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

<b>reconsider </b><font color="Maroon">c<sub>1</sub></font> = <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> = 0, <font color="Maroon">c<sub>3</sub></font> = 1 as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="lang1.html#K1">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K13">*</a>  ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> as    <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K13">*</a>  <b>by </b><i><a class="ref" href="relset_1.html#T8">RELSET_1:8</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>6</sub></font> =  <a href="lang1.html#G1">DTConstrStr</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> #);<br/>

<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">c<sub>6</sub></font> ;<br/>
<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="bintree1.html#D4" target="_self">BINTREE1:def 4</a>
<div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">c<sub>6</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>assume </b><a NAME="E1:17_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="lang1.html#R1">==&gt;</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><a NAME="E2:17_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="lang1.html#D1">LANG1:def 1</a></i>;<br/><a NAME="E3:17_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then </b><i><font color="Green">E9</font></i>: <font color="Maroon">c<sub>10</sub></font> = 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>
<br/>.= 
<span class="p1"><a href="lang1.html#K4">&lt;*</a><span class="default">1,1</span><a href="lang1.html#K4">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#D9">FINSEQ_1:def 9</a></i>
;<br/><b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>8</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>8</sub></font>;<br/><b>thus </b><a NAME="E5:17_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K4">&lt;*</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="lang1.html#K4">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="bintree1.html#E4:17_1_1_1"><i><font color="Green">E9</font></i></a></i>;<br/></div>
<b>end;</b></div>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>assume </b><a NAME="E1:17_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="lang1.html#R1">==&gt;</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><a NAME="E2:17_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="lang1.html#D1">LANG1:def 1</a></i>;<br/><a NAME="E3:17_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:17_1_1"/><b>then </b><i><font color="Green">E9</font></i>: 
<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="lang1.html#NM1">Symbol</a> of <font color="Maroon">c<sub>6</sub></font> : for <font color="Olive">b<sub>1</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a> holds not <font color="Olive">b<sub>1</sub></font> <a href="lang1.html#R1">==&gt;</a> <font color="Olive">b<sub>2</sub></font> </span>}</span> 
 
;<br/>
<a NAME="E9:17_1_1"/><b>then </b><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="lang1.html#K5">Terminals</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="lang1.html#D2">LANG1:def 2</a></i>;<br/>
<b>thus </b><a NAME="E10:17_1_1"/>
 <a href="lang1.html#K5">Terminals</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> 
 <b>by </b><i><a class="txt" href="bintree1.html#E8:17_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="lang1.html#D2">LANG1:def 2</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="dtconstr.html#D6">DTCONSTR:def 6</a><br/>

<a NAME="E11:17_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E12:17_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="lang1.html#R1">==&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="lang1.html#D1">LANG1:def 1</a></i>;<br/>
<a NAME="E13:17_1_1"/><b>then </b><i><font color="Green">E11</font></i>: 
( <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="lang1.html#NM1">Symbol</a> of <font color="Maroon">c<sub>6</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <font color="Olive">b<sub>1</sub></font> <a href="lang1.html#R1">==&gt;</a> <font color="Olive">b<sub>2</sub></font> </span>}</span>  &amp; <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">c<sub>6</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <font color="Olive">b<sub>1</sub></font> <a href="lang1.html#R1">==&gt;</a> <font color="Olive">b<sub>2</sub></font> </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="lang1.html#K6">NonTerminals</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="ref" href="lang1.html#D3">LANG1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E14:17_1_1"/>
 <a href="lang1.html#K6">NonTerminals</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="dtconstr.html#D7">DTCONSTR:def 7</a><br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="dtconstr.html#D8">DTCONSTR:def 8</a>
<div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">c<sub>6</sub></font>;<br/><b>assume </b><a NAME="E1:17_1_1_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="lang1.html#K6">NonTerminals</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:17_1_1_3"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">c<sub>6</sub></font> st <br/>( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; ex <font color="Olive">b<sub>2</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <font color="Olive">b<sub>1</sub></font> <a href="lang1.html#R1">==&gt;</a> <font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="bintree1.html#E13:17_1_1"><i><font color="Green">E11</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E4:17_1_1_3"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="lang1.html#R1">==&gt;</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><a NAME="E5:17_1_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="bintree1.html#E4:17_1_1_3"><i><font color="Green">E12</font></i></a>, <a class="ref" href="lang1.html#D1">LANG1:def 1</a></i>;<br/><a NAME="E6:17_1_1_3"/><b>then </b><i><font color="Green">E13</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> =  <a href="dtconstr.html#K4">TS</a> <font color="Maroon">c<sub>6</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="bintree1.html#E9:17_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="dtconstr.html#D4">DTCONSTR:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> =  <a href="trees_4.html#K2">root-tree</a> <font color="Maroon">c<sub>8</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>11</sub></font> <b>by </b><i><a class="txt" href="bintree1.html#E9:17_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="dtconstr.html#D4">DTCONSTR:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>12</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="lang1.html#K1">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>12</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> as    <a href="trees_4.html#M1">FinSequence</a> of  <a href="dtconstr.html#K4">TS</a> <font color="Maroon">c<sub>6</sub></font> ;<br/><b>take </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font>;<br/><a NAME="E10:17_1_1_3"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="trees_4.html#K2">root-tree</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="trees_4.html#K2">root-tree</a> 1</span>)</span></span><a href="lang1.html#K4">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D9">FINSEQ_1:def 9</a></i>;<br/><a NAME="E11:17_1_1_3"/><b>then </b>
(  <a href="dtconstr.html#K1">roots</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K4">&lt;*</a><span class="default">1,1</span><a href="lang1.html#K4">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><a href="trees_4.html#K2">root-tree</a> 1</span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="dtconstr.html#T6">DTCONSTR:6</a>, <a class="ref" href="finseq_1.html#D9">FINSEQ_1:def 9</a>, <a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>;<br/><b>hence </b><a NAME="E12:17_1_1_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1">roots</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="bintree1.html#E4:17_1_1_3"><i><font color="Green">E12</font></i></a>, <a class="txt" href="bintree1.html#E6:17_1_1_3"><i><font color="Green">E13</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E16:17_1_1"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="lang1.html#V1">strict</a>
 ;<br/>


</div>
