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<b>let </b><font color="Maroon">C</font> be  <a href="topreal2.html#V1">being_simple_closed_curve</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>set </b><font color="Maroon">s</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2;<br/>
<b>set </b><font color="Maroon">e</font> = <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">f</font> = <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1;<br/>
<a NAME="E1:2"/><i><font color="Green">E43</font></i>: 
 <a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1 <a href="hidden.html#R1">=</a>  <a href="jordan1a.html#K1">Center</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="jordan1b.html#T17">JORDAN1B:17</a></i>;<br/>
<a NAME="E2:2"/><b>then </b>
 <a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="jordan1a.html#D1">JORDAN1A:def 1</a></i>;<br/>
<a NAME="E3:2"/><b>then </b><i><font color="Green">E44</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E4:2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 4
 
<b>by </b><i><a class="ref" href="jordan8.html#T13">JORDAN8:13</a></i>;<br/>
<a NAME="E5:2"/><b>then </b><i><font color="Green">E45</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:2"/><b>then </b><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i>, <a class="ref" href="jordan1a.html#T59">JORDAN1A:59</a></i>;<br/>
<a NAME="E7:2"/><b>then </b><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1 <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K6">Vertical_Line</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="jordan1a.html#T17">JORDAN1A:17</a></i>;<br/>
<a NAME="E8:2"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i>, , <a class="ref" href="jordan1a.html#T59">JORDAN1A:59</a></i>;<br/>
<a NAME="E9:2"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K6">Vertical_Line</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="jordan1a.html#T17">JORDAN1A:17</a></i>;<br/>
<a NAME="E10:2"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="jordan6.html#K6">Vertical_Line</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i>, <a class="ref" href="jordan1a.html#T23">JORDAN1A:23</a></i>;<br/>
<a NAME="E11:2"/><i><font color="Green">E50</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="jordan8.html#D1">JORDAN8:def 1</a></i>;<br/>
<a NAME="E12:2"/>
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="jordan8.html#T13">JORDAN8:13</a></i>;<br/>
<a NAME="E13:2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span> <a href="int_1.html#K5">div</a> 2 <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="int_1.html#T83">INT_1:83</a></i>;<br/>
<a NAME="E14:2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>;<br/>
<a NAME="E15:2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E16:2"/><b>then </b>
 <a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="jordan1a.html#D1">JORDAN1A:def 1</a></i>;<br/>
<a NAME="E17:2"/><b>then </b><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i>, , , <a class="ref" href="goboard5.html#T5">GOBOARD5:5</a></i>;<br/>
<a NAME="E18:2"/><i><font color="Green">E55</font></i>: 
( <a href="pscomp_1.html#K14">proj2</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <a href="pscomp_1.html#K14">proj2</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</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"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p0">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon">Es</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E20:2"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">Es</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of 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> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Es</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span><a href="euclid.html#K23">]|</a></span>;<br/>
<b>consider </b><font color="Maroon">E</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</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> such that </b><br/><a NAME="E22:2"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">i</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</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"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon">Fs</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E24:2"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">Fs</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">i</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of 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> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span><a href="euclid.html#K23">]|</a></span>;<br/>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</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> such that </b><br/><a NAME="E26:2"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">i</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<b>set </b><font color="Maroon">e1</font> = <a href="pscomp_1.html#K14">proj2</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">f1</font> = <a href="pscomp_1.html#K14">proj2</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>;<br/>
<a NAME="E27:2"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>set </b><font color="Maroon">p</font> = <font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font>;<br/>
<a NAME="E1:2_1"/><i><font color="Green">E62</font></i>: 
 <a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="jordan1a.html#K1">Center</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan1b.html#T17">JORDAN1B:17</a></i>;<br/>
<a NAME="E2:2_1"/><b>then </b><i><font color="Green">E63</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#R2">meets</a>  <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan1b.html#T34">JORDAN1B:34</a></i>;<br/>
<a NAME="E3:2_1"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E4:2_1"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="jordan1a.html#T65">JORDAN1A:65</a></i>;<br/>
<a NAME="E5:2_1"/><b>then </b><i><font color="Green">E64</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#R2">meets</a>  <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E6:2_1"/>
<font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Es</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E7:2_1"/><b>then </b><i><font color="Green">E65</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 &amp; <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">Es</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> )
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<b>reconsider </b><font color="Maroon">DD</font> = <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> as  <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> <b>by </b><i><a class="ref" href="pscomp_1.html#T64">PSCOMP_1:64</a></i>;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <font color="Maroon">DD</font> as  <a href="rcomp_1.html#V1">compact</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="jct_misc.html#T24">JCT_MISC:24</a></i>;<br/>
<a NAME="E10:2_1"/>
<font color="Maroon">DD</font> <a href="tarski.html#R1">c=</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>
 
;<br/>
<a NAME="E11:2_1"/><b>then </b>
<font color="Maroon">DD</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <a href="pscomp_1.html#K14">proj2</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E12:2_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <a href="pscomp_1.html#K14">proj2</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">DD</font> <a href="hidden.html#R1">=</a> <font color="Maroon">DD</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E13:2_1"/>
<font color="Maroon">DD</font> <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="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E14:2_1"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">DD</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E15:2_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">D</font> <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#T151">RELAT_1:151</a></i>;<br/>
<a NAME="E16:2_1"/>
<font color="Maroon">Es</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">D</font>
 
<b>by </b><i/>;<br/>
<a NAME="E17:2_1"/><b>then </b>
<font color="Maroon">Es</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T32">RCOMP_1:32</a></i>;<br/>
<b>then consider </b><font color="Maroon">dd</font> being   <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> such that </b><br/><a NAME="E19:2_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">dd</font> <a href="hidden.html#R2">in</a> <font color="Maroon">DD</font>
 <b>and </b><br/><a NAME="E20:2_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">Es</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">dd</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E21:2_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">dd</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E22:2_1"/>
<font color="Maroon">dd</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E23:2_1"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">dd</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i>, , <a class="ref" href="jordan6.html#T34">JORDAN6:34</a></i>;<br/>
<a NAME="E24:2_1"/>
<font color="Maroon">dd</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i>, , <a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>;<br/>
<b>hence </b><a NAME="E25:2_1"/>
<font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="topreal3.html#T11">TOPREAL3:11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E28:2"/><i><font color="Green">E88</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>set </b><font color="Maroon">p</font> = <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font>;<br/>
<a NAME="E1:2_2"/><i><font color="Green">E89</font></i>: 
 <a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="jordan1a.html#K1">Center</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan1b.html#T17">JORDAN1B:17</a></i>;<br/>
<a NAME="E2:2_2"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Es</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:2_2"/><b>then </b><i><font color="Green">E91</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<font color="Maroon">Es</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
 <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"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p0">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:2_2"/><i><font color="Green">E92</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E7:2_2"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E8:2_2"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">j</font>
 <b>by </b><i>, , , , <a class="ref" href="jordan1f.html#T13">JORDAN1F:13</a></i>;<br/>
<a NAME="E9:2_2"/>
<font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E10:2_2"/><b>then </b><i><font color="Green">E95</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="subset_1.html#R2">meets</a>  <a href="jordan6.html#K9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, , , , , <a class="ref" href="jordan1j.html#T62">JORDAN1J:62</a></i>;<br/>
<a NAME="E11:2_2"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E12:2_2"/><b>then </b><i><font color="Green">E96</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">Fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> )
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<b>reconsider </b><font color="Maroon">DD</font> = <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> as  <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> <b>by </b><i><a class="ref" href="pscomp_1.html#T64">PSCOMP_1:64</a></i>;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <font color="Maroon">DD</font> as  <a href="rcomp_1.html#V1">compact</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="jct_misc.html#T24">JCT_MISC:24</a></i>;<br/>
<a NAME="E15:2_2"/>
<font color="Maroon">DD</font> <a href="tarski.html#R1">c=</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>
 
;<br/>
<a NAME="E16:2_2"/><b>then </b>
<font color="Maroon">DD</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <a href="pscomp_1.html#K14">proj2</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E17:2_2"/><b>then </b><i><font color="Green">E97</font></i>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <a href="pscomp_1.html#K14">proj2</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">DD</font> <a href="hidden.html#R1">=</a> <font color="Maroon">DD</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E18:2_2"/>
<font color="Maroon">DD</font> <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="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E19:2_2"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">DD</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E20:2_2"/><b>then </b><i><font color="Green">E98</font></i>: 
<font color="Maroon">D</font> <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#T151">RELAT_1:151</a></i>;<br/>
<a NAME="E21:2_2"/>
<font color="Maroon">Fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">D</font>
 
<b>by </b><i/>;<br/>
<a NAME="E22:2_2"/><b>then </b>
<font color="Maroon">Fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T32">RCOMP_1:32</a></i>;<br/>
<b>then consider </b><font color="Maroon">dd</font> being   <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> such that </b><br/><a NAME="E24:2_2"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">dd</font> <a href="hidden.html#R2">in</a> <font color="Maroon">DD</font>
 <b>and </b><br/><a NAME="E25:2_2"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">Fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">dd</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E26:2_2"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">dd</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E27:2_2"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">dd</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E28:2_2"/>
<font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K6">Vertical_Line</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i>, <a class="ref" href="jordan1a.html#T17">JORDAN1A:17</a></i>;<br/>
<a NAME="E29:2_2"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="jordan6.html#K6">Vertical_Line</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i>, <a class="ref" href="jordan1a.html#T23">JORDAN1A:23</a></i>;<br/>
<a NAME="E30:2_2"/><b>then </b><i><font color="Green">E103</font></i>: 
<font color="Maroon">dd</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i>, , <a class="ref" href="jordan6.html#T34">JORDAN6:34</a></i>;<br/>
<a NAME="E31:2_2"/>
<font color="Maroon">dd</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i>, , <a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>;<br/>
<b>hence </b><a NAME="E32:2_2"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="topreal3.html#T11">TOPREAL3:11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E29:2"/><i><font color="Green">E104</font></i>: 
<a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p4">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p4">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, , <a class="ref" href="sprect_1.html#T61">SPRECT_1:61</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>5</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K1">REAL</a>  = <a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon">S</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E31:2"/><i><font color="Green">E105</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">i</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<b>set </b><font color="Maroon">AA</font> = <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">BB</font> = <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font></span>)</span>;<br/>
<a NAME="E32:2"/>
( <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span> is <a href="compts_1.html#V6">compact</a> &amp; <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font></span>)</span> is <a href="compts_1.html#V6">compact</a> )
 
<b>by </b><i><a class="ref" href="compts_1.html#T20">COMPTS_1:20</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">A</font> = <a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p4">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p4">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span></span>)</span>, <font color="Maroon">B</font> = <a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p4">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p4">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font></span>)</span></span>)</span> as  <a href="rcomp_1.html#V1">compact</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="jct_misc.html#T24">JCT_MISC:24</a></i>;<br/>
<a NAME="E34:2"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_3"/>
<font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:2_3"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E4:2_3"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">p</font> being   <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> such that </b><br/><a NAME="E7:2_3"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>
 <b>and </b><br/><a NAME="E8:2_3"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<b>consider </b><font color="Maroon">q</font> being   <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> such that </b><br/><a NAME="E10:2_3"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font></span>)</span>
 <b>and </b><br/><a NAME="E11:2_3"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">q</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E12:2_3"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E13:2_3"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>
 
<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">E119</font></i>: <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
<b>by </b><i>, <a class="ref" href="jordan6.html#T34">JORDAN6:34</a></i>
<br/>.= 
<font color="Maroon">q</font> <a href="euclid.html#K21">`1</a> 
<b>by </b><i>, , <a class="ref" href="jordan6.html#T34">JORDAN6:34</a></i>
;<br/>
<font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  = 
<a href="pscomp_1.html#K14">proj2</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">q</font>
<b>by </b><i>, , <a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>
<br/>.= 
<font color="Maroon">q</font> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>
;<br/>
<a NAME="E16:2_3"/><b>then </b><i><font color="Green">E120</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 
<b>by </b><i>, <a class="ref" href="topreal3.html#T11">TOPREAL3:11</a></i>;<br/>
<a NAME="E17:2_3"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E18:2_3"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E19:2_3"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E20:2_3"/><b>then </b><i><font color="Green">E122</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="jordan6.html#T65">JORDAN6:65</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_3_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_3_3"/>
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font> or <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font> )
 </span><b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_3_3_1"><b>suppose </b></a><a NAME="E1:2_3_3_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><a NAME="E2:2_3_3_1"/><b>then </b><i><font color="Green">E123</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i>, , <a class="ref" href="jordan6.html#T34">JORDAN6:34</a></i>;<br/><a NAME="E3:2_3_3_1"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E4:2_3_3_1"/>
contradiction
 <b>by </b><i><a class="ref" href="sprect_1.html#T33">SPRECT_1:33</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_3_3_2"><b>suppose </b></a><a NAME="E1:2_3_3_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><a NAME="E2:2_3_3_2"/><b>then </b><i><font color="Green">E124</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i>, , <a class="ref" href="jordan6.html#T34">JORDAN6:34</a></i>;<br/><a NAME="E3:2_3_3_2"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E4:2_3_3_2"/>
contradiction
 <b>by </b><i><a class="ref" href="sprect_1.html#T33">SPRECT_1:33</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E35:2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E36:2"/><b>then </b><i><font color="Green">E125</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p4">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p4">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<a NAME="E37:2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E38:2"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p4">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p4">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<a NAME="E39:2"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> is <a href="jct_misc.html#V1">connected</a> &amp; <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font> &amp; <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:2_4"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:2_4"/>
<font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> is <a href="jct_misc.html#V1">connected</a>
 <b>by </b><i><a class="ref" href="jct_misc.html#T15">JCT_MISC:15</a></i>;<br/>

<b>reconsider </b><font color="Maroon">DD</font> = <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> as  <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> <b>by </b><i><a class="ref" href="pscomp_1.html#T64">PSCOMP_1:64</a></i>;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <font color="Maroon">DD</font> as  <a href="rcomp_1.html#V1">compact</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="jct_misc.html#T24">JCT_MISC:24</a></i>;<br/>
<a NAME="E5:2_4"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Es</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:2_4"/><b>then </b><i><font color="Green">E129</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E7:2_4"/><i><font color="Green">E130</font></i>: 
 <a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="jordan1a.html#K1">Center</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan1b.html#T17">JORDAN1B:17</a></i>;<br/>
<a NAME="E8:2_4"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E9:2_4"/><b>then </b><i><font color="Green">E132</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E10:2_4"/><i><font color="Green">E133</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#R2">meets</a>  <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan1b.html#T34">JORDAN1B:34</a></i>;<br/>
<a NAME="E11:2_4"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E12:2_4"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span>,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan1a.html#T65">JORDAN1A:65</a></i>;<br/>
<a NAME="E13:2_4"/><b>then </b><i><font color="Green">E134</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#R2">meets</a>  <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E14:2_4"/>
<font color="Maroon">DD</font> <a href="tarski.html#R1">c=</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>
 
;<br/>
<a NAME="E15:2_4"/><b>then </b>
<font color="Maroon">DD</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <a href="pscomp_1.html#K14">proj2</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E16:2_4"/><b>then </b><i><font color="Green">E135</font></i>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <a href="pscomp_1.html#K14">proj2</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">DD</font> <a href="hidden.html#R1">=</a> <font color="Maroon">DD</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E17:2_4"/>
<font color="Maroon">DD</font> <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="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E18:2_4"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">DD</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E19:2_4"/><b>then </b><i><font color="Green">E136</font></i>: 
<font color="Maroon">D</font> <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#T151">RELAT_1:151</a></i>;<br/>
<a NAME="E20:2_4"/>
<font color="Maroon">Es</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">D</font>
 
<b>by </b><i/>;<br/>
<a NAME="E21:2_4"/><b>then </b>
<font color="Maroon">Es</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T32">RCOMP_1:32</a></i>;<br/>
<b>then consider </b><font color="Maroon">dd</font> being   <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> such that </b><br/><a NAME="E23:2_4"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">dd</font> <a href="hidden.html#R2">in</a> <font color="Maroon">DD</font>
 <b>and </b><br/><a NAME="E24:2_4"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">Es</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">dd</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E25:2_4"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">dd</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E26:2_4"/><b>then </b><i><font color="Green">E140</font></i>: 
<font color="Maroon">dd</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i>, , <a class="ref" href="jordan6.html#T34">JORDAN6:34</a></i>;<br/>
<i><font color="Green">E141</font></i>: <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<font color="Maroon">Es</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<font color="Maroon">dd</font> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, <a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>
;<br/>
<a NAME="E28:2_4"/><b>then </b><i><font color="Green">E142</font></i>: 
<font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>
 
<b>by </b><i>, , <a class="ref" href="topreal3.html#T11">TOPREAL3:11</a></i>;<br/>
<a NAME="E29:2_4"/>
<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1 <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E30:2_4"/><b>then </b><i><font color="Green">E143</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="topreal1.html#T12">TOPREAL1:12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">DD</font> = <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> as  <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> <b>by </b><i><a class="ref" href="pscomp_1.html#T64">PSCOMP_1:64</a></i>;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <font color="Maroon">DD</font> as  <a href="rcomp_1.html#V1">compact</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="jct_misc.html#T24">JCT_MISC:24</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<font color="Maroon">Es</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
 <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"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p0">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E35:2_4"/><i><font color="Green">E144</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E36:2_4"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E37:2_4"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">j</font>
 <b>by </b><i>, , , , <a class="ref" href="jordan1f.html#T13">JORDAN1F:13</a></i>;<br/>
<a NAME="E38:2_4"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E39:2_4"/><b>then </b><i><font color="Green">E148</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="subset_1.html#R2">meets</a>  <a href="jordan6.html#K9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, , , <a class="txt" href="jordan11.html#E1"><i><font color="Green">Lemma42</font></i></a>, <a class="txt" href="jordan11.html#E1:2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="jordan1j.html#T62">JORDAN1J:62</a></i>;<br/>
<a NAME="E40:2_4"/>
<font color="Maroon">DD</font> <a href="tarski.html#R1">c=</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>
 
;<br/>
<a NAME="E41:2_4"/><b>then </b>
<font color="Maroon">DD</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <a href="pscomp_1.html#K14">proj2</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E42:2_4"/><b>then </b><i><font color="Green">E149</font></i>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <a href="pscomp_1.html#K14">proj2</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">DD</font> <a href="hidden.html#R1">=</a> <font color="Maroon">DD</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E43:2_4"/>
<font color="Maroon">DD</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="jordan11.html#E5:2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E44:2_4"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">DD</font>
 
<b>by </b><i><a class="txt" href="jordan11.html#E6:2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E45:2_4"/><b>then </b><i><font color="Green">E150</font></i>: 
<font color="Maroon">D</font> <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#T151">RELAT_1:151</a></i>;<br/>
<a NAME="E46:2_4"/>
<font color="Maroon">Fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">D</font>
 
<b>by </b><i/>;<br/>
<a NAME="E47:2_4"/><b>then </b>
<font color="Maroon">Fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="txt" href="jordan11.html#E7:2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="rcomp_1.html#T32">RCOMP_1:32</a></i>;<br/>
<b>then consider </b><font color="Maroon">dd</font> being   <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> such that </b><br/><a NAME="E49:2_4"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">dd</font> <a href="hidden.html#R2">in</a> <font color="Maroon">DD</font>
 <b>and </b><br/><a NAME="E50:2_4"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">Fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">dd</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E51:2_4"/><i><font color="Green">E153</font></i>: 
<font color="Maroon">dd</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan11.html#E8:2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E52:2_4"/>
<font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K6">Vertical_Line</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i>, <a class="ref" href="jordan1a.html#T17">JORDAN1A:17</a></i>;<br/>
<a NAME="E53:2_4"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="jordan6.html#K6">Vertical_Line</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i>, <a class="ref" href="jordan1a.html#T23">JORDAN1A:23</a></i>;<br/>
<a NAME="E54:2_4"/><b>then </b><i><font color="Green">E154</font></i>: 
<font color="Maroon">dd</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i>, <a class="txt" href="jordan11.html#E11:2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="jordan6.html#T34">JORDAN6:34</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<font color="Maroon">Fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<font color="Maroon">dd</font> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="txt" href="jordan11.html#E10:2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>
;<br/>
<a NAME="E56:2_4"/><b>then </b><i><font color="Green">E155</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan11.html#E11:2"><i><font color="Green">E50</font></i></a>, , <a class="ref" href="topreal3.html#T11">TOPREAL3:11</a></i>;<br/>
<a NAME="E57:2_4"/><b>then </b><i><font color="Green">E156</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>
 
<b>by </b><i>, , <a class="ref" href="topreal1.html#T12">TOPREAL1:12</a></i>;<br/>
<a NAME="E58:2_4"/>
<font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E59:2_4"/><b>then </b><i><font color="Green">E157</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="topreal1.html#T12">TOPREAL1:12</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<font color="Maroon">Es</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
 <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"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p0">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E62:2_4"/><i><font color="Green">E158</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E63:2_4"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E64:2_4"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">j</font>
 <b>by </b><i>, , , , <a class="ref" href="jordan1f.html#T13">JORDAN1F:13</a></i>;<br/>
<i><font color="Green">E161</font></i>: <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<font color="Maroon">Fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
 <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"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E67:2_4"/><i><font color="Green">E162</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E68:2_4"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E69:2_4"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">k</font>
 <b>by </b><i>, , , <a class="txt" href="jordan11.html#E3:2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="jordan11.html#E18:2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan11.html#E20:2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan11.html#E22:2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="jordan1i.html#T30">JORDAN1I:30</a></i>;<br/>
<a NAME="E70:2_4"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E71:2_4"/><i><font color="Green">E166</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan1a.html#K1">Center</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan1b.html#T12">JORDAN1B:12</a></i>;<br/>
<a NAME="E72:2_4"/><i><font color="Green">E167</font></i>: 
 <a href="jordan1a.html#K1">Center</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan1b.html#T14">JORDAN1B:14</a></i>;<br/>
<a NAME="E73:2_4"/>
for <font color="Olive">p</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> holds <br/><font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_4_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:2_4_6"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<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="E3:2_4_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <a href="pscomp_1.html#K14">proj2</a> 
 <b>and </b><br/><a NAME="E4:2_4_6"/><i><font color="Green">E169</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">DD</font>
 <b>and </b><br/><a NAME="E5:2_4_6"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</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="E7:2_4_6"/><i><font color="Green">E171</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i>, , , <a class="ref" href="topreal1.html#T10">TOPREAL1:10</a></i>;<br/>
<a NAME="E8:2_4_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E9:2_4_6"/><b>then </b>
<font color="Maroon">x</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i>, <a class="ref" href="topreal1.html#T10">TOPREAL1:10</a></i>;<br/>
<b>hence </b><a NAME="E10:2_4_6"/>
<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, <a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E74:2_4"/><b>then </b><i><font color="Green">E172</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">D</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan11.html#E7:2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="seq_4.html#T62">SEQ_4:62</a></i>;<br/>
<a NAME="E75:2_4"/><b>then </b><i><font color="Green">E173</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 
<b>by </b><i>, <a class="txt" href="jordan11.html#E18:2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan11.html#E22:2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan11.html#E24:2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan11.html#E26:2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan11.html#E27:2"><i><font color="Green">E61</font></i></a>, , , <a class="ref" href="goboard5.html#T5">GOBOARD5:5</a></i>;<br/>
<a NAME="E76:2_4"/><i><font color="Green">E174</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> holds <br/><font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_4_7"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">DD</font> = <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> as  <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> <b>by </b><i><a class="ref" href="pscomp_1.html#T64">PSCOMP_1:64</a></i>;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <font color="Maroon">DD</font> as  <a href="rcomp_1.html#V1">compact</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="jct_misc.html#T24">JCT_MISC:24</a></i>;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E3:2_4_7"/><i><font color="Green">E175</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">p</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> ;<br/>
<a NAME="E5:2_4_7"/><i><font color="Green">E176</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan11.html#E2"><i><font color="Green">Lemma68</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:2_4_7"/><b>then </b><i><font color="Green">E177</font></i>: 
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i>, , <a class="ref" href="goboard7.html#T5">GOBOARD7:5</a></i>;<br/>
<a NAME="E7:2_4_7"/><i><font color="Green">E178</font></i>: 
<font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan11.html#E24:2"><i><font color="Green">E58</font></i></a>, , , <a class="ref" href="topreal1.html#T10">TOPREAL1:10</a></i>;<br/>
<a NAME="E8:2_4_7"/>
<font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Es</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i/>;<br/>
<b>then </b><i><font color="Green">E179</font></i>: <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<font color="Maroon">Es</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
 <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">D</font>
<b>by </b><i/>
;<br/>
<a NAME="E10:2_4_7"/>
<font color="Maroon">D</font> is <a href="seq_4.html#V3">bounded</a>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T28">RCOMP_1:28</a></i>;<br/>
<a NAME="E11:2_4_7"/><b>then </b><i><font color="Green">E180</font></i>: 
<font color="Maroon">D</font> 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="E12:2_4_7"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan11.html#E2"><i><font color="Green">Lemma68</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E13:2_4_7"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">DD</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E14:2_4_7"/><b>then </b>
<a href="pscomp_1.html#K14">proj2</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E15:2_4_7"/><b>then </b>
<font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>;<br/>
<a NAME="E16:2_4_7"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i>, , <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>
<a NAME="E17:2_4_7"/><b>then </b>
<font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E18:2_4_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="topreal3.html#T11">TOPREAL3:11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E77:2_4"/><i><font color="Green">E181</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> holds <br/><font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_4_8"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">EE</font> = <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> as  <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> <b>by </b><i><a class="ref" href="pscomp_1.html#T64">PSCOMP_1:64</a></i>;<br/>
<b>reconsider </b><font color="Maroon">E0</font> = <a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <font color="Maroon">EE</font> as  <a href="rcomp_1.html#V1">compact</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="jct_misc.html#T24">JCT_MISC:24</a></i>;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E3:2_4_8"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">p</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> ;<br/>
<a NAME="E5:2_4_8"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:2_4_8"/><b>then </b><i><font color="Green">E186</font></i>: 
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i>, , <a class="ref" href="goboard7.html#T5">GOBOARD7:5</a></i>;<br/>
<a NAME="E7:2_4_8"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan11.html#E24:2"><i><font color="Green">E58</font></i></a>, , , <a class="ref" href="topreal1.html#T10">TOPREAL1:10</a></i>;<br/>
<a NAME="E8:2_4_8"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i/>;<br/>
<b>then </b><i><font color="Green">E188</font></i>: <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<font color="Maroon">Fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
 <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">E0</font>
<b>by </b><i/>
;<br/>
<a NAME="E10:2_4_8"/>
<font color="Maroon">E0</font> is <a href="seq_4.html#V3">bounded</a>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T28">RCOMP_1:28</a></i>;<br/>
<a NAME="E11:2_4_8"/><b>then </b><i><font color="Green">E189</font></i>: 
<font color="Maroon">E0</font> 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="E12:2_4_8"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E13:2_4_8"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">EE</font>
 
<b>by </b><i><a class="txt" href="jordan11.html#E17:2"><i><font color="Green">E54</font></i></a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E14:2_4_8"/><b>then </b>
<a href="pscomp_1.html#K14">proj2</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E15:2_4_8"/><b>then </b>
<font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>;<br/>
<a NAME="E16:2_4_8"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i>, , <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>
<a NAME="E17:2_4_8"/><b>then </b>
<font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E18:2_4_8"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="topreal3.html#T11">TOPREAL3:11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E78:2_4"/><i><font color="Green">E190</font></i>: 
( <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">j</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">j</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E79:2_4"/><i><font color="Green">E191</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">j</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_4_9"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:2_4_9"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">j</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_4_9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:2_4_9_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:2_4_9_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">j</font>
 
<b>by </b><i>, <a class="txt" href="jordan11.html#E22:2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan11.html#E27:2"><i><font color="Green">E61</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E3:2_4_9_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">j</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:2_4_9"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">j</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E3:2_4_9"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:2_4_9"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan11.html#E3:2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="jordan11.html#E22:2"><i><font color="Green">E57</font></i></a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E80:2_4"/><i><font color="Green">E192</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_4_10"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:2_4_10"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_4_10_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:2_4_10_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:2_4_10_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">k</font>
 
<b>by </b><i>, <a class="txt" href="jordan11.html#E22:2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan11.html#E27:2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan11.html#E2"><i><font color="Green">Lemma68</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:2_4_10_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:2_4_10"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E3:2_4_10"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:2_4_10"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan11.html#E27:2"><i><font color="Green">E61</font></i></a>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E81:2_4"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="subset_1.html#R2">meets</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="txt" href="jordan11.html#E20:2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan11.html#E22:2"><i><font color="Green">E57</font></i></a>, , <a class="txt" href="jordan11.html#E27:2"><i><font color="Green">E61</font></i></a>, , , <a class="ref" href="jordan1j.html#T64">JORDAN1J:64</a></i>;<br/>
<a NAME="E82:2_4"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T77">XBOOLE_1:77</a></i>;<br/>
<b>hence </b><a NAME="E83:2_4"/>
<font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="jordan1a.html#T24">JORDAN1A:24</a></i>;<br/>

<a NAME="E84:2_4"/>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="subset_1.html#R2">meets</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="txt" href="jordan11.html#E20:2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan11.html#E22:2"><i><font color="Green">E57</font></i></a>, , <a class="txt" href="jordan11.html#E27:2"><i><font color="Green">E61</font></i></a>, , , , <a class="ref" href="jordan1j.html#T63">JORDAN1J:63</a></i>;<br/>
<a NAME="E85:2_4"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T77">XBOOLE_1:77</a></i>;<br/>
<b>hence </b><a NAME="E86:2_4"/>
<font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="jordan1a.html#T24">JORDAN1A:24</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">r</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E41:2"/><i><font color="Green">E194</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p4">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p4">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>and </b><br/><a NAME="E42:2"/><i><font color="Green">E195</font></i>: 
not <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E43:2"/><i><font color="Green">E196</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font> )
 <b>by </b><i>, , , <a class="ref" href="jct_misc.html#T21">JCT_MISC:21</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">p</font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></span>;<br/>
<a NAME="E45:2"/><i><font color="Green">E197</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="jordan1a.html#T21">JORDAN1A:21</a></i>;<br/>
<a NAME="E46:2"/><i><font color="Green">E198</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<div><i><font color="Green">E199</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_5"/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/><a NAME="E2:2_5"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan6.html#T65">JORDAN6:65</a></i>;<br/><a NAME="E3:2_5"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:2_5"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p4">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p4">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p4">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p4">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>;<br/><a NAME="E5:2_5"/><b>then </b>
<a href="pscomp_1.html#K14">proj2</a>  <a href="seq_1.html#K2">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p0">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p0">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/><a NAME="E6:2_5"/><b>then </b>
<font color="Maroon">r</font> <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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p0">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p0">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pscomp_1.html#T128">PSCOMP_1:128</a></i>;<br/><b>hence </b><a NAME="E7:2_5"/>
contradiction
 <b>by </b><i>, <a class="ref" href="relat_1.html#T153">RELAT_1:153</a></i>;<br/></div><b>end;</b></div>
<a NAME="E48:2"/><i><font color="Green">E200</font></i>: 
 <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><a href="jordan10.html#K4">BDD-Family</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K1">BDD</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan10.html#T21">JORDAN10:21</a></i>;<br/>
<a NAME="E49:2"/>
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="hidden.html#R2">in</a>  <a href="jordan10.html#K4">BDD-Family</a> <font color="Maroon">C</font> holds <br/><span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<a NAME="E1:2_6"/><i><font color="Green">E202</font></i>: 
 <a href="jordan10.html#K4">BDD-Family</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><a href="jordan2c.html#K1">BDD</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Olive">k1</font></span>)</span></span>)</span></span>)</span></span>)</span> where <font color="Olive">k1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 
<b>by </b><i><a class="ref" href="jordan10.html#D2">JORDAN10:def 2</a></i>;<br/>
<b>assume </b><a NAME="E2:2_6"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="jordan10.html#K4">BDD-Family</a> <font color="Maroon">C</font>
 ;<br/>

<b>then consider </b><font color="Maroon">k1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:2_6"/><i><font color="Green">E204</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K1">BDD</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k1</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">k0</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:2_6"/><i><font color="Green">E206</font></i>: 
<font color="Maroon">k1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k0</font>
 <b>and </b><br/><a NAME="E7:2_6"/><i><font color="Green">E207</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font>
 <b>by </b><i/>;<br/>
<a NAME="E8:2_6"/><i><font color="Green">E208</font></i>: 
<font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Es</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k0</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E9:2_6"/><b>then </b><i><font color="Green">E209</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E10:2_6"/><i><font color="Green">E210</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k0</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E11:2_6"/><b>then </b><i><font color="Green">E211</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E12:2_6"/><i><font color="Green">E212</font></i>: 
( <a href="pscomp_1.html#K14">proj2</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <a href="pscomp_1.html#K14">proj2</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>;<br/>
<b>reconsider </b><font color="Maroon">DD</font> = <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> as  <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> <b>by </b><i><a class="ref" href="pscomp_1.html#T64">PSCOMP_1:64</a></i>;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <font color="Maroon">DD</font> as  <a href="rcomp_1.html#V1">compact</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="jct_misc.html#T24">JCT_MISC:24</a></i>;<br/>
<a NAME="E15:2_6"/><i><font color="Green">E213</font></i>: 
 <a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="jordan1a.html#K1">Center</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan1b.html#T17">JORDAN1B:17</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<font color="Maroon">Es</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k0</font>
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
 <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"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p0">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E18:2_6"/><i><font color="Green">E214</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E19:2_6"/><i><font color="Green">E215</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E20:2_6"/><i><font color="Green">E216</font></i>: 
<font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">j</font>
 <b>by </b><i>, , , , <a class="ref" href="jordan1f.html#T13">JORDAN1F:13</a></i>;<br/>
<a NAME="E21:2_6"/>
<font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E22:2_6"/><b>then </b><i><font color="Green">E217</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span> <a href="subset_1.html#R2">meets</a>  <a href="jordan6.html#K9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, , , <a class="txt" href="jordan11.html#E1"><i><font color="Green">Lemma42</font></i></a>, <a class="txt" href="jordan11.html#E1:2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="jordan1j.html#T62">JORDAN1J:62</a></i>;<br/>
<a NAME="E23:2_6"/>
<font color="Maroon">DD</font> <a href="tarski.html#R1">c=</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>
 
;<br/>
<a NAME="E24:2_6"/><b>then </b>
<font color="Maroon">DD</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <a href="pscomp_1.html#K14">proj2</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E25:2_6"/><b>then </b><i><font color="Green">E218</font></i>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <a href="pscomp_1.html#K14">proj2</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">DD</font> <a href="hidden.html#R1">=</a> <font color="Maroon">DD</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E26:2_6"/>
<font color="Maroon">DD</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="jordan11.html#E3:2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E27:2_6"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">DD</font>
 
<b>by </b><i><a class="txt" href="jordan11.html#E5:2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E28:2_6"/><b>then </b><i><font color="Green">E219</font></i>: 
<font color="Maroon">D</font> <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#T151">RELAT_1:151</a></i>;<br/>
<a NAME="E29:2_6"/>
<font color="Maroon">Fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k0</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">D</font>
 
<b>by </b><i/>;<br/>
<a NAME="E30:2_6"/><b>then </b>
<font color="Maroon">Fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="txt" href="jordan11.html#E6:2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="rcomp_1.html#T32">RCOMP_1:32</a></i>;<br/>
<b>then consider </b><font color="Maroon">dd</font> being   <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> such that </b><br/><a NAME="E32:2_6"/><i><font color="Green">E220</font></i>: 
<font color="Maroon">dd</font> <a href="hidden.html#R2">in</a> <font color="Maroon">DD</font>
 <b>and </b><br/><a NAME="E33:2_6"/><i><font color="Green">E221</font></i>: 
<font color="Maroon">Fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k0</font> <a href="hidden.html#R1">=</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">dd</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E34:2_6"/><i><font color="Green">E222</font></i>: 
<font color="Maroon">dd</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan11.html#E7:2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E35:2_6"/>
<font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K6">Vertical_Line</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i>, <a class="ref" href="jordan1a.html#T17">JORDAN1A:17</a></i>;<br/>
<a NAME="E36:2_6"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="jordan6.html#K6">Vertical_Line</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i>, <a class="ref" href="jordan1a.html#T23">JORDAN1A:23</a></i>;<br/>
<a NAME="E37:2_6"/><b>then </b><i><font color="Green">E223</font></i>: 
<font color="Maroon">dd</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i>, <a class="txt" href="jordan11.html#E10:2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="jordan6.html#T34">JORDAN6:34</a></i>;<br/>
<i><font color="Green">E224</font></i>: <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<font color="Maroon">Fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k0</font>
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<font color="Maroon">dd</font> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="txt" href="jordan11.html#E8:2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>
;<br/>
<a NAME="E39:2_6"/><b>then </b><i><font color="Green">E225</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan11.html#E10:2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan11.html#E11:2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="topreal3.html#T11">TOPREAL3:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">DD</font> = <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> as  <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> <b>by </b><i><a class="ref" href="pscomp_1.html#T64">PSCOMP_1:64</a></i>;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <font color="Maroon">DD</font> as  <a href="rcomp_1.html#V1">compact</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="jct_misc.html#T24">JCT_MISC:24</a></i>;<br/>
<a NAME="E42:2_6"/><i><font color="Green">E226</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#R2">meets</a>  <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan1b.html#T34">JORDAN1B:34</a></i>;<br/>
<a NAME="E43:2_6"/>
1 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E44:2_6"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span>,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan1a.html#T65">JORDAN1A:65</a></i>;<br/>
<a NAME="E45:2_6"/><b>then </b><i><font color="Green">E227</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#R2">meets</a>  <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E46:2_6"/>
<font color="Maroon">DD</font> <a href="tarski.html#R1">c=</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>
 
;<br/>
<a NAME="E47:2_6"/><b>then </b>
<font color="Maroon">DD</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <a href="pscomp_1.html#K14">proj2</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E48:2_6"/><b>then </b><i><font color="Green">E228</font></i>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <a href="pscomp_1.html#K14">proj2</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">DD</font> <a href="hidden.html#R1">=</a> <font color="Maroon">DD</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E49:2_6"/>
<font color="Maroon">DD</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="jordan11.html#E17:2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E50:2_6"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">DD</font>
 
<b>by </b><i><a class="txt" href="jordan11.html#E18:2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E51:2_6"/><b>then </b><i><font color="Green">E229</font></i>: 
<font color="Maroon">D</font> <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#T151">RELAT_1:151</a></i>;<br/>
<a NAME="E52:2_6"/>
<font color="Maroon">Es</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k0</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">D</font>
 
<b>by </b><i/>;<br/>
<a NAME="E53:2_6"/><b>then </b>
<font color="Maroon">Es</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="txt" href="jordan11.html#E20:2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="rcomp_1.html#T32">RCOMP_1:32</a></i>;<br/>
<b>then consider </b><font color="Maroon">dd</font> being   <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> such that </b><br/><a NAME="E55:2_6"/><i><font color="Green">E230</font></i>: 
<font color="Maroon">dd</font> <a href="hidden.html#R2">in</a> <font color="Maroon">DD</font>
 <b>and </b><br/><a NAME="E56:2_6"/><i><font color="Green">E231</font></i>: 
<font color="Maroon">Es</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k0</font> <a href="hidden.html#R1">=</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">dd</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E57:2_6"/><i><font color="Green">E232</font></i>: 
<font color="Maroon">dd</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan11.html#E22:2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E58:2_6"/><b>then </b><i><font color="Green">E233</font></i>: 
<font color="Maroon">dd</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i>, , <a class="ref" href="jordan6.html#T34">JORDAN6:34</a></i>;<br/>
<i><font color="Green">E234</font></i>: <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<font color="Maroon">Es</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k0</font>
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<font color="Maroon">dd</font> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="txt" href="jordan11.html#E24:2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>
;<br/>
<a NAME="E60:2_6"/><b>then </b><i><font color="Green">E235</font></i>: 
<font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="jordan11.html#E26:2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="topreal3.html#T11">TOPREAL3:11</a></i>;<br/>
<a NAME="E61:2_6"/><i><font color="Green">E236</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i>, , <a class="txt" href="jordan11.html#E27:2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="topreal1.html#T10">TOPREAL1:10</a></i>;<br/>
<a NAME="E62:2_6"/><b>then </b><i><font color="Green">E237</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan11.html#E10:2"><i><font color="Green">E49</font></i></a>, , <a class="ref" href="topreal1.html#T10">TOPREAL1:10</a></i>;<br/>
<a NAME="E63:2_6"/>
<font color="Maroon">r</font> <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"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E64:2_6"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, , <a class="ref" href="sprect_1.html#T61">SPRECT_1:61</a></i>;<br/>
<a NAME="E65:2_6"/><b>then </b><i><font color="Green">E238</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="jordan1a.html#T21">JORDAN1A:21</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<font color="Maroon">Es</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k0</font>
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
 <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"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p0">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E68:2_6"/><i><font color="Green">E239</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E69:2_6"/><i><font color="Green">E240</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E70:2_6"/><i><font color="Green">E241</font></i>: 
<font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">j</font>
 <b>by </b><i>, , , , <a class="ref" href="jordan1f.html#T13">JORDAN1F:13</a></i>;<br/>
<a NAME="E71:2_6"/><i><font color="Green">E242</font></i>: 
<font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E72:2_6"/><i><font color="Green">E243</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<i><font color="Green">E244</font></i>: <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<font color="Maroon">Fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k0</font>
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
 <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"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E75:2_6"/><i><font color="Green">E245</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E76:2_6"/><i><font color="Green">E246</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E77:2_6"/><i><font color="Green">E247</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">k</font>
 <b>by </b><i>, , , , , <a class="txt" href="jordan11.html#E2"><i><font color="Green">Lemma68</font></i></a>, , <a class="ref" href="jordan1i.html#T30">JORDAN1I:30</a></i>;<br/>
<b>reconsider </b><font color="Maroon">CC</font> = <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> as  <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> <b>by </b><i><a class="ref" href="pscomp_1.html#T64">PSCOMP_1:64</a></i>;<br/>
<b>reconsider </b><font color="Maroon">W</font> = <a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <font color="Maroon">CC</font> as  <a href="rcomp_1.html#V1">compact</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="jct_misc.html#T24">JCT_MISC:24</a></i>;<br/>
<a NAME="E80:2_6"/>
<font color="Maroon">CC</font> <a href="tarski.html#R1">c=</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>
 
;<br/>
<a NAME="E81:2_6"/><b>then </b>
<font color="Maroon">CC</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <a href="pscomp_1.html#K14">proj2</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E82:2_6"/><b>then </b><i><font color="Green">E250</font></i>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <a href="pscomp_1.html#K14">proj2</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">CC</font> <a href="hidden.html#R1">=</a> <font color="Maroon">CC</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E83:2_6"/>
<font color="Maroon">CC</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="jordan11.html#E3:2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E84:2_6"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">CC</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E85:2_6"/><b>then </b><i><font color="Green">E251</font></i>: 
<font color="Maroon">W</font> <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#T151">RELAT_1:151</a></i>;<br/>
<a NAME="E86:2_6"/><i><font color="Green">E252</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan1a.html#K1">Center</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan1b.html#T12">JORDAN1B:12</a></i>;<br/>
<a NAME="E87:2_6"/><i><font color="Green">E253</font></i>: 
 <a href="jordan1a.html#K1">Center</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan1b.html#T14">JORDAN1B:14</a></i>;<br/>
<a NAME="E88:2_6"/>
for <font color="Olive">p</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> holds <br/><font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_6_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:2_6_6"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 ;<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="E3:2_6_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <a href="pscomp_1.html#K14">proj2</a> 
 <b>and </b><br/><a NAME="E4:2_6_6"/><i><font color="Green">E254</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CC</font>
 <b>and </b><br/><a NAME="E5:2_6_6"/><i><font color="Green">E255</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</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="E7:2_6_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:2_6_6"/><b>then </b>
<font color="Maroon">x</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i>, <a class="ref" href="topreal1.html#T10">TOPREAL1:10</a></i>;<br/>
<b>hence </b><a NAME="E9:2_6_6"/>
<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, <a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E89:2_6"/><b>then </b>
 <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">W</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i>, <a class="ref" href="seq_4.html#T62">SEQ_4:62</a></i>;<br/>
<a NAME="E90:2_6"/><b>then </b><i><font color="Green">E256</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 
<b>by </b><i>, , <a class="txt" href="jordan11.html#E2"><i><font color="Green">Lemma68</font></i></a>, , , , , , <a class="ref" href="goboard5.html#T5">GOBOARD5:5</a></i>;<br/>
<a NAME="E91:2_6"/><i><font color="Green">E257</font></i>: 
<font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E92:2_6"/><b>then </b><i><font color="Green">E258</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="topreal1.html#T12">TOPREAL1:12</a></i>;<br/>
<a NAME="E93:2_6"/>
<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1 <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E94:2_6"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="topreal1.html#T12">TOPREAL1:12</a></i>;<br/>
<a NAME="E95:2_6"/><b>then </b><i><font color="Green">E259</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>
 
<b>by </b><i>, , <a class="ref" href="topreal1.html#T12">TOPREAL1:12</a></i>;<br/>
<a NAME="E96:2_6"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> iff <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_6_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:2_6_7"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> implies <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_6_7_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_6_7_1"/><i><font color="Green">E260</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">p</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> ;<br/>
<a NAME="E3:2_6_7_1"/><i><font color="Green">E261</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:2_6_7_1"/><b>then </b><i><font color="Green">E262</font></i>: 
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i>, , <a class="ref" href="goboard7.html#T5">GOBOARD7:5</a></i>;<br/>
<a NAME="E5:2_6_7_1"/><i><font color="Green">E263</font></i>: 
<font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i>, , <a class="ref" href="topreal1.html#T10">TOPREAL1:10</a></i>;<br/>
<a NAME="E6:2_6_7_1"/>
<font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Es</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k0</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i/>;<br/>
<b>then </b><i><font color="Green">E264</font></i>: <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<font color="Maroon">Es</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k0</font>
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
 <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">D</font>
<b>by </b><i/>
;<br/>
<a NAME="E8:2_6_7_1"/>
<font color="Maroon">D</font> is <a href="seq_4.html#V3">bounded</a>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T28">RCOMP_1:28</a></i>;<br/>
<a NAME="E9:2_6_7_1"/><b>then </b><i><font color="Green">E265</font></i>: 
<font color="Maroon">D</font> 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="E10:2_6_7_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E11:2_6_7_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">DD</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E12:2_6_7_1"/><b>then </b>
<a href="pscomp_1.html#K14">proj2</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E13:2_6_7_1"/><b>then </b>
<font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>;<br/>
<a NAME="E14:2_6_7_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i>, <a class="txt" href="jordan11.html#E28:2"><i><font color="Green">E88</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>
<a NAME="E15:2_6_7_1"/><b>then </b>
<font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E16:2_6_7_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font>
 <b>by </b><i>, <a class="ref" href="topreal3.html#T11">TOPREAL3:11</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:2_6_7"/><i><font color="Green">E266</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font>
 ;<br/>

<a NAME="E3:2_6_7"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<b>hence </b><a NAME="E4:2_6_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E97:2_6"/><b>then </b><i><font color="Green">E267</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">EE</font> = <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,1</span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> as  <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> <b>by </b><i><a class="ref" href="pscomp_1.html#T64">PSCOMP_1:64</a></i>;<br/>
<b>reconsider </b><font color="Maroon">E0</font> = <a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <font color="Maroon">EE</font> as  <a href="rcomp_1.html#V1">compact</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="jct_misc.html#T24">JCT_MISC:24</a></i>;<br/>
<a NAME="E100:2_6"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> iff <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_6_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:2_6_8"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> implies <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_6_8_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_6_8_1"/><i><font color="Green">E268</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">p</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> ;<br/>
<a NAME="E3:2_6_8_1"/><i><font color="Green">E269</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:2_6_8_1"/><b>then </b><i><font color="Green">E270</font></i>: 
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i>, , <a class="ref" href="goboard7.html#T5">GOBOARD7:5</a></i>;<br/>
<a NAME="E5:2_6_8_1"/><i><font color="Green">E271</font></i>: 
<font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i>, , <a class="ref" href="topreal1.html#T10">TOPREAL1:10</a></i>;<br/>
<a NAME="E6:2_6_8_1"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k0</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i/>;<br/>
<b>then </b><i><font color="Green">E272</font></i>: <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<font color="Maroon">Fs</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k0</font>
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
 <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">E0</font>
<b>by </b><i/>
;<br/>
<a NAME="E8:2_6_8_1"/>
<font color="Maroon">E0</font> is <a href="seq_4.html#V3">bounded</a>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T28">RCOMP_1:28</a></i>;<br/>
<a NAME="E9:2_6_8_1"/><b>then </b><i><font color="Green">E273</font></i>: 
<font color="Maroon">E0</font> 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="E10:2_6_8_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E11:2_6_8_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">EE</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E12:2_6_8_1"/><b>then </b>
<a href="pscomp_1.html#K14">proj2</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E13:2_6_8_1"/><b>then </b>
<font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>;<br/>
<a NAME="E14:2_6_8_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan11.html#E28:2"><i><font color="Green">E88</font></i></a>, , <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>
<a NAME="E15:2_6_8_1"/><b>then </b>
<font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E16:2_6_8_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font>
 <b>by </b><i>, <a class="ref" href="topreal3.html#T11">TOPREAL3:11</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:2_6_8"/><i><font color="Green">E274</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font>
 ;<br/>

<a NAME="E3:2_6_8"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<b>hence </b><a NAME="E4:2_6_8"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E101:2_6"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E102:2_6"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k0</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3">RightComp</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, , , <a class="txt" href="jordan11.html#E2"><i><font color="Green">Lemma68</font></i></a>, , , , , , </i>;<br/>
<a NAME="E103:2_6"/><b>then </b><i><font color="Green">E275</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3">RightComp</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E104:2_6"/>
<font color="Maroon">k0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E105:2_6"/><b>then </b><i><font color="Green">E276</font></i>: 
 <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k0</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan1h.html#T56">JORDAN1H:56</a></i>;<br/>
<a NAME="E106:2_6"/>
 <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k0</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="jordan1h.html#T56">JORDAN1H:56</a></i>;<br/>
<a NAME="E107:2_6"/><b>then </b>
 <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E108:2_6"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3">RightComp</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><font color="Maroon">k0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3">RightComp</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T49">PRE_TOPC:49</a></i>;<br/>
<a NAME="E109:2_6"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3">RightComp</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k1</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E110:2_6"/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="gobrd14.html#T47">GOBRD14:47</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E50:2"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><a href="jordan10.html#K4">BDD-Family</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E51:2"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">C</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E53:2"/><i><font color="Green">E277</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E54:2"/><i><font color="Green">E278</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E55:2"/><i><font color="Green">E279</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E56:2"/><i><font color="Green">E280</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E57:2"/><i><font color="Green">E281</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>and </b><br/><a NAME="E58:2"/><i><font color="Green">E282</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i</font>,<font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E59:2"/><i><font color="Green">E283</font></i>: 
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan1c.html#T35">JORDAN1C:35</a></i>;<br/>
<b>take </b>
<font color="Maroon">n</font>
;<br/>

<b>take </b>
<font color="Maroon">j</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="jordan1h.html#D3">JORDAN1H:def 3</a><br/>

<b>thus </b><a NAME="E60:2"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/>

<a NAME="E61:2"/><i><font color="Green">E284</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, , , <a class="ref" href="jordan1b.html#T44">JORDAN1B:44</a></i>;<br/>
<a NAME="E62:2"/><i><font color="Green">E285</font></i>: 
 <a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="jordan1a.html#K1">Center</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan1b.html#T17">JORDAN1B:17</a></i>;<br/>
<a NAME="E63:2"/>
2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="jordan1h.html#T58">JORDAN1H:58</a></i>;<br/>
<a NAME="E64:2"/><b>then </b><i><font color="Green">E286</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> &amp;  <a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="jordan1h.html#T58">JORDAN1H:58</a></i>;<br/>
<a NAME="E65:2"/>
4 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="jordan8.html#T13">JORDAN8:13</a></i>;<br/>
<a NAME="E66:2"/><b>then </b><i><font color="Green">E287</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E67:2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan8.html#D1">JORDAN8:def 1</a></i>;<br/>
<a NAME="E68:2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i>, , , , , , , <a class="ref" href="jordan1a.html#T57">JORDAN1A:57</a></i>;<br/>
<b>hence </b><a NAME="E69:2"/>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j</font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , , <a class="ref" href="jordan1b.html#T23">JORDAN1B:23</a></i>;<br/>


</div>
