<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="bintree2.html#V1" target="_self">full</a> <a href="trees_1.html#NM1">Tree</a>;<br/>

<a NAME="E1:29"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> <a href="finseq_2.html#K3">*</a> 
 
<b>by </b><i><a class="ref" href="bintree2.html#D2" target="_self">Def2</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="trees_2.html#K2">-level</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="trees_2.html#K2">-level</a> 0</span>)</span> = 
 <a href="card_1.html#K4">card</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="qc_lang4.html#T17">QC_LANG4:17</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="card_1.html#T79">CARD_1:79</a></i>
<br/>.= 
2 <a href="series_1.html#K3">to_power</a> 0
<b>by </b><i><a class="ref" href="power.html#T29">POWER:29</a></i>
;<br/>
<a NAME="E3:29"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E4:29"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:29_2"/><i><font color="Green">E25</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="trees_2.html#K2">-level</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="bintree2.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0 ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="bintree2.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1 ) </span>}</span> ;<br/>
<a NAME="E2:29_2"/><i><font color="Green">E26</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="euclid.html#T2">EUCLID:2</a></i>;<br/>
<a NAME="E3:29_2"/><i><font color="Green">E27</font></i>: 
 <a href="euclid.html#K5">0*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="bintree2.html#E1:29"><i><font color="Green">E22</font></i></a>, <a class="ref" href="binari_3.html#T5">BINARI_3:5</a>, <a class="ref" href="margrel1.html#D12">MARGREL1:def 12</a></i>;<br/>
<a NAME="E4:29_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/>
<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>

<br/>.= 
0
<b>by </b><i><a class="txt" href="bintree2.html#E4:29_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/>
<a NAME="E6:29_2"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="euclid.html#K5">0*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="bintree2.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0 ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="bintree2.html#E2:29_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="bintree2.html#E3:29_2"><i><font color="Green">E27</font></i></a></i>;<br/>
<i><font color="Green">E30</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default">1</span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_2.html#T19">FINSEQ_2:19</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="euclid.html#T2">EUCLID:2</a></i>
;<br/>
<a NAME="E8:29_2"/>
 <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> <a href="finseq_2.html#K3">*</a> 
 
<b>by </b><i><a class="ref" href="binari_3.html#T5">BINARI_3:5</a>, <a class="ref" href="margrel1.html#D12">MARGREL1:def 12</a></i>;<br/>
<a NAME="E9:29_2"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="trees_4.html#M1">FinSequence</a> of <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E10:29_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">1</span><a href="binarith.html#K13">*&gt;</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:29_2_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">1</span><a href="binarith.html#K13">*&gt;</a></span>
 ;<br/>

<a NAME="E2:29_2_3"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/>
<a NAME="E3:29_2_3"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:29_2_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:29_2"/><b>then </b><i><font color="Green">E32</font></i>: 
<span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">1</span><a href="binarith.html#K13">*&gt;</a></span> is    <a href="trees_4.html#M1">FinSequence</a> of <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E12:29_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">1</span><a href="binarith.html#K13">*&gt;</a></span> is    <a href="trees_4.html#M1">FinSequence</a> of <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="txt" href="bintree2.html#E9:29_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="bintree2.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>
<a NAME="E13:29_2"/><b>then </b><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">1</span><a href="binarith.html#K13">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="bintree2.html#E1:29"><i><font color="Green">E22</font></i></a>, <a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E14:29_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="euclid.html#T2">EUCLID:2</a></i>;<br/>
<a NAME="E15:29_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default">1</span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/>
<a NAME="E16:29_2"/><b>then </b><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">1</span><a href="binarith.html#K13">*&gt;</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="bintree2.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="bintree2.html#E7:29_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="bintree2.html#E13:29_2"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E17:29_2"/><i><font color="Green">E35</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="bintree2.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0 ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="trees_2.html#K2">-level</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:29_2_4"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="bintree2.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0 ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="bintree2.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:29_2_4"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E4:29_2_4"/><i><font color="Green">E37</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>and </b><br/><a NAME="E5:29_2_4"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/>
<a NAME="E6:29_2_4"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="bintree2.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> :  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 </span>}</span> 
 
<b>by </b><i><a class="txt" href="bintree2.html#E4:29_2_4"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:29_2_4"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="trees_2.html#K2">-level</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="bintree2.html#E3:29_2_4"><i><font color="Green">E36</font></i></a>, <a class="ref" href="trees_2.html#D6">TREES_2:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:29_2"/><i><font color="Green">E36</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="bintree2.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1 ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="trees_2.html#K2">-level</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:29_2_5"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="bintree2.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1 ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="bintree2.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:29_2_5"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E4:29_2_5"/><i><font color="Green">E38</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>and </b><br/><a NAME="E5:29_2_5"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/>
<a NAME="E6:29_2_5"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="bintree2.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> :  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 </span>}</span> 
 
<b>by </b><i><a class="txt" href="bintree2.html#E4:29_2_5"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:29_2_5"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="trees_2.html#K2">-level</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="bintree2.html#E3:29_2_5"><i><font color="Green">E37</font></i></a>, <a class="ref" href="trees_2.html#D6">TREES_2:def 6</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="bintree2.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0 ) </span>}</span> , <font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="bintree2.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1 ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="bintree2.html#E6:29_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="bintree2.html#E16:29_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="bintree2.html#E17:29_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<a NAME="E20:29_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="trees_2.html#K2">-level</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="bintree2.html#E17:29_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="bintree2.html#E18:29_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E21:29_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="trees_2.html#K2">-level</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:29_2_6"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="trees_2.html#K2">-level</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>

<a NAME="E2:29_2_6"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="bintree2.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> :  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 </span>}</span> 
 
<b>by </b><i><a class="ref" href="trees_2.html#D6">TREES_2:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="bintree2.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:29_2_6"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E5:29_2_6"/><i><font color="Green">E40</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>
<a NAME="E6:29_2_6"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_6_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_2_6_1"/><i><font color="Green">E41</font></i>: 
not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:29_2_6_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/>
<a NAME="E3:29_2_6_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="bintree2.html#E5:29_2_6"><i><font color="Green">E40</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:29_2_6_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="margrel1.html#K6">BOOLEAN</a> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<a NAME="E5:29_2_6_1"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="margrel1.html#D12">MARGREL1:def 12</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:29_2_6_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="bintree2.html#E4:29_2_6"><i><font color="Green">E39</font></i></a>, <a class="txt" href="bintree2.html#E5:29_2_6"><i><font color="Green">E40</font></i></a>, <a class="txt" href="bintree2.html#E1:29_2_6_1"><i><font color="Green">E41</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:29_2_6"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:29_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#R1">misses</a> <font color="Maroon">c<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_2_7"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:29_2_7"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E4:29_2_7"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="bintree2.html#E3:29_2_7"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="bintree2.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:29_2_7"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E7:29_2_7"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>and </b><br/><a NAME="E8:29_2_7"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="bintree2.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E10:29_2_7"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E11:29_2_7"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>and </b><br/><a NAME="E12:29_2_7"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="bintree2.html#E4:29_2_7"><i><font color="Green">E41</font></i></a></i>;<br/>
<b>thus </b><a NAME="E13:29_2_7"/>
contradiction
 <b>by </b><i><a class="txt" href="bintree2.html#E6:29_2_7"><i><font color="Green">E42</font></i></a>, <a class="txt" href="bintree2.html#E8:29_2_7"><i><font color="Green">E43</font></i></a>, <a class="txt" href="bintree2.html#E10:29_2_7"><i><font color="Green">E44</font></i></a>, <a class="txt" href="bintree2.html#E12:29_2_7"><i><font color="Green">E45</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="trees_2.html#K2">-level</a> <font color="Maroon">c<sub>2</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="bintree2.html#E1:29"><i><font color="Green">E22</font></i></a>, <a class="ref" href="bintree2.html#T10" target="_self">Th10</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">0</span><a href="binarith.html#K13">*&gt;</a></span> );<br/>
<a NAME="E24:29_2"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font>ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font>;<br/>

<a NAME="E1:29_2_8"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="trees_2.html#K2">-level</a> <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<a NAME="E2:29_2_8"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="bintree2.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> :  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="trees_2.html#D6">TREES_2:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="bintree2.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:29_2_8"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E5:29_2_8"/><i><font color="Green">E42</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">0</span><a href="binarith.html#K13">*&gt;</a></span>;<br/>
<a NAME="E6:29_2_8"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">0</span><a href="binarith.html#K13">*&gt;</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_8_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:29_2_8_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">0</span><a href="binarith.html#K13">*&gt;</a></span>
 ;<br/>

<a NAME="E2:29_2_8_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/>
<a NAME="E3:29_2_8_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:29_2_8_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:29_2_8"/><b>then </b>
<span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">0</span><a href="binarith.html#K13">*&gt;</a></span> is    <a href="trees_4.html#M1">FinSequence</a> of <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E8:29_2_8"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">0</span><a href="binarith.html#K13">*&gt;</a></span> is    <a href="trees_4.html#M1">FinSequence</a> of <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="ref" href="margrel1.html#D12">MARGREL1:def 12</a>, <a class="txt" href="bintree2.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>
<a NAME="E9:29_2_8"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">0</span><a href="binarith.html#K13">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="bintree2.html#E1:29"><i><font color="Green">E22</font></i></a>, <a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E10:29_2_8"/><i><font color="Green">E44</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default">0</span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="bintree2.html#E5:29_2_8"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_2.html#T19">FINSEQ_2:19</a></i>;<br/>
<a NAME="E11:29_2_8"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default">0</span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="bintree2.html#E5:29_2_8"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/>
<a NAME="E12:29_2_8"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">0</span><a href="binarith.html#K13">*&gt;</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="bintree2.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0 ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="bintree2.html#E9:29_2_8"><i><font color="Green">E43</font></i></a>, <a class="txt" href="bintree2.html#E10:29_2_8"><i><font color="Green">E44</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">0</span><a href="binarith.html#K13">*&gt;</a></span> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>11</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/>

<b>thus </b><a NAME="E14:29_2_8"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">0</span><a href="binarith.html#K13">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="bintree2.html#E4:29_2_8"><i><font color="Green">E41</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">1</span><a href="binarith.html#K13">*&gt;</a></span> );<br/>
<a NAME="E25:29_2"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font>ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font>;<br/>

<a NAME="E1:29_2_9"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="trees_2.html#K2">-level</a> <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<a NAME="E2:29_2_9"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="bintree2.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> :  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="trees_2.html#D6">TREES_2:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="bintree2.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:29_2_9"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E5:29_2_9"/><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">1</span><a href="binarith.html#K13">*&gt;</a></span>;<br/>
<a NAME="E6:29_2_9"/>
<font color="Maroon">c<sub>9</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">1</span><a href="binarith.html#K13">*&gt;</a></span> is    <a href="trees_4.html#M1">FinSequence</a> of <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="txt" href="bintree2.html#E11:29_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="margrel1.html#D12">MARGREL1:def 12</a>, <a class="txt" href="bintree2.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>
<a NAME="E7:29_2_9"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">1</span><a href="binarith.html#K13">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="bintree2.html#E1:29"><i><font color="Green">E22</font></i></a>, <a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E8:29_2_9"/><i><font color="Green">E45</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default">1</span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="bintree2.html#E5:29_2_9"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_2.html#T19">FINSEQ_2:19</a></i>;<br/>
<a NAME="E9:29_2_9"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default">1</span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="bintree2.html#E5:29_2_9"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/>
<a NAME="E10:29_2_9"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">1</span><a href="binarith.html#K13">*&gt;</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="bintree2.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="bintree2.html#E7:29_2_9"><i><font color="Green">E44</font></i></a>, <a class="txt" href="bintree2.html#E8:29_2_9"><i><font color="Green">E45</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">1</span><a href="binarith.html#K13">*&gt;</a></span> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>11</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/>

<b>thus </b><a NAME="E12:29_2_9"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">1</span><a href="binarith.html#K13">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="bintree2.html#E4:29_2_9"><i><font color="Green">E42</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E27:29_2"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="bintree2.html#E24:29_2"><i><font color="Green">E40</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E29:29_2"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> holds <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="bintree2.html#E25:29_2"><i><font color="Green">E41</font></i></a>);<br/></i>
<a NAME="E30:29_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_10"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:29_2_10"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E2:29_2_10"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E3:29_2_10"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="bintree2.html#E1:29_2_10"><i><font color="Green">E45</font></i></a>, <a class="txt" href="bintree2.html#E2:29_2_10"><i><font color="Green">E46</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E6:29_2_10"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>and </b><br/><a NAME="E7:29_2_10"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">0</span><a href="binarith.html#K13">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="bintree2.html#E27:29_2"><i><font color="Green">E42</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E9:29_2_10"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E10:29_2_10"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">0</span><a href="binarith.html#K13">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="bintree2.html#E27:29_2"><i><font color="Green">E42</font></i></a></i>;<br/><b>thus </b><a NAME="E11:29_2_10"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="bintree2.html#E3:29_2_10"><i><font color="Green">E47</font></i></a>, <a class="txt" href="bintree2.html#E6:29_2_10"><i><font color="Green">E48</font></i></a>, <a class="txt" href="bintree2.html#E7:29_2_10"><i><font color="Green">E49</font></i></a>, <a class="txt" href="bintree2.html#E9:29_2_10"><i><font color="Green">E50</font></i></a>, <a class="txt" href="bintree2.html#E10:29_2_10"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_2.html#T20">FINSEQ_2:20</a></i>;<br/></div><b>end;</b></div>
<a NAME="E32:29_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E33:29_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="txt" href="bintree2.html#E30:29_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="card_1.html#T60">CARD_1:60</a></i>;<br/>
<a NAME="E34:29_2"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>7</sub></font>, <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T45">FUNCT_2:45</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_11"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_2_11"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="bintree2.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:29_2_11"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E4:29_2_11"/><i><font color="Green">E47</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>and </b><br/><a NAME="E5:29_2_11"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="margrel1.html#K6">BOOLEAN</a> , <font color="Maroon">c<sub>13</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="margrel1.html#K6">BOOLEAN</a> <b> such that </b><br/><a NAME="E7:29_2_11"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="binarith.html#K13">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="bintree2.html#E4:29_2_11"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_2.html#T22">FINSEQ_2:22</a></i>;<br/><a NAME="E8:29_2_11"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="bintree2.html#E4:29_2_11"><i><font color="Green">E47</font></i></a>, <a class="txt" href="bintree2.html#E7:29_2_11"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finseq_2.html#T19">FINSEQ_2:19</a></i>;<br/><a NAME="E9:29_2_11"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></font> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font>;<br/><a NAME="E11:29_2_11"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bintree2.html#E1:29"><i><font color="Green">E22</font></i></a>, <a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a>, <a class="ref" href="margrel1.html#D12">MARGREL1:def 12</a></i>;<br/><a NAME="E12:29_2_11"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="bintree2.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> :  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="bintree2.html#E9:29_2_11"><i><font color="Green">E51</font></i></a></i>;<br/><b>hence </b><a NAME="E13:29_2_11"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="trees_2.html#D6">TREES_2:def 6</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="bintree2.html#E12:29_2_11"><i><font color="Green">E52</font></i></a>, <a class="ref" href="trees_2.html#D6">TREES_2:def 6</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E16:29_2_11"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>and </b><br/><a NAME="E17:29_2_11"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">0</span><a href="binarith.html#K13">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="bintree2.html#E27:29_2"><i><font color="Green">E42</font></i></a></i>;<br/><b>thus </b><a NAME="E18:29_2_11"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="bintree2.html#E3:29_2_11"><i><font color="Green">E46</font></i></a>, <a class="txt" href="bintree2.html#E5:29_2_11"><i><font color="Green">E48</font></i></a>, <a class="txt" href="bintree2.html#E7:29_2_11"><i><font color="Green">E49</font></i></a>, <a class="txt" href="bintree2.html#E8:29_2_11"><i><font color="Green">E50</font></i></a>, <a class="txt" href="bintree2.html#E16:29_2_11"><i><font color="Green">E53</font></i></a>, <a class="txt" href="bintree2.html#E17:29_2_11"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/></div><b>end;</b></div>
<a NAME="E36:29_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T16">FUNCT_2:16</a></i>;<br/>
<a NAME="E37:29_2"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="bintree2.html#E34:29_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="card_1.html#T81">CARD_1:81</a></i>;<br/>
<a NAME="E38:29_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_12"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:29_2_12"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E2:29_2_12"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E3:29_2_12"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="bintree2.html#E1:29_2_12"><i><font color="Green">E48</font></i></a>, <a class="txt" href="bintree2.html#E2:29_2_12"><i><font color="Green">E49</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E6:29_2_12"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>and </b><br/><a NAME="E7:29_2_12"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">1</span><a href="binarith.html#K13">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="bintree2.html#E29:29_2"><i><font color="Green">E43</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E9:29_2_12"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E10:29_2_12"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">1</span><a href="binarith.html#K13">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="bintree2.html#E29:29_2"><i><font color="Green">E43</font></i></a></i>;<br/><b>thus </b><a NAME="E11:29_2_12"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="bintree2.html#E3:29_2_12"><i><font color="Green">E50</font></i></a>, <a class="txt" href="bintree2.html#E6:29_2_12"><i><font color="Green">E51</font></i></a>, <a class="txt" href="bintree2.html#E7:29_2_12"><i><font color="Green">E52</font></i></a>, <a class="txt" href="bintree2.html#E9:29_2_12"><i><font color="Green">E53</font></i></a>, <a class="txt" href="bintree2.html#E10:29_2_12"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_2.html#T20">FINSEQ_2:20</a></i>;<br/></div><b>end;</b></div>
<a NAME="E40:29_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E41:29_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="txt" href="bintree2.html#E38:29_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="card_1.html#T60">CARD_1:60</a></i>;<br/>
<a NAME="E42:29_2"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>7</sub></font>, <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T45">FUNCT_2:45</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_13"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_2_13"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="bintree2.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:29_2_13"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E4:29_2_13"/><i><font color="Green">E50</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>and </b><br/><a NAME="E5:29_2_13"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="margrel1.html#K6">BOOLEAN</a> , <font color="Maroon">c<sub>13</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="margrel1.html#K6">BOOLEAN</a> <b> such that </b><br/><a NAME="E7:29_2_13"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="binarith.html#K13">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="bintree2.html#E4:29_2_13"><i><font color="Green">E50</font></i></a>, <a class="ref" href="finseq_2.html#T22">FINSEQ_2:22</a></i>;<br/><a NAME="E8:29_2_13"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="bintree2.html#E4:29_2_13"><i><font color="Green">E50</font></i></a>, <a class="txt" href="bintree2.html#E7:29_2_13"><i><font color="Green">E52</font></i></a>, <a class="ref" href="finseq_2.html#T19">FINSEQ_2:19</a></i>;<br/><a NAME="E9:29_2_13"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></font> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font>;<br/><a NAME="E11:29_2_13"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bintree2.html#E1:29"><i><font color="Green">E22</font></i></a>, <a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a>, <a class="ref" href="margrel1.html#D12">MARGREL1:def 12</a></i>;<br/><a NAME="E12:29_2_13"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="bintree2.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> :  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="bintree2.html#E9:29_2_13"><i><font color="Green">E54</font></i></a></i>;<br/><b>hence </b><a NAME="E13:29_2_13"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="trees_2.html#D6">TREES_2:def 6</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="bintree2.html#E12:29_2_13"><i><font color="Green">E55</font></i></a>, <a class="ref" href="trees_2.html#D6">TREES_2:def 6</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E16:29_2_13"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>and </b><br/><a NAME="E17:29_2_13"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">1</span><a href="binarith.html#K13">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="bintree2.html#E29:29_2"><i><font color="Green">E43</font></i></a></i>;<br/><b>thus </b><a NAME="E18:29_2_13"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="bintree2.html#E3:29_2_13"><i><font color="Green">E49</font></i></a>, <a class="txt" href="bintree2.html#E5:29_2_13"><i><font color="Green">E51</font></i></a>, <a class="txt" href="bintree2.html#E7:29_2_13"><i><font color="Green">E52</font></i></a>, <a class="txt" href="bintree2.html#E8:29_2_13"><i><font color="Green">E53</font></i></a>, <a class="txt" href="bintree2.html#E16:29_2_13"><i><font color="Green">E56</font></i></a>, <a class="txt" href="bintree2.html#E17:29_2_13"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/></div><b>end;</b></div>
<a NAME="E44:29_2"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T16">FUNCT_2:16</a></i>;<br/>
<b>thus </b>2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T32">POWER:32</a></i>
<br/>.= 
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="bintree2.html#E1:29_2"><i><font color="Green">E25</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="bintree2.html#E37:29_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="bintree2.html#E42:29_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="bintree2.html#E44:29_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="card_1.html#T81">CARD_1:81</a></i>
<br/>.= 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="bintree2.html#E22:29_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="card_2.html#T53">CARD_2:53</a></i>
<br/>.= 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="trees_2.html#K2">-level</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="bintree2.html#E20:29_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="bintree2.html#E21:29_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>
;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E5:29"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="bintree2.html#E3:29"><i><font color="Green">E23</font></i></a>, <a class="txt" href="bintree2.html#E4:29"><i><font color="Green">E24</font></i></a>);</i><br/>


</div>
