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<b>let </b><font color="Maroon" title="c22">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"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="jordan24.html#R1" title="JORDAN24:pred.1">realize-max-dist-in</a> <font color="Maroon" title="c22">C</font> implies for <font color="Olive" title="b1">Jc</font>, <font color="Olive" title="b2">Jd</font> being <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> <a href="jordan21.html#V1" title="JORDAN21:attr.1">with_the_max_arc</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="Olive" title="b1">Jc</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Olive" title="b2">Jd</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c22">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Jc</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">Jd</font> &amp; <font color="Olive" title="b1">Jc</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">Jd</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp;  <a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c22">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Jc</font> &amp;  <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c22">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">Jd</font> &amp;  <a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">W-bound</a> <font color="Maroon" title="c22">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> <font color="Olive" title="b1">Jc</font> &amp;  <a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">E-bound</a> <font color="Maroon" title="c22">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> <font color="Olive" title="b1">Jc</font> holds <br/> <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c22">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K2" title="CONNSP_3:func.2">Down</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Olive" title="b1">Jc</font></span>)</span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Olive" title="b1">Jc</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:148"/><i><font color="Green" title="E171">A1</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="jordan24.html#R1" title="JORDAN24:pred.1">realize-max-dist-in</a> <font color="Maroon" title="c22">C</font>
 ; <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">Jc</font>, <font color="Olive" title="b2">Jd</font> being <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> <a href="jordan21.html#V1" title="JORDAN21:attr.1">with_the_max_arc</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="Olive" title="b1">Jc</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Olive" title="b2">Jd</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c22">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Jc</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">Jd</font> &amp; <font color="Olive" title="b1">Jc</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">Jd</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp;  <a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c22">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Jc</font> &amp;  <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c22">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">Jd</font> &amp;  <a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">W-bound</a> <font color="Maroon" title="c22">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> <font color="Olive" title="b1">Jc</font> &amp;  <a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">E-bound</a> <font color="Maroon" title="c22">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> <font color="Olive" title="b1">Jc</font> holds <br/> <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c22">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K2" title="CONNSP_3:func.2">Down</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Olive" title="b1">Jc</font></span>)</span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Olive" title="b1">Jc</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span></span><br/>

<b>let </b><font color="Maroon" title="c23">Jc</font>, <font color="Maroon" title="c24">Jd</font> be  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> <a href="jordan21.html#V1" title="JORDAN21:attr.1">with_the_max_arc</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c23">Jc</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c24">Jd</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c22">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">Jc</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c24">Jd</font> &amp; <font color="Maroon" title="c23">Jc</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c24">Jd</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp;  <a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c22">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">Jc</font> &amp;  <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c22">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">Jd</font> &amp;  <a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">W-bound</a> <font color="Maroon" title="c22">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> <font color="Maroon" title="c23">Jc</font> &amp;  <a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">E-bound</a> <font color="Maroon" title="c22">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> <font color="Maroon" title="c23">Jc</font> implies  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c22">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K2" title="CONNSP_3:func.2">Down</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c23">Jc</font></span>)</span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c24">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c23">Jc</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:148"/><i><font color="Green" title="E172">A2</font></i>: 
<font color="Maroon" title="c23">Jc</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>and </b><br/><a NAME="E3:148"/><i><font color="Green" title="E173">A3</font></i>: 
<font color="Maroon" title="c24">Jd</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>and </b><br/><a NAME="E4:148"/><i><font color="Green" title="E174">A4</font></i>: 
<font color="Maroon" title="c22">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">Jc</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c24">Jd</font>
 <b>and </b><br/><a NAME="E5:148"/><i><font color="Green" title="E175">A5</font></i>: 
<font color="Maroon" title="c23">Jc</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c24">Jd</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <b>and </b><br/><a NAME="E6:148"/><i><font color="Green" title="E176">A6</font></i>: 
 <a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c22">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">Jc</font>
 <b>and </b><br/><a NAME="E7:148"/><i><font color="Green" title="E177">A7</font></i>: 
 <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c22">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">Jd</font>
 <b>and </b><br/><a NAME="E8:148"/><i><font color="Green" title="E178">A8</font></i>: 
 <a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">W-bound</a> <font color="Maroon" title="c22">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> <font color="Maroon" title="c23">Jc</font>
 <b>and </b><br/><a NAME="E9:148"/><i><font color="Green" title="E179">A9</font></i>: 
 <a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">E-bound</a> <font color="Maroon" title="c22">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> <font color="Maroon" title="c23">Jc</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c22">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K2" title="CONNSP_3:func.2">Down</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c23">Jc</font></span>)</span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c24">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c23">Jc</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c25">Ux</font> =  <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K2" title="CONNSP_3:func.2">Down</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c23">Jc</font></span>)</span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c24">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c23">Jc</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> as   <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> <b>by </b><i><a class="ref" href="pre_topc.html#T39" title="PRE_TOPC:th.39">PRE_TOPC:39</a></i>;<br/>
<a NAME="E11:148"/>
<font color="Maroon" title="c25">Ux</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c22">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="148_1"><b>proof </b></a><div class="add">

<a NAME="E1:148_1"/>
<font color="Maroon" title="c25">Ux</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c22">C</font>
 
<b>by </b><i><a class="txt" href="#E1:148"><i><font color="Green" title="E171">A1</font></i></a>, <a class="txt" href="#E2:148"><i><font color="Green" title="E172">A2</font></i></a>, <a class="txt" href="#E3:148"><i><font color="Green" title="E173">A3</font></i></a>, <a class="txt" href="#E4:148"><i><font color="Green" title="E174">A4</font></i></a>, <a class="txt" href="#E5:148"><i><font color="Green" title="E175">A5</font></i></a>, <a class="txt" href="#E6:148"><i><font color="Green" title="E176">A6</font></i></a>, <a class="txt" href="#E7:148"><i><font color="Green" title="E177">A7</font></i></a>, <a class="txt" href="#E8:148"><i><font color="Green" title="E178">A8</font></i></a>, <a class="txt" href="#E9:148"><i><font color="Green" title="E179">A9</font></i></a>, <a class="ref" href="jordan.html#T95" target="_self" title="JORDAN:th.95">Th95</a></i>;<br/>
<b>hence </b><a NAME="E2:148_1"/>
<font color="Maroon" title="c25">Ux</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c22">C</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T26" title="JORDAN2C:th.26">JORDAN2C:26</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c22">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c25">Ux</font></span><br/>

<b>set </b><font color="Maroon" title="c26">F</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">B</font> where <font color="Olive" title="b1">B</font> is   <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> : <font color="Olive" title="b1">B</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c22">C</font> </span>}</span> ;<br/>
<b>let </b><font color="Maroon" title="c27">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c27">q</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="c22">C</font> or <font color="Maroon" title="c27">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">Ux</font> )</span><br/>

<b>assume </b><a NAME="E3:148_1"/>
<font color="Maroon" title="c27">q</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="c22">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c27">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">Ux</font></span><br/>

<b>then consider </b><font color="Maroon" title="c28">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:148_1"/><i><font color="Green" title="E180">A10</font></i>: 
<font color="Maroon" title="c27">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">Z</font>
 <b>and </b><br/><a NAME="E6:148_1"/><i><font color="Green" title="E181">A11</font></i>: 
<font color="Maroon" title="c28">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">B</font> where <font color="Olive" title="b1">B</font> is   <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> : <font color="Olive" title="b1">B</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c22">C</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E7:148_1"/>
 ex <font color="Olive" title="b1">B</font> being  <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 <br/>( <font color="Maroon" title="c28">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> &amp; <font color="Olive" title="b1">B</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c22">C</font> )
 
<b>by </b><i><a class="txt" href="#E6:148_1"><i><font color="Green" title="E181">A11</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:148_1"/>
<font color="Maroon" title="c27">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">Ux</font>
 <b>by </b><i><a class="txt" href="#E1:148"><i><font color="Green" title="E171">A1</font></i></a>, <a class="txt" href="#E2:148"><i><font color="Green" title="E172">A2</font></i></a>, <a class="txt" href="#E3:148"><i><font color="Green" title="E173">A3</font></i></a>, <a class="txt" href="#E4:148"><i><font color="Green" title="E174">A4</font></i></a>, <a class="txt" href="#E5:148"><i><font color="Green" title="E175">A5</font></i></a>, <a class="txt" href="#E6:148"><i><font color="Green" title="E176">A6</font></i></a>, <a class="txt" href="#E7:148"><i><font color="Green" title="E177">A7</font></i></a>, <a class="txt" href="#E8:148"><i><font color="Green" title="E178">A8</font></i></a>, <a class="txt" href="#E9:148"><i><font color="Green" title="E179">A9</font></i></a>, <a class="txt" href="#E5:148_1"><i><font color="Green" title="E180">A10</font></i></a>, <a class="ref" href="jordan.html#T95" target="_self" title="JORDAN:th.95">Th95</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:148"/>
 <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c22">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K2" title="CONNSP_3:func.2">Down</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c23">Jc</font></span>)</span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c24">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c23">Jc</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
