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

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being    <a href="filerec1.html#NM1" title="FILEREC1:NM.1">File</a> of <font color="Maroon" title="c1">D</font><br/>  for <font color="Olive" title="b3">n</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  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">n</font> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> <a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> (<font color="Olive" title="b3">n</font>,<font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">g</font> be    <a href="filerec1.html#NM1" title="FILEREC1:NM.1">File</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</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  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">n</font> &amp; <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> <a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> (<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">n</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:39"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">n</font>
 <span class="kw">and </span><br/><a NAME="E2:39"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
<font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font></span><br/>

<a NAME="E3:39"/><span class="lab"><font color="Green" title="E26">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39"><span class="lab"><font color="Green" title="E25">A2</font></span></a></span>;<br/>
<a NAME="E4:39"/>
 <a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:39_1"/>
 <a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:39_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c3">g</font> <a href="finseq_8.html#R2" title="FINSEQ_8:pred.2">is_substring_of</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_8.html#D10" title="FINSEQ_8:def.10">FINSEQ_8:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:39_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="finseq_8.html#D7" title="FINSEQ_8:def.7">FINSEQ_8:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A4</font></span>: 
<font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_8.html#D10" title="FINSEQ_8:def.10">FINSEQ_8:def 10</a></span>;<br/>
<a NAME="E6:39"/>
<font color="Maroon" title="c2">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="finseq_8.html#D8" title="FINSEQ_8:def.8">FINSEQ_8:def 8</a></span>;<br/>
<a NAME="E7:39"/><span class="kw">then </span>
<font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="finseq_8.html#D10" title="FINSEQ_8:def.10">FINSEQ_8:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:39"/>
 <a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
