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<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">k1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:16"/><i><font color="Green">E50</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k1</font> &amp; <font color="Maroon">k1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k2</font> &amp; <font color="Maroon">k2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; ( <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">k2</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">k2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">k1</font></span>)</span> <a href="real_1.html#K3">+</a> 1 or <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">k2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k1</font> ) )
 ;<br/>

<a NAME="E2:16"/><b>then </b><i><font color="Green">E51</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k2</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E3:16"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">k1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:16"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E5:16"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E6:16"/><b>then </b><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E7:16"/><i><font color="Green">E62</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f</font>,<font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k2</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E8:16"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">k2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">k1</font></span>)</span> <a href="real_1.html#K3">+</a> 1 implies <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">k2</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">k1</font>
<b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
<b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
;<br/>
<b>hence </b><a NAME="E10:16"/><i><font color="Green">E64</font></i>: 
( <span class="p1">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f</font>,<font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f</font>,<font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span> )
 <b>by </b><i>, , , <a class="ref" href="jordan3.html#T1" target="_self">Th1</a>, <a class="ref" href="jordan3.html#T2" target="_self">Th2</a>, </i>;<br/>

<b>hence </b><a NAME="E11:16"/>
<span class="p1">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f</font>,<font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>

<b>thus </b><a NAME="E12:16"/>
<span class="p1">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f</font>,<font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">k1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan3.html#T4" target="_self">Th4</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>


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