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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#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">CR</font>, <font color="Olive" title="b2">r1</font>, <font color="Olive" title="b3">r2</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="b2">r1</font> <a href="finseq_8.html#R5" title="FINSEQ_8:pred.5">is_terminated_by</a> <font color="Olive" title="b1">CR</font> &amp; <font color="Olive" title="b3">r2</font> <a href="finseq_8.html#R5" title="FINSEQ_8:pred.5">is_terminated_by</a> <font color="Olive" title="b1">CR</font> holds <br/><font color="Olive" title="b1">CR</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <font color="Olive" title="b3">r2</font> <a href="finseq_8.html#R2" title="FINSEQ_8:pred.2">is_substring_of</a>  <a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">r1</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <font color="Olive" title="b3">r2</font></span>)</span>,<font color="Olive" title="b1">CR</font>),1</span><br/><span class="kw">let </span><font color="Maroon" title="c2">CR</font>, <font color="Maroon" title="c3">r1</font>, <font color="Maroon" title="c4">r2</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"> ( <font color="Maroon" title="c3">r1</font> <a href="finseq_8.html#R5" title="FINSEQ_8:pred.5">is_terminated_by</a> <font color="Maroon" title="c2">CR</font> &amp; <font color="Maroon" title="c4">r2</font> <a href="finseq_8.html#R5" title="FINSEQ_8:pred.5">is_terminated_by</a> <font color="Maroon" title="c2">CR</font> implies <font color="Maroon" title="c2">CR</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <font color="Maroon" title="c4">r2</font> <a href="finseq_8.html#R2" title="FINSEQ_8:pred.2">is_substring_of</a>  <a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <font color="Maroon" title="c4">r2</font></span>)</span>,<font color="Maroon" title="c2">CR</font>),1 )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:38"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<font color="Maroon" title="c3">r1</font> <a href="finseq_8.html#R5" title="FINSEQ_8:pred.5">is_terminated_by</a> <font color="Maroon" title="c2">CR</font>
 <span class="kw">and </span><br/><a NAME="E2:38"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
<font color="Maroon" title="c4">r2</font> <a href="finseq_8.html#R5" title="FINSEQ_8:pred.5">is_terminated_by</a> <font color="Maroon" title="c2">CR</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">CR</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <font color="Maroon" title="c4">r2</font> <a href="finseq_8.html#R2" title="FINSEQ_8:pred.2">is_substring_of</a>  <a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <font color="Maroon" title="c4">r2</font></span>)</span>,<font color="Maroon" title="c2">CR</font>),1</span><br/>

<a NAME="E3:38"/>
<font color="Maroon" title="c4">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> (<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c2">CR</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="filerec1.html#T28" target="_self" title="FILEREC1:th.28">Th29</a></span>;<br/>
<a NAME="E4:38"/><span class="kw">then </span>
<font color="Maroon" title="c4">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_8.html#K5" title="FINSEQ_8:func.5">ovlcon</a> (<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c2">CR</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_8.html#D11" title="FINSEQ_8:def.11">FINSEQ_8:def 11</a></span>;<br/>
<a NAME="E5:38"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A3</font></span>: 
<font color="Maroon" title="c4">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r2</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">CR</font> <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> <span class="p3">(<span class="default"><a href="finseq_8.html#K4" title="FINSEQ_8:func.4">ovlpart</a> (<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c2">CR</font>)</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_8.html#D3" title="FINSEQ_8:def.3">FINSEQ_8:def 3</a></span>;<br/>
<a NAME="E6:38"/>
<font color="Maroon" title="c3">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> (<font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c2">CR</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="filerec1.html#T28" target="_self" title="FILEREC1:th.28">Th29</a></span>;<br/>
<a NAME="E7:38"/><span class="kw">then </span>
<font color="Maroon" title="c3">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_8.html#K5" title="FINSEQ_8:func.5">ovlcon</a> (<font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c2">CR</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_8.html#D11" title="FINSEQ_8:def.11">FINSEQ_8:def 11</a></span>;<br/>
<a NAME="E8:38"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A4</font></span>: 
<font color="Maroon" title="c3">r1</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <font color="Maroon" title="c4">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</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="c3">r1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p5">(<span class="default"><a href="finseq_8.html#K4" title="FINSEQ_8:func.4">ovlpart</a> (<font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c2">CR</font>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <font color="Maroon" title="c2">CR</font></span>)</span> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r2</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">CR</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <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"><a href="finseq_8.html#K4" title="FINSEQ_8:func.4">ovlpart</a> (<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c2">CR</font>)</span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:38"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="ref" href="finseq_8.html#T11" title="FINSEQ_8:th.11">FINSEQ_8:11</a></span>;<br/>
 <a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <font color="Maroon" title="c4">r2</font></span>)</span>,<font color="Maroon" title="c2">CR</font>) = 
 <a href="finseq_8.html#K5" title="FINSEQ_8:func.5">ovlcon</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <font color="Maroon" title="c4">r2</font></span>)</span>,<font color="Maroon" title="c2">CR</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_8.html#D11" title="FINSEQ_8:def.11">FINSEQ_8:def 11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <font color="Maroon" title="c4">r2</font></span>)</span> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">CR</font> <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> <span class="p3">(<span class="default"><a href="finseq_8.html#K4" title="FINSEQ_8:func.4">ovlpart</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <font color="Maroon" title="c4">r2</font></span>)</span>,<font color="Maroon" title="c2">CR</font>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_8.html#D3" title="FINSEQ_8:def.3">FINSEQ_8: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="c3">r1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p0">(<span class="default"><a href="finseq_8.html#K4" title="FINSEQ_8:func.4">ovlpart</a> (<font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c2">CR</font>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">CR</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <font color="Maroon" title="c4">r2</font></span>)</span></span>)</span> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">CR</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <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"><a href="finseq_8.html#K4" title="FINSEQ_8:func.4">ovlpart</a> (<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c2">CR</font>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">CR</font> <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> <span class="p3">(<span class="default"><a href="finseq_8.html#K4" title="FINSEQ_8:func.4">ovlpart</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <font color="Maroon" title="c4">r2</font></span>)</span>,<font color="Maroon" title="c2">CR</font>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:38"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="ref" href="filerec1.html#T1" target="_self" title="FILEREC1:th.1">Th1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</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="c3">r1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p5">(<span class="default"><a href="finseq_8.html#K4" title="FINSEQ_8:func.4">ovlpart</a> (<font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c2">CR</font>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">CR</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <font color="Maroon" title="c4">r2</font></span>)</span></span>)</span> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">CR</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <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"><a href="finseq_8.html#K4" title="FINSEQ_8:func.4">ovlpart</a> (<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c2">CR</font>)</span>)</span></span>)</span></span>)</span> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">CR</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <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"><a href="finseq_8.html#K4" title="FINSEQ_8:func.4">ovlpart</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <font color="Maroon" title="c4">r2</font></span>)</span>,<font color="Maroon" title="c2">CR</font>)</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T32" title="FINSEQ_1:th.32">FINSEQ_1:32</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E10:38"/>
<font color="Maroon" title="c2">CR</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <font color="Maroon" title="c4">r2</font> <a href="finseq_8.html#R2" title="FINSEQ_8:pred.2">is_substring_of</a>  <a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <font color="Maroon" title="c4">r2</font></span>)</span>,<font color="Maroon" title="c2">CR</font>),1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="filerec1.html#T35" target="_self" title="FILEREC1:th.35">Th36</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/>


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