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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">f</font>, <font color="Olive" title="b2">CR</font> being    <a href="filerec1.html#NM1" title="FILEREC1:NM.1">File</a> of <font color="Maroon" title="c1">D</font> holds  <font color="Olive" title="b1">f</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <font color="Olive" title="b2">CR</font> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">CR</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="c2">f</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <font color="Maroon" title="c3">CR</font> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> </span><br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:26_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</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="c2">f</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"><span class="kw">suppose </span></a><a NAME="E1:26_1_1"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <font color="Maroon" title="c3">CR</font> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> </span><br/><div class="add"><a NAME="E2:26_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></span>;<br/><a NAME="E3:26_1_1"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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">CR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:26_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A2</font></span>: 
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">f</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <font color="Maroon" title="c3">CR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>;<br/><a NAME="E5:26_1_1"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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> <font color="Maroon" title="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E6:26_1_1"/><span class="kw">then </span>
 <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <font color="Maroon" title="c3">CR</font></span>)</span>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_8.html#T1" title="FINSEQ_8:th.1">FINSEQ_8:1</a></span>;<br/><span class="kw">hence </span><a NAME="E7:26_1_1"/>
<font color="Maroon" title="c2">f</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <font color="Maroon" title="c3">CR</font> <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="#E4:26_1_1"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="ref" href="finseq_8.html#D8" title="FINSEQ_8:def.8">FINSEQ_8:def 8</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_2"><span class="kw">suppose </span></a><a NAME="E1:26_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <font color="Maroon" title="c3">CR</font> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:26_1_2"/>
<font color="Maroon" title="c2">f</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <font color="Maroon" title="c3">CR</font> <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="ref" href="finseq_8.html#D8" title="FINSEQ_8:def.8">FINSEQ_8:def 8</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></div><span class="kw">end;</span></div>

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