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<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">r1</font> be    <a href="filerec1.html#NM1" target="_self">File</a> of <font color="Maroon">D</font>, <font color="Maroon">r2</font> be    <a href="filerec1.html#NM1" target="_self">File</a> of <font color="Maroon">D</font>, <font color="Maroon">r3</font> be    <a href="filerec1.html#NM1" target="_self">File</a> of <font color="Maroon">D</font>, <font color="Maroon">f</font> be    <a href="filerec1.html#NM1" target="_self">File</a> of <font color="Maroon">D</font>;<br/>



<b>assume </b><a NAME="E1:37"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="finseq_8.html#K1">^</a> <font color="Maroon">r2</font></span>)</span> <a href="finseq_8.html#K1">^</a> <font color="Maroon">r3</font>
 ;<br/>

<a NAME="E2:37"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="finseq_8.html#K1">^</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="finseq_8.html#K1">^</a> <font color="Maroon">r3</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/>
<a NAME="E3:37"/><i><font color="Green">E35</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">r3</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/>
<a NAME="E4:37"/><i><font color="Green">E36</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="finseq_8.html#K1">^</a> <font color="Maroon">r2</font></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">r1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1"><b>suppose </b></a><a NAME="E1:37_1_1_1"/><i><font color="Green">E37</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">r2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:37_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">r2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <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="E3:37_1_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r1</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">r1</font> <a href="finseq_8.html#K1">^</a> <font color="Maroon">r2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E4:37_1_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 0 <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">r1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r3</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a>, </i>;<br/><a NAME="E5:37_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">r1</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/><a NAME="E6:37_1_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <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="E7:37_1_1_1"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r2</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r1</font></span>)</span> <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><i><font color="Green">E54</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r1</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">r2</font>

;<br/><i><font color="Green">E55</font></i>:  <a href="jordan3.html#K1">mid</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="finseq_8.html#K1">^</a> <font color="Maroon">r2</font></span>)</span> <a href="finseq_8.html#K1">^</a> <font color="Maroon">r3</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="finseq_8.html#K1">^</a> <font color="Maroon">r2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r1</font> <a href="finseq_8.html#K1">^</a> <font color="Maroon">r2</font></span>)</span> <a href="finseq_8.html#K1">^</a> <font color="Maroon">r3</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">r1</font> <a href="finseq_8.html#K1">^</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="filerec1.html#T2" target="_self">Th2</a>, <a class="ref" href="jordan3.html#D1">JORDAN3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r1</font> <a href="finseq_8.html#K1">^</a> <font color="Maroon">r2</font></span>)</span> <a href="finseq_8.html#K1">^</a> <font color="Maroon">r3</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="finseq_1.html#K17">|</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"><font color="Maroon">r1</font> <a href="finseq_8.html#K1">^</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r1</font> <a href="finseq_8.html#K1">^</a> <font color="Maroon">r2</font></span>)</span> <a href="finseq_8.html#K1">^</a> <font color="Maroon">r3</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="finseq_1.html#K17">|</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"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r1</font> <a href="finseq_8.html#K1">^</a> <font color="Maroon">r2</font></span>)</span> <a href="finseq_8.html#K1">^</a> <font color="Maroon">r3</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="finseq_1.html#K17">|</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> <font color="Maroon">r1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="filerec1.html#T4" target="_self">Th4</a>, <a class="ref" href="nat_2.html#T9">NAT_2:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="finseq_8.html#K1">^</a> <span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="finseq_8.html#K1">^</a> <font color="Maroon">r3</font></span>)</span></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="finseq_8.html#K1">^</a> <font color="Maroon">r3</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#T40">FINSEQ_5:40</a></i>
<br/>.= 
<font color="Maroon">r2</font>
<b>by </b><i><a class="ref" href="finseq_5.html#T26">FINSEQ_5:26</a></i>
;<br/><a NAME="E10:37_1_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 implies  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( 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"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="finseq_8.html#K1">^</a> <font color="Maroon">r2</font></span>)</span> <a href="finseq_8.html#K1">^</a> <font color="Maroon">r3</font></span>)</span> &amp;  <a href="jordan3.html#K1">mid</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="finseq_8.html#K1">^</a> <font color="Maroon">r2</font></span>)</span> <a href="finseq_8.html#K1">^</a> <font color="Maroon">r3</font></span>)</span>,<font color="Olive">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_3"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E2:37_1_1_1_3"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/>
<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> <font color="Maroon">r1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r2</font></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="finseq_8.html#K1">^</a> <font color="Maroon">r2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
;<br/>
<a NAME="E4:37_1_1_1_3"/><b>then </b>
( 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"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="finseq_8.html#K1">^</a> <font color="Maroon">r2</font></span>)</span> <a href="finseq_8.html#K1">^</a> <font color="Maroon">r3</font></span>)</span> &amp;  <a href="jordan3.html#K1">mid</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="finseq_8.html#K1">^</a> <font color="Maroon">r2</font></span>)</span> <a href="finseq_8.html#K1">^</a> <font color="Maroon">r3</font></span>)</span>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> )
 
<b>by </b><i><a class="txt" href="filerec1.html#E1:37"><i><font color="Green">E33</font></i></a>, , <a class="ref" href="filerec1.html#T3" target="_self">Th3</a>, , , <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<b>hence </b><a NAME="E5:37_1_1_1_3"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 implies  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( 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"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="finseq_8.html#K1">^</a> <font color="Maroon">r2</font></span>)</span> <a href="finseq_8.html#K1">^</a> <font color="Maroon">r3</font></span>)</span> &amp;  <a href="jordan3.html#K1">mid</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="finseq_8.html#K1">^</a> <font color="Maroon">r2</font></span>)</span> <a href="finseq_8.html#K1">^</a> <font color="Maroon">r3</font></span>)</span>,<font color="Olive">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> ) )
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E11:37_1_1_1"/>
<font color="Maroon">r2</font> <a href="finseq_8.html#R2">is_substring_of</a> <font color="Maroon">f</font>,1
 <b>by </b><i>, <a class="ref" href="finseq_8.html#D7">FINSEQ_8:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2"><b>suppose </b></a><a NAME="E1:37_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:37_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><b>hence </b><a NAME="E3:37_1_1_2"/>
<font color="Maroon">r2</font> <a href="finseq_8.html#R2">is_substring_of</a> <font color="Maroon">f</font>,1
 <b>by </b><i><a class="ref" href="finseq_8.html#D7">FINSEQ_8:def 7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:37"/>
( <font color="Maroon">r1</font> <a href="finseq_8.html#R2">is_substring_of</a> <font color="Maroon">f</font>,1 &amp; <font color="Maroon">r2</font> <a href="finseq_8.html#R2">is_substring_of</a> <font color="Maroon">f</font>,1 &amp; <font color="Maroon">r3</font> <a href="finseq_8.html#R2">is_substring_of</a> <font color="Maroon">f</font>,1 )
 <b>by </b><i>, , </i>;<br/>


</div>
