<br/>
<a NAME="E2:18"/>
<font color="Maroon" title="c2">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">north_halfline</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E3:18"/><span class="kw">then </span>
<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="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<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> is  <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="jct_misc.html#T14" title="JCT_MISC:th.14">JCT_MISC:14</a>, <a class="ref" href="jordan2c.html#T12" title="JORDAN2C:th.12">JORDAN2C:12</a></span>;<br/>
<a NAME="E4:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A2</font></span>: 
<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="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<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> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D11" title="XXREAL_2:def.11">XXREAL_2:def 11</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:18"/><span class="lab"><font color="Green" title="E16">A3</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">C</font>
 ; <a class="txt" 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="c2">C</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">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="c2">C</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">north_halfline</a> <font color="Maroon" title="c1">p</font> &amp;  <a href="jordan18.html#K2" title="JORDAN18:func.2">South-Bound</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">C</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> &amp;  <a href="jordan18.html#K2" title="JORDAN18:func.2">South-Bound</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">C</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K6" title="TOPREAL1:func.6">south_halfline</a> <font color="Maroon" title="c1">p</font> )</span><br/>

<a NAME="E6:18"/><span class="kw">then </span>
 not  <a href="topreal1.html#K4" title="TOPREAL1:func.4">north_halfline</a> <font color="Maroon" title="c1">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c2">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan18.html#T11" target="_self" title="JORDAN18:th.11">Th12</a></span>;<br/>
<a NAME="E7:18"/><span class="kw">then </span>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">north_halfline</a> <font color="Maroon" title="c1">p</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c2">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#T129" title="JORDAN2C:th.129">JORDAN2C:129</a></span>;<br/>
<a NAME="E8:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A4</font></span>: 
 not <font color="Maroon" title="c2">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">north_halfline</a> <font color="Maroon" title="c1">p</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<a NAME="E9:18"/>
<font color="Maroon" title="c2">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">north_halfline</a> <font color="Maroon" title="c1">p</font></span>)</span> is  <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="jordan2c.html#T12" title="JORDAN2C:th.12">JORDAN2C:12</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E10:18"/><span class="kw">then </span>
<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="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<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> is  <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jct_misc.html#T13" title="JCT_MISC:th.13">JCT_MISC:13</a></span>;<br/>
<a NAME="E11:18"/><span class="kw">then </span>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">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="c2">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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E8:18"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E4:18"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="jordan18.html#E2"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a>, <a class="ref" href="rcomp_1.html#T13" title="RCOMP_1:th.13">RCOMP_1:13</a>, <a class="ref" href="relat_1.html#T119" title="RELAT_1:th.119">RELAT_1:119</a></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">set</a> <span class="kw"> such that </span><br/><a NAME="E13:18"/><span class="lab"><font color="Green" title="E18">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="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 <span class="kw">and </span><br/><a NAME="E14:18"/><span class="lab"><font color="Green" title="E19">A6</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<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 class="kw">and </span><br/><a NAME="E15:18"/><span class="lab"><font color="Green" title="E20">A7</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">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="c2">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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T64" title="FUNCT_2:th.64">FUNCT_2:64</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">x</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> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:18"><span class="lab"><font color="Green" title="E18">A5</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E21">A8</font></span>: <font color="Maroon" title="c5">x</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  = 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">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="c2">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:18"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="pscomp_1.html#D6" title="PSCOMP_1:def.6">PSCOMP_1:def 6</a></span>
<br/>.= 
<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="c2">C</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>
;<br/>
<a NAME="E18:18"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">north_halfline</a> <font color="Maroon" title="c1">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:18"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c5">x</font> <a href="euclid.html#K17" title="EUCLID:func.17">`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="topreal1.html#D10" title="TOPREAL1:def.10">TOPREAL1:def 10</a></span>
<br/>.= 
<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="c2">C</font>)</span>)</span> <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" title="EUCLID:th.52">EUCLID:52</a></span>
;<br/>
<a NAME="E20:18"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan18.html#K1" title="JORDAN18:func.1">North-Bound</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">C</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:18"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="ref" href="topreal3.html#T6" title="TOPREAL3:th.6">TOPREAL3:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:18"/>
(  <a href="jordan18.html#K1" title="JORDAN18:func.1">North-Bound</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">C</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">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="c2">C</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">north_halfline</a> <font color="Maroon" title="c1">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:18"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="jordan18.html#K2" title="JORDAN18:func.2">South-Bound</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">C</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> &amp;  <a href="jordan18.html#K2" title="JORDAN18:func.2">South-Bound</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">C</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K6" title="TOPREAL1:func.6">south_halfline</a> <font color="Maroon" title="c1">p</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">X</font> = <font color="Maroon" title="c2">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K6" title="TOPREAL1:func.6">south_halfline</a> <font color="Maroon" title="c1">p</font></span>)</span>;<br/>
<a NAME="E22:18"/>
<font color="Maroon" title="c2">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K6" title="TOPREAL1:func.6">south_halfline</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E23:18"/><span class="kw">then </span>
<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="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<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> is  <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="jct_misc.html#T14" title="JCT_MISC:th.14">JCT_MISC:14</a>, <a class="ref" href="jordan2c.html#T12" title="JORDAN2C:th.12">JORDAN2C:12</a></span>;<br/>
<a NAME="E24:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A9</font></span>: 
<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="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<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> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D11" title="XXREAL_2:def.11">XXREAL_2:def 11</a></span>;<br/>
<a NAME="E25:18"/>
 not  <a href="topreal1.html#K6" title="TOPREAL1:func.6">south_halfline</a> <font color="Maroon" title="c1">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c2">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:18"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="jordan18.html#T12" target="_self" title="JORDAN18:th.12">Th13</a></span>;<br/>
<a NAME="E26:18"/><span class="kw">then </span>
 <a href="topreal1.html#K6" title="TOPREAL1:func.6">south_halfline</a> <font color="Maroon" title="c1">p</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c2">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#T128" title="JORDAN2C:th.128">JORDAN2C:128</a></span>;<br/>
<a NAME="E27:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A10</font></span>: 
 not <font color="Maroon" title="c2">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K6" title="TOPREAL1:func.6">south_halfline</a> <font color="Maroon" title="c1">p</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<a NAME="E28:18"/>
<font color="Maroon" title="c2">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K6" title="TOPREAL1:func.6">south_halfline</a> <font color="Maroon" title="c1">p</font></span>)</span> is  <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="jordan2c.html#T12" title="JORDAN2C:th.12">JORDAN2C:12</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E29:18"/><span class="kw">then </span>
<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="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<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> is  <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jct_misc.html#T13" title="JCT_MISC:th.13">JCT_MISC:13</a></span>;<br/>
<a NAME="E30:18"/><span class="kw">then </span>
 <a href="seq_4.html#K4" title="SEQ_4:func.4">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="c2">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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E27:18"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="txt" href="#E24:18"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="txt" href="jordan18.html#E2"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a>, <a class="ref" href="rcomp_1.html#T12" title="RCOMP_1:th.12">RCOMP_1:12</a>, <a class="ref" href="relat_1.html#T119" title="RELAT_1:th.119">RELAT_1:119</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E32:18"/><span class="lab"><font color="Green" title="E24">A11</font></span>: 
<font color="Maroon" title="c7">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="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 <span class="kw">and </span><br/><a NAME="E33:18"/><span class="lab"><font color="Green" title="E25">A12</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<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 class="kw">and </span><br/><a NAME="E34:18"/><span class="lab"><font color="Green" title="E26">A13</font></span>: 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">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="c2">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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T64" title="FUNCT_2:th.64">FUNCT_2:64</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c7">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> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:18"><span class="lab"><font color="Green" title="E24">A11</font></span></a></span>;<br/>
<a NAME="E36:18"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K6" title="TOPREAL1:func.6">south_halfline</a> <font color="Maroon" title="c1">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:18"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E27">A14</font></span>: <font color="Maroon" title="c8">x</font> <a href="euclid.html#K17" title="EUCLID:func.17">`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="topreal1.html#D12" title="TOPREAL1:def.12">TOPREAL1:def 12</a></span>
<br/>.= 
<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="c2">C</font>)</span>)</span> <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" title="EUCLID:th.52">EUCLID:52</a></span>
;<br/>
<font color="Maroon" title="c8">x</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  = 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">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="c2">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="txt" href="#E34:18"><span class="lab"><font color="Green" title="E26">A13</font></span></a>, <a class="ref" href="pscomp_1.html#D6" title="PSCOMP_1:def.6">PSCOMP_1:def 6</a></span>
<br/>.= 
<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="c2">C</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>
;<br/>
<a NAME="E39:18"/><span class="kw">then </span>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan18.html#K2" title="JORDAN18:func.2">South-Bound</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">C</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:18"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="ref" href="topreal3.html#T6" title="TOPREAL3:th.6">TOPREAL3:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E40:18"/>
(  <a href="jordan18.html#K2" title="JORDAN18:func.2">South-Bound</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">C</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> &amp;  <a href="jordan18.html#K2" title="JORDAN18:func.2">South-Bound</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">C</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K6" title="TOPREAL1:func.6">south_halfline</a> <font color="Maroon" title="c1">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:18"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
