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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> ;<br/><b>let </b><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">g</font> be    <a href="filerec1.html#NM1" title="FILEREC1:NM.1">File</a> of <font color="Maroon" title="c1">D</font>;<br/><b>let </b><font color="Maroon" title="c4">n</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> ;<br/>





<b>assume </b><a NAME="E1:40"/><i><font color="Green" title="E24">A1</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c4">n</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> )
 ;<br/>

<a NAME="E2:40"/><i><font color="Green" title="E25">A2</font></i>: 
 <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#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/>
<b>assume </b><a NAME="E3:40"/><i><font color="Green" title="E26">A3</font></i>: 
 <a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font>
 ;<br/>

<a NAME="E4:40"/><b>then </b><i><font color="Green" title="E27">A4</font></i>: 
 <a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&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">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E5:40"/>
 <a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">f</font>,<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> 
 
<b>by </b><i><a class="txt" href="filerec1.html#E3:40"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/>
<a NAME="E6:40"/><b>then </b><i><font color="Green" title="E28">A5</font></i>: 
( <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> &amp; <font color="Maroon" title="c2">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a>  &amp; ( for <font color="Olive" title="b1">j</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="b1">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Olive" title="b1">j</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>  &amp; <font color="Maroon" title="c2">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a>  holds <br/><font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> ) )
 
<b>by </b><i><a class="ref" href="finseq_8.html#D10" title="FINSEQ_8:def.10">FINSEQ_8:def 10</a></i>;<br/>
<a NAME="E7:40"/><i><font color="Green" title="E29">A6</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="txt" href="filerec1.html#E4:40"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E8:40"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <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="E9:40"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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="filerec1.html#E7:40"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<a NAME="E10:40"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_5.html#T35" title="FINSEQ_5:th.35">FINSEQ_5:35</a></i>;<br/>
<a NAME="E11:40"/><b>then </b>
 <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="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E12:40"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="filerec1.html#E2:40"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="filerec1.html#E6:40"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="finseq_8.html#D8" title="FINSEQ_8:def.8">FINSEQ_8:def 8</a></i>;<br/>
<a NAME="E13:40"/><b>then </b>
<font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>
<b>hence </b><a NAME="E14:40"/>
contradiction
 <b>by </b><i><a class="txt" href="filerec1.html#E1:40"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="filerec1.html#E3:40"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="filerec1.html#T38" target="_self" title="FILEREC1:th.38">Th38</a></i>;<br/>


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