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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"> <span class="p1">(<span class="default"><a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">N-bound</a> <font color="Maroon" title="c1">C</font></span><br/>

<b>set </b><font color="Maroon" title="c2">w</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2;<br/>
<a NAME="E1:56"/><i><font color="Green" title="E38">A1</font></i>: 
(  <a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">W-bound</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">W-bound</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font></span>)</span> &amp;  <a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">E-bound</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">E-bound</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="jordan21.html#T26" target="_self" title="JORDAN21:th.26">Th26</a>, <a class="ref" href="jordan21.html#T27" target="_self" title="JORDAN21:th.27">Th27</a></i>;<br/>
<a NAME="E2:56"/><i><font color="Green" title="E39">A2</font></i>: 
not <a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="jordan21.html#T30" target="_self" title="JORDAN21:th.30">Th30</a></i>;<br/>
<a NAME="E3:56"/><i><font color="Green" title="E40">A3</font></i>: 
<a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> is <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a>
 
<b>by </b><i><a class="ref" href="jordan21.html#T13" target="_self" title="JORDAN21:th.13">Th13</a></i>;<br/>
<a NAME="E4:56"/>
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="jordan6.html#T76" title="JORDAN6:th.76">JORDAN6:76</a></i>;<br/>
<a NAME="E5:56"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></i>;<br/>
<a NAME="E6:56"/><b>then </b><i><font color="Green" title="E41">A4</font></i>: 
 <a href="pscomp_1.html#NK2" title="PSCOMP_1:NK.2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#NK2" title="PSCOMP_1:NK.2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:56"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="#E3:56"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a>, <a class="ref" href="seq_4.html#T65" title="SEQ_4:th.65">SEQ_4:65</a></i>;<br/>
<a NAME="E7:56"/><i><font color="Green" title="E42">A5</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#NK2" title="PSCOMP_1:NK.2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:56"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E8:56"/><i><font color="Green" title="E43">A6</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#NK2" title="PSCOMP_1:NK.2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E9:56"/>
<span class="p1">(<span class="default"><a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">N-bound</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="jordan21.html#T52" target="_self" title="JORDAN21:th.52">Th52</a></i>;<br/>
<b>hence </b><a NAME="E10:56"/>
<span class="p1">(<span class="default"><a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">N-bound</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="#E6:56"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="#E7:56"><i><font color="Green" title="E42">A5</font></i></a>, <a class="txt" href="#E8:56"><i><font color="Green" title="E43">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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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