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

<b>let </b><font color="Maroon" title="c1">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f1</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f1</font> holds <br/>(  <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">k</font>,<font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">k</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">k</font>,<font color="Olive" title="b2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><b>let </b><font color="Maroon" title="c2">f1</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f1</font> holds <br/>(  <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c2">f1</font>,<font color="Olive" title="b1">k</font>,<font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c2">f1</font>,<font color="Olive" title="b1">k</font>,<font color="Olive" title="b1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><b>let </b><font color="Maroon" title="c3">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f1</font> implies (  <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )</span><br/>





<b>assume </b><a NAME="E1:68"/><i><font color="Green" title="E60">A1</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<a NAME="E2:68"/><i><font color="Green" title="E61">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="rfinseq.html#T42" title="RFINSEQ:th.42">RFINSEQ:42</a></i>;<br/>
<a NAME="E3:68"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font>
 
<b>by </b><i><a class="txt" href="#E1:68"><i><font color="Green" title="E60">A1</font></i></a>, <a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a></i>;<br/>
<a NAME="E4:68"/><b>then </b><i><font color="Green" title="E62">A3</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">k</font>
 
<b>by </b><i><a class="txt" href="#E1:68"><i><font color="Green" title="E60">A1</font></i></a>, <a class="ref" href="jordan3.html#T23" target="_self" title="JORDAN3:th.23">Th23</a></i>;<br/>
<a NAME="E5:68"/><i><font color="Green" title="E63">A4</font></i>: 
<font color="Maroon" title="c2">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">k</font>
 
<b>by </b><i><a class="txt" href="#E1:68"><i><font color="Green" title="E60">A1</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E6:68"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f1</font>
 
<b>by </b><i><a class="txt" href="#E1:68"><i><font color="Green" title="E60">A1</font></i></a>, <a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a></i>;<br/>
<a NAME="E7:68"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E8:68"/><b>then </b><i><font color="Green" title="E64">A5</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E2:68"><i><font color="Green" title="E61">A2</font></i></a>, <a class="ref" href="nat_d.html#T39" title="NAT_D:th.39">NAT_D:39</a></i>;<br/>
<a NAME="E9:68"/><b>then </b><i><font color="Green" title="E65">A6</font></i>: 
not <font color="Maroon" title="c2">f1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
<b>by </b><i><a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></i>
<br/>.= 
1

;<br/>
<b>then </b> <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c3">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1
<b>by </b><i><a class="ref" href="jordan3.html#D1" target="_self" title="JORDAN3:def.1">Def1</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
<b>by </b><i><a class="txt" href="#E9:68"><i><font color="Green" title="E65">A6</font></i></a>, <a class="ref" href="finseq_5.html#T23" title="FINSEQ_5:th.23">FINSEQ_5:23</a></i>
;<br/>
<b>hence </b><a NAME="E12:68"/>
(  <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="#E4:68"><i><font color="Green" title="E62">A3</font></i></a>, <a class="txt" href="#E5:68"><i><font color="Green" title="E63">A4</font></i></a>, <a class="txt" href="#E8:68"><i><font color="Green" title="E64">A5</font></i></a>, <a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
