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<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">f</font>, <font color="Olive" title="b2">g</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="b3">n</font>, <font color="Olive" title="b4">m</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 <font color="Olive" title="b4">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b3">n</font> &amp; <font color="Olive" title="b2">g</font> <a href="finseq_8.html#R2" title="FINSEQ_8:pred.2">is_substring_of</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b4">m</font> holds <br/><font color="Olive" title="b2">g</font> <a href="finseq_8.html#R2" title="FINSEQ_8:pred.2">is_substring_of</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">n</font></span><br/><b>let </b><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">g</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">n</font>, <font color="Olive" title="b2">m</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 <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">n</font> &amp; <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="Olive" title="b2">m</font> holds <br/><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="Olive" title="b1">n</font></span><br/><b>let </b><font color="Maroon" title="c4">n</font>, <font color="Maroon" title="c5">m</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"> ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">n</font> &amp; <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="c5">m</font> implies <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><br/>





<b>assume </b><a NAME="E1:26"/><i><font color="Green" title="E14">A1</font></i>: 
( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">n</font> &amp; <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="c5">m</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1"><b>case </b></a><a NAME="E1:26_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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><br/><div class="add"><b>then consider </b><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#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:26_1_1_1"/><i><font color="Green" title="E15">A2</font></i>: 
( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">i</font> &amp; <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> <font color="Maroon" title="c2">f</font> &amp;  <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">i</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> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> )
 <b>by </b><i><a class="txt" href="#E1:26"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="finseq_8.html#D7" target="_self" title="FINSEQ_8:def.7">Def7</a></i>;<br/><a NAME="E4:26_1_1_1"/>
<font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">i</font>
 <b>by </b><i><a class="txt" href="#E1:26"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="#E3:26_1_1_1"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:26_1_1_1"/>
<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>
 <b>by </b><i><a class="txt" href="#E3:26_1_1_1"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="finseq_8.html#D7" target="_self" title="FINSEQ_8:def.7">Def7</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><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_2"><b>case </b></a><a NAME="E1:26_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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><br/><div class="add"><b>hence </b><a NAME="E2:26_1_1_2"/>
<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>
 <b>by </b><i><a class="ref" href="finseq_8.html#D7" target="_self" title="FINSEQ_8:def.7">Def7</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><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:26"/>
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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