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<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#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font>, <font color="Olive" title="b3">r3</font>, <font color="Olive" title="b4">f</font> being    <a href="filerec1.html#NM1" title="FILEREC1:NM.1">File</a> of <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">r1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">r2</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b3">r3</font> holds <br/>( <font color="Olive" title="b1">r1</font> <a href="finseq_8.html#R2" title="FINSEQ_8:pred.2">is_substring_of</a> <font color="Olive" title="b4">f</font>,1 &amp; <font color="Olive" title="b2">r2</font> <a href="finseq_8.html#R2" title="FINSEQ_8:pred.2">is_substring_of</a> <font color="Olive" title="b4">f</font>,1 &amp; <font color="Olive" title="b3">r3</font> <a href="finseq_8.html#R2" title="FINSEQ_8:pred.2">is_substring_of</a> <font color="Olive" title="b4">f</font>,1 )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">r1</font>, <font color="Maroon" title="c3">r2</font>, <font color="Maroon" title="c4">r3</font>, <font color="Maroon" title="c5">f</font> be    <a href="filerec1.html#NM1" title="FILEREC1:NM.1">File</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">r2</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">r3</font> implies ( <font color="Maroon" title="c2">r1</font> <a href="finseq_8.html#R2" title="FINSEQ_8:pred.2">is_substring_of</a> <font color="Maroon" title="c5">f</font>,1 &amp; <font color="Maroon" title="c3">r2</font> <a href="finseq_8.html#R2" title="FINSEQ_8:pred.2">is_substring_of</a> <font color="Maroon" title="c5">f</font>,1 &amp; <font color="Maroon" title="c4">r3</font> <a href="finseq_8.html#R2" title="FINSEQ_8:pred.2">is_substring_of</a> <font color="Maroon" title="c5">f</font>,1 ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:36"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">r2</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">r3</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">r1</font> <a href="finseq_8.html#R2" title="FINSEQ_8:pred.2">is_substring_of</a> <font color="Maroon" title="c5">f</font>,1 &amp; <font color="Maroon" title="c3">r2</font> <a href="finseq_8.html#R2" title="FINSEQ_8:pred.2">is_substring_of</a> <font color="Maroon" title="c5">f</font>,1 &amp; <font color="Maroon" title="c4">r3</font> <a href="finseq_8.html#R2" title="FINSEQ_8:pred.2">is_substring_of</a> <font color="Maroon" title="c5">f</font>,1 )</span><br/>

<a NAME="E2:36"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">r2</font></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">r1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.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="c3">r2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E23">A3</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="36_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">r2</font> <a href="finseq_8.html#R2" title="FINSEQ_8:pred.2">is_substring_of</a> <font color="Maroon" title="c5">f</font>,1</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:36_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:36_1_1_1"/><span class="lab"><font color="Green" title="E24">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">r2</font> <a href="finseq_8.html#R2" title="FINSEQ_8:pred.2">is_substring_of</a> <font color="Maroon" title="c5">f</font>,1</span><br/><div class="add"><span class="lab"><font color="Green" title="E25">A5</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">r1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r2</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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">r1</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">r1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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">r1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r2</font>

;<br/><a NAME="E3:36_1_1_1"/><span class="lab"><font color="Green" title="E26">A6</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">r1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.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="c3">r2</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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">r1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36_1_1_1"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E4:36_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36_1_1_1"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E5:36_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A7</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">r1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">r2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A8</font></span>:  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">r1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">r2</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">r3</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">r1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">r1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">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" title="c2">r1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">r2</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">r3</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</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" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">r1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">r1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">r2</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">r1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_6.html#D3" onmouseover="rs('finseq_6/D3')" onmouseout="rh()">FINSEQ_6:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">r1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">r2</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">r3</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">r1</font></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</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" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">r1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">r2</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">r1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" onmouseover="rs('nat_d/T34')" onmouseout="rh()">NAT_D:34</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">r1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">r2</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">r3</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">r1</font></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</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" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">r1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r2</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">r1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">r1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">r2</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">r3</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">r1</font></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</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" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">r1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r2</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">r1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36_1_1_1"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="ref" href="nat_2.html#T7" onmouseover="rs('nat_2/T7')" onmouseout="rh()">NAT_2:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">r1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">r2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">r3</font></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">r1</font></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36_1_1_1"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="ref" href="finseq_1.html#T32" onmouseover="rs('finseq_1/T32')" onmouseout="rh()">FINSEQ_1:32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">r3</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r2</font></span>)</span>

<br/>.= 
<font color="Maroon" title="c3">r2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T23" onmouseover="rs('finseq_5/T23')" onmouseout="rh()">FINSEQ_5:23</a></span>
;<br/><a NAME="E7:36_1_1_1"/><span class="lab"><font color="Green" title="E29">A9</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">r1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E8:36_1_1_1"/><span class="lab"><font color="Green" title="E30">A10</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">r1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">r1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r2</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.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="c4">r3</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="txt" href="#E5:36_1_1_1"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/><a NAME="E9:36_1_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  ex <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">r1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">r2</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">r3</font></span>)</span> &amp;  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">r1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">r2</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">r3</font></span>)</span>,<font color="Olive" title="b1">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r2</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r2</font> ) )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="lab"><font color="Green" title="E31">A11</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">r1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.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="c3">r2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">r1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.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="c3">r2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" onmouseover="rs('nat_d/T34')" onmouseout="rh()">NAT_D:34</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">r2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:36_1_1_1_3"/><span class="lab"><font color="Green" title="E32">A12</font></span>: 
<font color="Maroon" title="c6">i</font> <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">r1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/>
<a NAME="E4:36_1_1_1_3"/>
<font color="Maroon" title="c6">i</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">r1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">r2</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">r3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="txt" href="#E8:36_1_1_1"><span class="lab"><font color="Green" title="E30">A10</font></span></a>, <a class="txt" href="#E3:36_1_1_1_3"><span class="lab"><font color="Green" title="E32">A12</font></span></a>, <a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:36_1_1_1_3"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  ex <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">r1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">r2</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">r3</font></span>)</span> &amp;  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">r1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">r2</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">r3</font></span>)</span>,<font color="Olive" title="b1">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r2</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r2</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:36_1_1_1"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="txt" href="#E6:36_1_1_1"><span class="lab"><font color="Green" title="E28">A8</font></span></a>, <a class="txt" href="#E3:36_1_1_1_3"><span class="lab"><font color="Green" title="E32">A12</font></span></a>, <a class="txt" href="#E1:36_1_1_1_3"><span class="lab"><font color="Green" title="E31">A11</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><span class="kw">hence </span><a NAME="E10:36_1_1_1"/>
<font color="Maroon" title="c3">r2</font> <a href="finseq_8.html#R2" title="FINSEQ_8:pred.2">is_substring_of</a> <font color="Maroon" title="c5">f</font>,1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="finseq_8.html#D7" onmouseover="rs('finseq_8/D7')" onmouseout="rh()">FINSEQ_8:def 7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:36_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">r2</font> <a href="finseq_8.html#R2" title="FINSEQ_8:pred.2">is_substring_of</a> <font color="Maroon" title="c5">f</font>,1</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:36_1_1_2"/>
<font color="Maroon" title="c3">r2</font> <a href="finseq_8.html#R2" title="FINSEQ_8:pred.2">is_substring_of</a> <font color="Maroon" title="c5">f</font>,1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_8.html#D7" onmouseover="rs('finseq_8/D7')" onmouseout="rh()">FINSEQ_8:def 7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E4:36"/>
<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">r3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="finseq_1.html#T32" onmouseover="rs('finseq_1/T32')" onmouseout="rh()">FINSEQ_1:32</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:36"/>
( <font color="Maroon" title="c2">r1</font> <a href="finseq_8.html#R2" title="FINSEQ_8:pred.2">is_substring_of</a> <font color="Maroon" title="c5">f</font>,1 &amp; <font color="Maroon" title="c3">r2</font> <a href="finseq_8.html#R2" title="FINSEQ_8:pred.2">is_substring_of</a> <font color="Maroon" title="c5">f</font>,1 &amp; <font color="Maroon" title="c4">r3</font> <a href="finseq_8.html#R2" title="FINSEQ_8:pred.2">is_substring_of</a> <font color="Maroon" title="c5">f</font>,1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="ref" href="filerec1.html#T34" target="_self" onmouseover="rs('filerec1/T34')" onmouseout="rh()">Th34</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
