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<b>let </b><font color="Maroon">T</font> be   <a href="trees_1.html#NM1">Tree</a>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>set </b><font color="Maroon">p</font> = <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">T</font></span><a href="finseq_1.html#K9">*&gt;</a></span>;<br/>
<a NAME="E1:129"/><i><font color="Green">E39</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">T</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">T</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<b>thus </b><a NAME="E2:129"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K14" target="_self">^</a> <font color="Maroon">T</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies  ex <font color="Olive">p</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Olive">p</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="129_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:129_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K14" target="_self">^</a> <font color="Maroon">T</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">q</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E3:129_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">T</font></span><a href="finseq_1.html#K9">*&gt;</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">T</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="ref" href="trees_3.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E4:129_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1
 
;<br/>
<a NAME="E5:129_1"/><b>then </b>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i><a class="ref" href="trees_3.html#T16" target="_self">Th16</a>, <a class="ref" href="trees_3.html#T17" target="_self">Th17</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E6:129_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 
;<br/>
<b>hence </b><a NAME="E7:129_1"/>
 ex <font color="Olive">p</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Olive">p</font> )
 <b>by </b><i><a class="ref" href="trees_3.html#T16" target="_self">Th16</a>, <a class="ref" href="trees_3.html#T17" target="_self">Th17</a></i>;<br/>


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

<a NAME="E3:129"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="nat_1.html#K1">+</a> 1
 
;<br/>
<b>hence </b><a NAME="E4:129"/>
( ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  ex <font color="Olive">p</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Olive">p</font> ) ) implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K14" target="_self">^</a> <font color="Maroon">T</font> )
 <b>by </b><i><a class="ref" href="trees_3.html#T16" target="_self">Th16</a>, <a class="ref" href="trees_3.html#D5" target="_self">Def5</a></i>;<br/>


</div>
