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<b>let </b><font color="Maroon">p</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">k</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:88"/><i><font color="Green">E40</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>and </b><br/><a NAME="E2:88"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E3:88"/>
( <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> implies <font color="Maroon">p</font> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">A</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:88_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>thus </b><font color="Maroon">p</font> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">A</font> = 
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="finseq_3.html#T30" target="_self">Th30</a>, <a class="ref" href="finseq_3.html#T31" target="_self">Th31</a>, <a class="ref" href="finseq_3.html#T4" target="_self">Th4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">A</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">A</font></span>)</span> <a href="finseq_1.html#K7">^</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="finseq_3.html#T33" target="_self">Th33</a>, <a class="ref" href="finseq_3.html#T20" target="_self">Th20</a></i>
<br/>.= 
<font color="Maroon">q</font> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>
;<br/>


</div><b>end;</b></div>

<b>assume </b><b>that </b><br/><a NAME="E4:88"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">p</font> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E5:88"/><i><font color="Green">E50</font></i>: 
not <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<font color="Maroon">q</font> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">A</font> = 
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="finseq_3.html#T30" target="_self">Th30</a>, <a class="ref" href="finseq_3.html#T31" target="_self">Th31</a>, <a class="ref" href="finseq_3.html#T33" target="_self">Th33</a>, <a class="ref" href="finseq_3.html#T4" target="_self">Th4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">A</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">A</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_3.html#T38" target="_self">Th38</a>, <a class="ref" href="finseq_3.html#T19" target="_self">Th19</a></i>
;<br/>
<a NAME="E7:88"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="trees_1.html#T5">TREES_1:5</a></i>;<br/>
<b>hence </b><a NAME="E8:88"/>
contradiction
 ;<br/>


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