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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">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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></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"><b>suppose </b></a><a NAME="E1:26_1_1"/><i><font color="Green" title="E15">A1</font></i>: 
 <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:()"><i><font color="Red">::  thesis: </font></i></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"/><i><font color="Green" title="E16">A2</font></i>: 
 <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> 
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E3:26_1_1"/><i><font color="Green" title="E17">A3</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="#E1:26_1_1"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:26_1_1"/><b>then </b>
<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>
 <b>by </b><i><a class="txt" href="#E2:26_1_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E5:26_1_1"/><b>then </b>
( 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> &amp;  <a href="jordan3.html#K1" title="JORDAN3:func.1">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> )
 <b>by </b><i><a class="txt" href="#E3:26_1_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a>, <a class="ref" href="finseq_8.html#T1" title="FINSEQ_8:th.1">FINSEQ_8:1</a></i>;<br/><b>hence </b><a NAME="E6:26_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> 
 <b>by </b><i><a class="ref" href="finseq_8.html#D8" title="FINSEQ_8:def.8">FINSEQ_8:def 8</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_2"><b>suppose </b></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:()"><i><font color="Red">::  thesis: </font></i></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"><b>hence </b><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> 
 <b>by </b><i><a class="ref" href="finseq_8.html#D8" title="FINSEQ_8:def.8">FINSEQ_8:def 8</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>

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