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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="bintree1.html#V1">binary</a> <a href="trees_1.html#NM1">Tree</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="finseq_1.html#NM1">FinSequence</a>] means ( <font color="Maroon">a<sub>1</sub></font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> implies  <a href="relat_1.html#K2">rng</a> <font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a>  );<br/>
<a NAME="E1:2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T27">FINSEQ_1:27</a></i>;<br/>
<a NAME="E2:2"/><b>then </b><i><font color="Green">E5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K5">NAT</a> ]
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E3:2"/><i><font color="Green">E6</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">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="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:2_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>]
 ;<br/>

<b>assume </b><a NAME="E2:2_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>3</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>4</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="trees_1.html#T46">TREES_1:46</a></i>;<br/>
<a NAME="E4:2_1"/>
<font color="Maroon">c<sub>3</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>4</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="bintree2.html#E2:2_1"><i><font color="Green">E8</font></i></a></i>;<br/>
<a NAME="E5:2_1"/><b>then </b><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>3</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>4</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="trees_2.html#D5">TREES_2:def 5</a></i>;<br/>
<a NAME="E6:2_1"/><b>then </b>
not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="bintree1.html#T5">BINTREE1:5</a></i>;<br/>
<a NAME="E7:2_1"/><b>then </b>
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>5</sub></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">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K13">&lt;*</a><span class="default">0</span><a href="binarith.html#K13">*&gt;</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K13">&lt;*</a><span class="default">1</span><a href="binarith.html#K13">*&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/>
<a NAME="E8:2_1"/><b>then </b>
( <font color="Maroon">c<sub>3</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>4</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">0</span><a href="binarith.html#K13">*&gt;</a></span> or <font color="Maroon">c<sub>3</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>4</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</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#E5:2_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E9:2_1"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_2.html#T20">FINSEQ_2:20</a></i>;<br/>
<a NAME="E10:2_1"/><b>then </b><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>4</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/>
<a NAME="E11:2_1"/><i><font color="Green">E11</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</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="E1:2_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<b>hence </b><a NAME="E2:2_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="margrel1.html#K6">BOOLEAN</a> 
 <b>by </b><i><a class="txt" href="bintree2.html#E10:2_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="margrel1.html#D12">MARGREL1:def 12</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:2_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></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="E13:2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a> 
 
<b>by </b><i><a class="txt" href="bintree2.html#E1:2_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="bintree2.html#E2:2_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="bintree2.html#E11:2_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="trees_1.html#T46">TREES_1:46</a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E14:2_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">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="finseq_2.html#S2">FINSEQ_2:sch 2</a>(<a class="txt" href="bintree2.html#E2:2"><i><font color="Green">E5</font></i></a>, <a class="txt" href="bintree2.html#E3:2"><i><font color="Green">E6</font></i></a>);<br/></i>
<a NAME="E5:2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a> 
 
;<br/>
<b>hence </b><a NAME="E6:2"/>
<font color="Maroon">c<sub>2</sub></font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="margrel1.html#K6">BOOLEAN</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>


</div>
