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

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

<b>thus </b><a NAME="E1:9"/>
( <font color="Maroon">T</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_1.html#K13">*</a>  implies for <font color="Olive">t</font> being   <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> holds   <a href="trees_2.html#K1">succ</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span><a href="domain_1.html#K7">}</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">T</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_1.html#K13">*</a> 
 ;<br/>

<a NAME="E2:9_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">T</font> is <a href="bintree1.html#V1">binary</a>
 
<b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">t</font> be    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font>;<br/>

<a NAME="E3:9_1"/>
not <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <font color="Maroon">T</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:9_1"/>
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i>, <a class="ref" href="bintree1.html#D2">BINTREE1:def 2</a></i>;<br/>


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

<b>assume </b><a NAME="E2:9"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">t</font> being   <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> holds   <a href="trees_2.html#K1">succ</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 ;<br/>

<b>thus </b><a NAME="E3:9"/>
<font color="Maroon">T</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_1.html#K13">*</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:9_2"/>
<font color="Maroon">T</font> <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> <a href="finseq_1.html#K13">*</a> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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/>

<a NAME="E1:9_2_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">T</font> is <a href="bintree1.html#V1">binary</a>
 
<b>by </b><i>, </i>;<br/>
<b>assume </b><a NAME="E2:9_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 ;<br/>

<a NAME="E3:9_2_1"/><b>then </b>
<font color="Maroon">x</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>, </i>;<br/>
<b>hence </b><a NAME="E4:9_2_1"/>
<font color="Maroon">x</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_1.html#K13">*</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>


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

<b>let </b><font color="Maroon">x</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="E2:9_2"/>
<font color="Maroon">x</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_1.html#K13">*</a> 
 ;<br/>

<a NAME="E3:9_2"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">x</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/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="finseq_1.html#NM1">FinSequence</a>] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>;<br/>
<a NAME="E4:9_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K6">&lt;*&gt;</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="trees_1.html#T47">TREES_1:47</a></i>;<br/>
<a NAME="E5:9_2"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">p</font> being   <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><br/> for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</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>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">n</font></span><a href="binarith.html#K11">*&gt;</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <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>;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</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>;<br/>



<b>assume </b><a NAME="E1:9_2_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">p</font> as    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E3:9_2_2"/><i><font color="Green">E50</font></i>: 
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:9_2_2"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E5:9_2_2"/><b>then </b>
<font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="binarith.html#K11">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">p1</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:9_2_2"/>
<font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="binarith.html#K11">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:9_2"/>
for <font color="Olive">p</font> being   <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> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>]
 
<b>from </b><i><a class="ref" href="finseq_2.html#S2">FINSEQ_2:sch 2</a>(, );<br/></i>
<b>hence </b><a NAME="E7:9_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 <b>by </b><i/>;<br/>


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


</div>
