<?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="filerec1.html#NM1" target="_self">File</a> of <font color="Maroon">D</font>, <font color="Maroon">g</font> be    <a href="filerec1.html#NM1" target="_self">File</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:40"/><i><font color="Green">E33</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 ;<br/>

<a NAME="E2:40"/><i><font color="Green">E34</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<b>assume </b><a NAME="E3:40"/><i><font color="Green">E35</font></i>: 
 <a href="finseq_8.html#K7">instr</a> <font color="Maroon">n</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E4:40"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="finseq_8.html#K7">instr</a> <font color="Maroon">n</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E5:40"/>
 <a href="finseq_8.html#K7">instr</a> <font color="Maroon">n</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a>, </i>;<br/>
<a NAME="E6:40"/><b>then </b><i><font color="Green">E37</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_8.html#K7">instr</a> <font color="Maroon">n</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font> &amp; <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> <font color="Maroon">n</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Olive">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  holds <br/><font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_8.html#K7">instr</a> <font color="Maroon">n</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font> ) )
 
<b>by </b><i><a class="ref" href="finseq_8.html#D10">FINSEQ_8:def 10</a></i>;<br/>
<a NAME="E7:40"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_8.html#K7">instr</a> <font color="Maroon">n</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</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="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a>, <a class="ref" href="filerec1.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E8:40"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_8.html#K7">instr</a> <font color="Maroon">n</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E9:40"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_8.html#K7">instr</a> <font color="Maroon">n</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="filerec1.html#T3" target="_self">Th3</a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E10:40"/><b>then </b>
<font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> <font color="Maroon">n</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_5.html#T35">FINSEQ_5:35</a></i>;<br/>
<a NAME="E11:40"/><b>then </b>
 <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"><span class="p3">(<span class="default"><a href="finseq_8.html#K7">instr</a> <font color="Maroon">n</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E12:40"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="finseq_8.html#D8">FINSEQ_8:def 8</a>, </i>;<br/>
<a NAME="E13:40"/><b>then </b>
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<b>hence </b><a NAME="E14:40"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


</div>
