<?xml version="1.0"?>
<div class="add">

<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">k11</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k21</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:12"/><i><font color="Green">E50</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k11</font> &amp; <font color="Maroon">k11</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">k21</font> &amp; <font color="Maroon">k21</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 ;<br/>

<div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>now </b></a><div class="add"><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> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_1_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">k1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k2</font> &amp; 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>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 ;<br/><b>then </b><i><font color="Green">E53</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> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">k1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><span class="p3">(<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</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">k1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><span class="p4">(<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</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
;<br/><a NAME="E3:12_1_1"/>
<font color="Maroon">k1</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><b>then </b><i><font color="Green">E54</font></i>: 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">k1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">k1</font> <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/>.= 
<font color="Maroon">k1</font>

;<br/><a NAME="E5:12_1_1"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">k1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">k1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">k1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i/>;<br/><a NAME="E6:12_1_1"/><b>then </b><i><font color="Green">E62</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">k1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E7:12_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">k1</font> <a href="binarith.html#K3">-'</a> 1 <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>, </i>;<br/><a NAME="E8:12_1_1"/><b>then </b><i><font color="Green">E64</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">k1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">k1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/><a NAME="E9:12_1_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <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="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E10:12_1_1"/><b>then </b>
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<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</span>)</span> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">k1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="jordan3.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E11:12_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">k1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<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</span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E12:12_1_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">k1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><span class="p3">(<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</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="E13:12_1_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">k1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><span class="p4">(<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</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E14:12_1_1"/><b>then </b>
<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> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">k1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><b>hence </b><a NAME="E15:12_1_1"/>
<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> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k1</font>
 <b>by </b><i>, , <a class="ref" href="jordan3.html#T2" target="_self">Th2</a>, <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:12_1"/>
( <font color="Maroon">k1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k2</font> &amp; 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>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> implies <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> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k1</font> )
 ;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E3:12"/>
<span class="p1">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f</font>,<font color="Maroon">k11</font>,<font color="Maroon">k21</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k11</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_2_1"/>
( <font color="Maroon">k11</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k21</font> or <font color="Maroon">k11</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k21</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1"><b>suppose </b></a><a NAME="E1:12_2_1_1"/>
<font color="Maroon">k11</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k21</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:12_2_1_1"/>
<span class="p1">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f</font>,<font color="Maroon">k11</font>,<font color="Maroon">k21</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k11</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_2"><b>suppose </b></a><a NAME="E1:12_2_1_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">k11</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k21</font>
 ;<br/><div class="add"><a NAME="E2:12_2_1_2"/><i><font color="Green">E67</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <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="E3:12_2_1_2"/>
<font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> 1 <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>, </i>;<br/><a NAME="E4:12_2_1_2"/><b>then </b><i><font color="Green">E68</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/><a NAME="E5:12_2_1_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">k21</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E6:12_2_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">k11</font> <a href="real_1.html#K5">-</a> <font color="Maroon">k21</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E7:12_2_1_2"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k11</font> <a href="real_1.html#K5">-</a> <font color="Maroon">k21</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">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">k11</font> <a href="real_1.html#K5">-</a> <font color="Maroon">k21</font></span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<font color="Maroon">k11</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">k21</font> <a href="real_1.html#K5">-</a> 1</span>)</span>

<br/>.= 
<font color="Maroon">k11</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
;<br/><a NAME="E9:12_2_1_2"/><b>then </b><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></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"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E10:12_2_1_2"/><b>then </b><i><font color="Green">E73</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/><a NAME="E11:12_2_1_2"/><b>then </b><i><font color="Green">E74</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E12:12_2_1_2"/><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan3.html#T2" target="_self">Th2</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><i><font color="Green">E76</font></i>: <span class="p1">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">k21</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">k11</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">k21</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="jordan3.html#T1" target="_self">Th1</a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<font color="Maroon">k11</font>

;<br/><i><font color="Green">E77</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p4">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="jordan3.html#T4" target="_self">Th4</a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="jordan3.html#T5" target="_self">Th5</a>, <a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jordan3.html#T7" target="_self">Th7</a>, <a class="ref" href="finseq_5.html#T30">FINSEQ_5:30</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k11</font>
<b>by </b><i>, <a class="ref" href="jordan3.html#T9" target="_self">Th9</a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
;<br/><a NAME="E15:12_2_1_2"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan3.html#T4" target="_self">Th4</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p4">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p0">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#T61">FINSEQ_5:61</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k11</font>
<b>by </b><i><a class="ref" href="jordan3.html#T10" target="_self">Th10</a></i>
;<br/><b>hence </b><a NAME="E17:12_2_1_2"/>
<span class="p1">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f</font>,<font color="Maroon">k11</font>,<font color="Maroon">k21</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k11</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E4:12"/>
( <font color="Maroon">k11</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k21</font> implies (  <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">k11</font>,<font color="Maroon">k21</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k11</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">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="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f</font>,<font color="Maroon">k11</font>,<font color="Maroon">k21</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f</font>,<font color="Maroon">k11</font>,<font color="Maroon">k21</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">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="Olive">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k11</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_3"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">k11</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k21</font>
 ;<br/>

<a NAME="E2:12_3"/><b>then </b><i><font color="Green">E83</font></i>: 
 <a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f</font>,<font color="Maroon">k11</font>,<font color="Maroon">k21</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k11</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:12_3"/>
<font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> 1 <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>, </i>;<br/>
<a NAME="E4:12_3"/><b>then </b><i><font color="Green">E84</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E5:12_3"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">k11</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E6:12_3"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k11</font> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E7:12_3"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">k21</font> <a href="real_1.html#K5">-</a> <font color="Maroon">k11</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E8:12_3"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k11</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k21</font> <a href="real_1.html#K5">-</a> <font color="Maroon">k11</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">k21</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k11</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">k21</font> <a href="real_1.html#K5">-</a> <font color="Maroon">k11</font></span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="ref" href="jordan3.html#T1" target="_self">Th1</a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<font color="Maroon">k21</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">k11</font> <a href="real_1.html#K5">-</a> 1</span>)</span>

<br/>.= 
<font color="Maroon">k21</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
;<br/>
<a NAME="E10:12_3"/><b>then </b><i><font color="Green">E96</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k11</font></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"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E11:12_3"/><b>then </b><i><font color="Green">E97</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k11</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k11</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E12:12_3"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">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="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f</font>,<font color="Maroon">k11</font>,<font color="Maroon">k21</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f</font>,<font color="Maroon">k11</font>,<font color="Maroon">k21</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">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="Olive">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k11</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_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>assume </b><a NAME="E1:12_3_2"/><i><font color="Green">E98</font></i>: 
( 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>  <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">k11</font>,<font color="Maroon">k21</font></span>)</span> )
 ;<br/>

<a NAME="E2:12_3_2"/><b>then </b><i><font color="Green">E99</font></i>: 
<font color="Maroon">i</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">k11</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">k21</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">k11</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">k11</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="jordan3.html#T2" target="_self">Th2</a>, <a class="ref" href="jordan3.html#T5" target="_self">Th5</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E3:12_3_2"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k11</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k11</font></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"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="jordan3.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan3.html#T7" target="_self">Th7</a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E4:12_3_2"/><i><font color="Green">E101</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <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="jordan3.html#T7" target="_self">Th7</a>, <a class="ref" href="jordan3.html#T9" target="_self">Th9</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E5:12_3_2"/>
<font color="Maroon">k11</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k11</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E6:12_3_2"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k11</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<i><font color="Green">E103</font></i>: <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
<font color="Maroon">i</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">k11</font> <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/>.= 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k11</font></span>)</span> <a href="real_1.html#K5">-</a> 1

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k11</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
<b>by </b><i><a class="ref" href="jordan3.html#T12" target="_self">Th12</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
;<br/>
<span class="p1">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f</font>,<font color="Maroon">k11</font>,<font color="Maroon">k21</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k11</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="jordan3.html#T10" target="_self">Th10</a>, <a class="ref" href="finseq_3.html#T121">FINSEQ_3:121</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jordan3.html#T11" target="_self">Th11</a>, </i>
;<br/>
<b>hence </b><a NAME="E9:12_3_2"/>
<span class="p1">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f</font>,<font color="Maroon">k11</font>,<font color="Maroon">k21</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">k11</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="jordan3.html#T13" target="_self">Th13</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:12_3"/>
(  <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">k11</font>,<font color="Maroon">k21</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k11</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">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="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f</font>,<font color="Maroon">k11</font>,<font color="Maroon">k21</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f</font>,<font color="Maroon">k11</font>,<font color="Maroon">k21</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">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="Olive">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k11</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> ) )
 <b>by </b><i>, <a class="ref" href="jordan3.html#T4" target="_self">Th4</a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>


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

<b>assume </b><a NAME="E5:12"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">k11</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k21</font>
 ;<br/>

<a NAME="E6:12"/><b>then </b><i><font color="Green">E105</font></i>: 
 <a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f</font>,<font color="Maroon">k11</font>,<font color="Maroon">k21</font> <a href="hidden.html#R1">=</a>  <a href="finseq_5.html#K4">Rev</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E7:12"/><b>then </b><i><font color="Green">E106</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">k11</font>,<font color="Maroon">k21</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/>
<a NAME="E8:12"/>
<font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> 1 <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>, </i>;<br/>
<a NAME="E9:12"/><b>then </b><i><font color="Green">E107</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E10:12"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">k21</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E11:12"/><b>then </b><i><font color="Green">E109</font></i>: 
<font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k21</font> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E12:12"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">k11</font> <a href="real_1.html#K5">-</a> <font color="Maroon">k21</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E13:12"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k11</font> <a href="real_1.html#K5">-</a> <font color="Maroon">k21</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">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">k11</font> <a href="real_1.html#K5">-</a> <font color="Maroon">k21</font></span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="ref" href="jordan3.html#T2" target="_self">Th2</a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<font color="Maroon">k11</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">k21</font> <a href="real_1.html#K5">-</a> 1</span>)</span>

<br/>.= 
<font color="Maroon">k11</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
;<br/>
<a NAME="E15:12"/><b>then </b><i><font color="Green">E112</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></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"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E16:12"/><b>then </b><i><font color="Green">E113</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<b>thus </b><a NAME="E17:12"/><i><font color="Green">E114</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">k11</font>,<font color="Maroon">k21</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="jordan3.html#T5" target="_self">Th5</a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>

<b>thus </b><a NAME="E18:12"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">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="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f</font>,<font color="Maroon">k11</font>,<font color="Maroon">k21</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f</font>,<font color="Maroon">k11</font>,<font color="Maroon">k21</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">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">k11</font> <a href="binarith.html#K3">-'</a> <font color="Olive">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_5"><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>assume </b><a NAME="E1:12_5"/><i><font color="Green">E115</font></i>: 
( 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>  <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">k11</font>,<font color="Maroon">k21</font></span>)</span> )
 ;<br/>

<a NAME="E2:12_5"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="jordan3.html#T5" target="_self">Th5</a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E3:12_5"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">i</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">k21</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">k11</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">k21</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">k21</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="jordan3.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan3.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan3.html#T10" target="_self">Th10</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E4:12_5"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E5:12_5"/><b>then </b><i><font color="Green">E118</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k11</font>
 
<b>by </b><i><a class="ref" href="jordan3.html#T1" target="_self">Th1</a>, <a class="ref" href="jordan3.html#T12" target="_self">Th12</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<i><font color="Green">E119</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="jordan3.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan3.html#T10" target="_self">Th10</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"><span class="p3">(<span class="default"><font color="Maroon">k11</font> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">k21</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">k21</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="jordan3.html#T4" target="_self">Th4</a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">k11</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> 1

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="jordan3.html#T13" target="_self">Th13</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
;<br/>
<a NAME="E7:12_5"/>
<span class="p1">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E8:12_5"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="jordan3.html#T10" target="_self">Th10</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E9:12_5"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E10:12_5"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
;<br/>
<a NAME="E11:12_5"/><b>then </b><i><font color="Green">E120</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="jordan3.html#T11" target="_self">Th11</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E12:12_5"/><i><font color="Green">E121</font></i>: 
1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k11</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k11</font>
 
<b>by </b><i><a class="ref" href="jordan3.html#T10" target="_self">Th10</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E13:12_5"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k11</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E14:12_5"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k11</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E15:12_5"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k11</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E16:12_5"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">k11</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
;<br/>
<a NAME="E17:12_5"/><b>then </b><i><font color="Green">E122</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <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="jordan3.html#T13" target="_self">Th13</a>, , <a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<span class="p1">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f</font>,<font color="Maroon">k11</font>,<font color="Maroon">k21</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p0">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
<b>by </b><i>, , <a class="ref" href="jordan3.html#T10" target="_self">Th10</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p0">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="jordan3.html#T10" target="_self">Th10</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">k21</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="jordan3.html#T7" target="_self">Th7</a>, , <a class="ref" href="finseq_3.html#T121">FINSEQ_3:121</a></i>
<br/>.= 
<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">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , </i>
;<br/>
<b>hence </b><a NAME="E19:12_5"/>
<span class="p1">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f</font>,<font color="Maroon">k11</font>,<font color="Maroon">k21</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">k11</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>


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


</div>
