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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> =  <a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>assume </b><a NAME="E1:9"/>
<font color="Maroon">c<sub>1</sub></font> <a href="jordan1h.html#R1">is_sufficiently_large_for</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:9"/><b>then </b><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="jordan11.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E3:9"/><i><font color="Green">E11</font></i>: 
 <a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="jordan11.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E4:9"/><b>then </b><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="jordan11.html#E2:9"><i><font color="Green">E10</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p2">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p2">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan11.html#E3:9"><i><font color="Green">E11</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p3">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1

<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K5">-</a> 1
<b>by </b><i><a class="txt" href="jordan11.html#E2:9"><i><font color="Green">E10</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1
<b>by </b><i><a class="txt" href="jordan11.html#E4:9"><i><font color="Green">E12</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
;<br/>
<a NAME="E6:9"/><b>then </b>
2 <a href="card_4.html#K1">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p3">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="newton.html#T13">NEWTON:13</a></i>;<br/>
<b>hence </b><a NAME="E7:9"/>
 <a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p4">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p4">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2
 ;<br/>


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