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<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>set </b><font color="Maroon">p</font> = <font color="Maroon">i</font> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p1">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 0</span>)</span>;<br/>
<b>let </b><font color="Maroon">q</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="trees_2.html#D1">TREES_2:def 1</a><br/>

<a NAME="E1:120"/><i><font color="Green">E39</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p2">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 0</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></i>;<br/>
<a NAME="E2:120"/><b>then </b>
 <a href="trees_1.html#K2">elementary_tree</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a>  <a href="trees_3.html#K13" target="_self">tree</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p2">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 0</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="trees_3.html#T9" target="_self">Th9</a></i>;<br/>
<b>hence </b><a NAME="E3:120"/>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K2">elementary_tree</a> <font color="Maroon">i</font> implies <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K13" target="_self">tree</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p2">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 0</span>)</span></span>)</span> )
 ;<br/>

<b>assume </b><a NAME="E4:120"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K13" target="_self">tree</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p2">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 0</span>)</span></span>)</span>
 ;<br/>

<a NAME="E5:120"/><b>then </b><i><font color="Green">E40</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ex <font color="Olive">r</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p2">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 0</span>)</span></span>)</span> &amp; <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p2">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 0</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Olive">r</font> ) )
 
<b>by </b><i><a class="ref" href="trees_3.html#D5" target="_self">Def5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1"><b>now </b></a><div class="add"><b>given </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">r</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><a NAME="E1:120_1"/><i><font color="Green">E47</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">(<span class="default"><font color="Maroon">i</font> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p2">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 0</span>)</span></span>)</span> &amp; <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p2">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 0</span>)</span></span>)</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">q</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">r</font> )
 ;<br/><a NAME="E2:120_1"/>
<font color="Maroon">i</font> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p1">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">i</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p1">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 0</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_2.html#D2">FINSEQ_2:def 2</a></i>;<br/><a NAME="E3:120_1"/><b>then </b>
(  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p2">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 0</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 0</span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p2">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 0</span>)</span></span>)</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> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p2">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 0</span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="trees_3.html#T15" target="_self">Th15</a>, <a class="ref" href="trees_3.html#D3" target="_self">Def3</a>, <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/><a NAME="E4:120_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p2">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 0</span>)</span></span>)</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> <a href="hidden.html#R1">=</a>  <a href="trees_1.html#K2">elementary_tree</a> 0
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:120_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="trees_3.html#T15" target="_self">Th15</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="trees_1.html#T56">TREES_1:56</a></i>;<br/><a NAME="E6:120_1"/><b>then </b>
<font color="Maroon">q</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>
 <b>by </b><i><a class="ref" href="trees_3.html#T15" target="_self">Th15</a>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/><b>hence </b><a NAME="E7:120_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K2">elementary_tree</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="trees_3.html#T13" target="_self">Th13</a>, <a class="ref" href="trees_3.html#T15" target="_self">Th15</a>, <a class="ref" href="trees_1.html#T55">TREES_1:55</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:120"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K2">elementary_tree</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="trees_3.html#T14" target="_self">Th14</a>, <a class="ref" href="trees_1.html#T47">TREES_1:47</a></i>;<br/>


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