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

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">C</font> be  <a href="compts_1.html#V6">compact</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>



<b>assume </b><a NAME="E1:18"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">C</font>
 ;<br/>

<a NAME="E2:18"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">C</font> is <a href="jordan2c.html#V1">Bounded</a>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T73">JORDAN2C:73</a></i>;<br/>
<b>set </b><font color="Maroon">X</font> = <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K6">north_halfline</a> <font color="Maroon">p</font></span>)</span>;<br/>
<a NAME="E3:18"/>
not  <a href="topreal1.html#K6">north_halfline</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">C</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:18"/><b>then </b>
 <a href="topreal1.html#K6">north_halfline</a> <font color="Maroon">p</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T137">JORDAN2C:137</a></i>;<br/>
<a NAME="E5:18"/><b>then </b>
not <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K6">north_halfline</a> <font color="Maroon">p</font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E6:18"/><b>then </b><i><font color="Green">E57</font></i>: 
<a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K6">north_halfline</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T152">RELAT_1:152</a></i>;<br/>
<a NAME="E7:18"/>
<font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K6">north_halfline</a> <font color="Maroon">p</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E8:18"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K6">north_halfline</a> <font color="Maroon">p</font></span>)</span> is <a href="jordan2c.html#V1">Bounded</a>
 
<b>by </b><i>, <a class="ref" href="jordan2c.html#T16">JORDAN2C:16</a></i>;<br/>
<a NAME="E9:18"/>
<font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K6">north_halfline</a> <font color="Maroon">p</font></span>)</span> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T35">TOPS_1:35</a></i>;<br/>
<a NAME="E10:18"/><b>then </b><i><font color="Green">E67</font></i>: 
<a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K6">north_halfline</a> <font color="Maroon">p</font></span>)</span></span>)</span> is <a href="rcomp_1.html#V2">closed</a>
 
<b>by </b><i>, <a class="ref" href="jct_misc.html#T22">JCT_MISC:22</a></i>;<br/>
<a NAME="E11:18"/>
<a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K6">north_halfline</a> <font color="Maroon">p</font></span>)</span></span>)</span> is <a href="seq_4.html#V3">bounded</a>
 
<b>by </b><i>, <a class="ref" href="jct_misc.html#T23">JCT_MISC:23</a></i>;<br/>
<a NAME="E12:18"/><b>then </b>
<a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K6">north_halfline</a> <font color="Maroon">p</font></span>)</span></span>)</span> is <a href="seq_4.html#V2">bounded_below</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#D3">SEQ_4:def 3</a></i>;<br/>
<a NAME="E13:18"/><b>then </b>
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="topreal1.html#K6">north_halfline</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K6">north_halfline</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="rcomp_1.html#T31">RCOMP_1:31</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:18"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 <b>and </b><br/><a NAME="E16:18"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K6">north_halfline</a> <font color="Maroon">p</font></span>)</span>
 <b>and </b><br/><a NAME="E17:18"/><i><font color="Green">E72</font></i>: 
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="topreal1.html#K6">north_halfline</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i/>;<br/>
<a NAME="E19:18"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K6">north_halfline</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then </b><i><font color="Green">E73</font></i>: <font color="Maroon">x</font> <a href="euclid.html#K21">`1</a>  = 
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="ref" href="topreal1.html#D12">TOPREAL1:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan18.html#K1" target="_self">North-Bound</a> <font color="Maroon">p</font>,<font color="Maroon">C</font></span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i/>
;<br/>
<font color="Maroon">x</font> <a href="euclid.html#K22">`2</a>  = 
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="topreal1.html#K6">north_halfline</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan18.html#K1" target="_self">North-Bound</a> <font color="Maroon">p</font>,<font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i/>
;<br/>
<a NAME="E22:18"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="jordan18.html#K1" target="_self">North-Bound</a> <font color="Maroon">p</font>,<font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="topreal3.html#T11">TOPREAL3:11</a></i>;<br/>
<b>hence </b><a NAME="E23:18"/>
(  <a href="jordan18.html#K1" target="_self">North-Bound</a> <font color="Maroon">p</font>,<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp;  <a href="jordan18.html#K1" target="_self">North-Bound</a> <font color="Maroon">p</font>,<font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K6">north_halfline</a> <font color="Maroon">p</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>

<b>set </b><font color="Maroon">X</font> = <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K8">south_halfline</a> <font color="Maroon">p</font></span>)</span>;<br/>
<a NAME="E24:18"/>
not  <a href="topreal1.html#K8">south_halfline</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">C</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E25:18"/><b>then </b>
 <a href="topreal1.html#K8">south_halfline</a> <font color="Maroon">p</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T136">JORDAN2C:136</a></i>;<br/>
<a NAME="E26:18"/><b>then </b>
not <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K8">south_halfline</a> <font color="Maroon">p</font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E27:18"/><b>then </b><i><font color="Green">E74</font></i>: 
<a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K8">south_halfline</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T152">RELAT_1:152</a></i>;<br/>
<a NAME="E28:18"/>
<font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K8">south_halfline</a> <font color="Maroon">p</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E29:18"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K8">south_halfline</a> <font color="Maroon">p</font></span>)</span> is <a href="jordan2c.html#V1">Bounded</a>
 
<b>by </b><i>, <a class="ref" href="jordan2c.html#T16">JORDAN2C:16</a></i>;<br/>
<a NAME="E30:18"/>
<font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K8">south_halfline</a> <font color="Maroon">p</font></span>)</span> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T35">TOPS_1:35</a></i>;<br/>
<a NAME="E31:18"/><b>then </b><i><font color="Green">E76</font></i>: 
<a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K8">south_halfline</a> <font color="Maroon">p</font></span>)</span></span>)</span> is <a href="rcomp_1.html#V2">closed</a>
 
<b>by </b><i>, <a class="ref" href="jct_misc.html#T22">JCT_MISC:22</a></i>;<br/>
<a NAME="E32:18"/>
<a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K8">south_halfline</a> <font color="Maroon">p</font></span>)</span></span>)</span> is <a href="seq_4.html#V3">bounded</a>
 
<b>by </b><i>, <a class="ref" href="jct_misc.html#T23">JCT_MISC:23</a></i>;<br/>
<a NAME="E33:18"/><b>then </b>
<a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K8">south_halfline</a> <font color="Maroon">p</font></span>)</span></span>)</span> is <a href="seq_4.html#V1">bounded_above</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#D3">SEQ_4:def 3</a></i>;<br/>
<a NAME="E34:18"/><b>then </b>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="topreal1.html#K8">south_halfline</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K8">south_halfline</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="rcomp_1.html#T30">RCOMP_1:30</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E36:18"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 <b>and </b><br/><a NAME="E37:18"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K8">south_halfline</a> <font color="Maroon">p</font></span>)</span>
 <b>and </b><br/><a NAME="E38:18"/><i><font color="Green">E79</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="topreal1.html#K8">south_halfline</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i/>;<br/>
<a NAME="E40:18"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K8">south_halfline</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then </b><i><font color="Green">E80</font></i>: <font color="Maroon">x</font> <a href="euclid.html#K21">`1</a>  = 
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="ref" href="topreal1.html#D14">TOPREAL1:def 14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan18.html#K2" target="_self">South-Bound</a> <font color="Maroon">p</font>,<font color="Maroon">C</font></span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i/>
;<br/>
<font color="Maroon">x</font> <a href="euclid.html#K22">`2</a>  = 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="topreal1.html#K8">south_halfline</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan18.html#K2" target="_self">South-Bound</a> <font color="Maroon">p</font>,<font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i/>
;<br/>
<a NAME="E43:18"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="jordan18.html#K2" target="_self">South-Bound</a> <font color="Maroon">p</font>,<font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="topreal3.html#T11">TOPREAL3:11</a></i>;<br/>
<b>hence </b><a NAME="E44:18"/>
(  <a href="jordan18.html#K2" target="_self">South-Bound</a> <font color="Maroon">p</font>,<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp;  <a href="jordan18.html#K2" target="_self">South-Bound</a> <font color="Maroon">p</font>,<font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K8">south_halfline</a> <font color="Maroon">p</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div>
