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

<b>set </b><font color="Maroon">c<sub>1</sub></font> = <font color="Maroon">F<sub>1</sub></font>();<br/>
<b>set </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">F<sub>2</sub></font>();<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">F<sub>3</sub></font>();<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">F<sub>2</sub></font>(),  <a href="finseq_1.html#NM1">FinSequence</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>3</sub></font>() = <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">a<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="mcart_1.html#K15">pr1</a> <span class="p2">(<span class="default"><a href="trees_3.html#K16">roots</a> <font color="Maroon">a<sub>2</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="mcart_1.html#K16">pr2</a> <span class="p2">(<span class="default"><a href="trees_3.html#K16">roots</a> <font color="Maroon">a<sub>2</sub></font></span>)</span></span>)</span>);<br/>
<a NAME="E1:17_1"/>
 <a href="card_3.html#K3">Union</a> <font color="Maroon">F<sub>1</sub></font>() <a href="tarski.html#R1">c=</a>  <a href="trees_3.html#K5">FinTrees</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</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="E1:17_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:17_1_1"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="dtconstr.html#E1:17"><i><font color="Green">E8</font></i></a>, <a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K5">FinTrees</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<a NAME="E4:17_1_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_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> ;<br/>

<b>assume </b><a NAME="E1:17_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> 0
 ;<br/>

<a NAME="E2:17_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">F<sub>2</sub></font>()ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>3</sub></font>() st <br/>( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K2">root-tree</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; ( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="lang1.html#K5">Terminals</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) or ( <font color="Olive">b<sub>1</sub></font> <a href="lang1.html#R1">==&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive">b<sub>1</sub></font>,<a href="xboole_0.html#K1">{}</a> ,<a href="xboole_0.html#K1">{}</a> ) ) ) )
 
<b>by </b><i><a class="txt" href="dtconstr.html#E2:17"><i><font color="Green">E9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:17_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K5">FinTrees</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E13</font></i>: <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>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:17_1_1_2"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>6</sub></font>]
 ;<br/><b>thus </b><a NAME="E2:17_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_2_1"><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> ;<br/>

<b>assume </b><a NAME="E1:17_1_1_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>

<a NAME="E2:17_1_1_2_1"/><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="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>)</span><a href="domain_1.html#K1">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">F<sub>2</sub></font>(), B is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_2.html#K3">*</a>  : ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="trees_3.html#K5">FinTrees</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K2" target="_self">pr1</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1" target="_self">roots</a> <font color="Olive">b<sub>3</sub></font></span>)</span> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="dtconstr.html#E3:17"><i><font color="Green">E10</font></i></a></i>;<br/>
<a NAME="E3:17_1_1_2_1"/><b>then </b><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>)</span><a href="domain_1.html#K1">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">F<sub>2</sub></font>(), B is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_2.html#K3">*</a>  : ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="trees_3.html#K5">FinTrees</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K2" target="_self">pr1</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1" target="_self">roots</a> <font color="Olive">b<sub>3</sub></font></span>)</span> ) </span>}</span>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>assume </b><a NAME="E4:17_1_1_2_1"/><i><font color="Green">E16</font></i>: 
not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K5">FinTrees</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">F<sub>2</sub></font>(), <font color="Maroon">c<sub>9</sub></font> being    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E6:17_1_1_2_1"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>)</span><a href="domain_1.html#K1">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">c<sub>9</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="trees_3.html#K5">FinTrees</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span> 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; <font color="Maroon">c<sub>8</sub></font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K2" target="_self">pr1</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1" target="_self">roots</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="dtconstr.html#E1:17_1_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="dtconstr.html#E3:17_1_1_2_1"><i><font color="Green">E15</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="trees_3.html#K5">FinTrees</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span> <b>by </b><i><a class="txt" href="dtconstr.html#E6:17_1_1_2_1"><i><font color="Green">E17</font></i></a></i>;<br/>
<a NAME="E8:17_1_1_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>)</span><a href="domain_1.html#K1">]</a></span> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="dtconstr.html#E6:17_1_1_2_1"><i><font color="Green">E17</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:17_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="dtconstr.html#E4:17_1_1_2_1"><i><font color="Green">E16</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E6:17_1_1"/>
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="dtconstr.html#E4:17_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="dtconstr.html#E5:17_1_1"><i><font color="Green">E13</font></i></a>);<br/></i>
<b>hence </b><a NAME="E7:17_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K5">FinTrees</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="dtconstr.html#E3:17_1_1"><i><font color="Green">E11</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="card_3.html#K3">Union</a> <font color="Maroon">F<sub>1</sub></font>() as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="trees_3.html#K5">FinTrees</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>thus </b><a NAME="E3:17_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">F<sub>2</sub></font>();<br/><b>assume </b><a NAME="E1:17_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="lang1.html#K5">Terminals</a> <font color="Maroon">F<sub>2</sub></font>()
 ;<br/><a NAME="E2:17_1_2"/><b>then </b>
 <a href="trees_4.html#K2">root-tree</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c<sub>5</sub></font>)</span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> 0
 <b>by </b><i><a class="txt" href="dtconstr.html#E2:17"><i><font color="Green">E9</font></i></a></i>;<br/><b>hence </b><a NAME="E3:17_1_2"/>
 <a href="trees_4.html#K2">root-tree</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c<sub>5</sub></font>)</span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="dtconstr.html#E1:17"><i><font color="Green">E8</font></i></a>, <a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">F<sub>2</sub></font>();<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="trees_4.html#M1">FinSequence</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><b>assume </b><a NAME="E1:17_1_3"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K2" target="_self">pr1</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1" target="_self">roots</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 ;<br/><b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="dtconstr.html#K2" target="_self">pr1</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1" target="_self">roots</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="dtconstr.html#K3" target="_self">pr2</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1" target="_self">roots</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>;<br/><a NAME="E2:17_1_3"/><i><font color="Green">E12</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>2</sub></font>;<br/><a NAME="E3:17_1_3"/><i><font color="Green">E13</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> holds <br/>ex <font color="Olive">b<sub>2</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>,<font color="Olive">b<sub>2</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:17_1_3_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:17_1_3_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="dtconstr.html#E2:17_1_3"><i><font color="Green">E12</font></i></a>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E3:17_1_3_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="dtconstr.html#E1:17"><i><font color="Green">E8</font></i></a>, <a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/>
<b>hence </b><a NAME="E4:17_1_3_1"/>
ex <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="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:17_1_3"/><i><font color="Green">E14</font></i>: 
(  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S5">FINSEQ_1:sch 5</a>(<a class="txt" href="dtconstr.html#E3:17_1_3"><i><font color="Green">E13</font></i></a>);<br/></i><div><i><font color="Green">E15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_3_2"><b>now </b></a><div class="add"><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a<sub>2</sub></font>;<br/><b>assume </b><a NAME="E1:17_1_3_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:17_1_3_2"/><b>then </b><i><font color="Green">E16</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> is <a href="finset_1.html#V1">finite</a> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E3:17_1_3_2"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>] )
 ;<br/><a NAME="E4:17_1_3_2"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> 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>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>]
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:17_1_3_2"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>10</sub></font>,<font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="cqc_sim1.html#S4">CQC_SIM1:sch 4</a>(<a class="txt" href="dtconstr.html#E2:17_1_3_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="dtconstr.html#E3:17_1_3_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="dtconstr.html#E4:17_1_3_2"><i><font color="Green">E18</font></i></a>);<br/></i><b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font>;<br/><b>thus </b><a NAME="E7:17_1_3_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_3_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</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="E1:17_1_3_2_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:17_1_3_2_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="dtconstr.html#E3:17_1_3_2_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/>
<a NAME="E5:17_1_3_2_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="dtconstr.html#E2:17_1_3"><i><font color="Green">E12</font></i></a>, <a class="txt" href="dtconstr.html#E5:17_1_3"><i><font color="Green">E14</font></i></a>, <a class="txt" href="dtconstr.html#E3:17_1_3_2_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="dtconstr.html#E2:17_1_3_2"><i><font color="Green">E16</font></i></a></i>;<br/>
<a NAME="E7:17_1_3_2_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="dtconstr.html#E6:17_1_3_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="dtconstr.html#E5:17_1_3_2_1"><i><font color="Green">E21</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E9:17_1_3_2_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">F<sub>5</sub></font>(<font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="mcart_1.html#K15">pr1</a> <span class="p2">(<span class="default"><a href="trees_3.html#K16">roots</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="mcart_1.html#K16">pr2</a> <span class="p2">(<span class="default"><a href="trees_3.html#K16">roots</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span>)</span><a href="domain_1.html#K1">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">F<sub>2</sub></font>(), B is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="finseq_2.html#K3">*</a>  : ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="trees_3.html#K5">FinTrees</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K2" target="_self">pr1</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1" target="_self">roots</a> <font color="Olive">b<sub>3</sub></font></span>)</span> ) </span>}</span> ;<br/>
<a NAME="E10:17_1_3_2_1"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_3_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:17_1_3_2_1_1"/>
<font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>)
 
<b>by </b><i><a class="txt" href="dtconstr.html#E3:17"><i><font color="Green">E10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:17_1_3_2_1_1"/>
<font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:17_1_3_2_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_3_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:17_1_3_2_1_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<b>hence </b><a NAME="E2:17_1_3_2_1_2"/>
<font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dtconstr.html#E10:17_1_3_2_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="measure2.html#T22">MEASURE2:22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:17_1_3_2_1"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="dtconstr.html#E9:17_1_3_2_1"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E13:17_1_3_2_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="dtconstr.html#E2:17_1_3"><i><font color="Green">E12</font></i></a>, <a class="txt" href="dtconstr.html#E5:17_1_3"><i><font color="Green">E14</font></i></a>, <a class="txt" href="dtconstr.html#E3:17_1_3_2_1"><i><font color="Green">E20</font></i></a></i>;<br/>
<b>hence </b><a NAME="E14:17_1_3_2_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="dtconstr.html#E12:17_1_3_2_1"><i><font color="Green">E24</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_3_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_1_3_3"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:17_1_3_3"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <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="E3:17_1_3_3"/><b>then </b>
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T26">FINSEQ_1:26</a></i>;<br/><a NAME="E4:17_1_3_3"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="dtconstr.html#E2:17_1_3"><i><font color="Green">E12</font></i></a>, <a class="txt" href="dtconstr.html#E5:17_1_3"><i><font color="Green">E14</font></i></a>, <a class="ref" href="finseq_1.html#T26">FINSEQ_1:26</a></i>;<br/><a NAME="E5:17_1_3_3"/><b>then </b><i><font color="Green">E16</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <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/><b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="nat_1.html#NM1">Nat</a>;<br/><b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font>;<br/><b>thus </b><a NAME="E7:17_1_3_3"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="dtconstr.html#E5:17_1_3_3"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E9:17_1_3"/><i><font color="Green">E16</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="dtconstr.html#E6:17_1_3"><i><font color="Green">E15</font></i></a></i>;<br/><a NAME="E10:17_1_3"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> &amp; <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F<sub>1</sub></font>() )
 <b>by </b><i><a class="txt" href="dtconstr.html#E1:17"><i><font color="Green">E8</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>;<br/><a NAME="E11:17_1_3"/><b>then </b>
<font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="trees_3.html#K5">FinTrees</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="trees_4.html#M1">FinSequence</a> of <font color="Maroon">c<sub>11</sub></font> <b>by </b><i><a class="txt" href="dtconstr.html#E9:17_1_3"><i><font color="Green">E16</font></i></a>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>11</sub></font> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/><a NAME="E15:17_1_3"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="dtconstr.html#K2" target="_self">pr1</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K1" target="_self">roots</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="dtconstr.html#K3" target="_self">pr2</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K1" target="_self">roots</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>)</span><a href="domain_1.html#K1">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>)</span><a href="domain_1.html#K1">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">F<sub>2</sub></font>(), B is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>11</sub></font> <a href="finseq_2.html#K3">*</a>  : ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="trees_3.html#K5">FinTrees</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K2" target="_self">pr1</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1" target="_self">roots</a> <font color="Olive">b<sub>3</sub></font></span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="dtconstr.html#E1:17_1_3"><i><font color="Green">E11</font></i></a></i>;<br/><a NAME="E16:17_1_3"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="dtconstr.html#K2" target="_self">pr1</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K1" target="_self">roots</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="dtconstr.html#K3" target="_self">pr2</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K1" target="_self">roots</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>)</span><a href="domain_1.html#K1">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>)</span><a href="domain_1.html#K1">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">F<sub>2</sub></font>(), B is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>11</sub></font> <a href="finseq_2.html#K3">*</a>  : ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="trees_3.html#K5">FinTrees</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K2" target="_self">pr1</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1" target="_self">roots</a> <font color="Olive">b<sub>3</sub></font></span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E17:17_1_3"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="dtconstr.html#K2" target="_self">pr1</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K1" target="_self">roots</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="dtconstr.html#K3" target="_self">pr2</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K1" target="_self">roots</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>)</span><a href="domain_1.html#K1">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="dtconstr.html#E3:17"><i><font color="Green">E10</font></i></a></i>;<br/><b>hence </b><a NAME="E18:17_1_3"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="dtconstr.html#K2" target="_self">pr1</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K1" target="_self">roots</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="dtconstr.html#K3" target="_self">pr2</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K1" target="_self">roots</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>)</span><a href="domain_1.html#K1">]</a></span> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="dtconstr.html#E1:17"><i><font color="Green">E8</font></i></a>, <a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="trees_3.html#K5">FinTrees</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>;<br/>

<b>assume </b><a NAME="E6:17_1"/><i><font color="Green">E11</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">F<sub>2</sub></font>() st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="lang1.html#K5">Terminals</a> <font color="Maroon">F<sub>2</sub></font>() holds <br/> <a href="trees_4.html#K2">root-tree</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>)</span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">F<sub>2</sub></font>()<br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="trees_4.html#M1">FinSequence</a> of <font color="Maroon">c<sub>5</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K2" target="_self">pr1</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1" target="_self">roots</a> <font color="Olive">b<sub>2</sub></font></span>)</span> holds <br/><span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>)</span><a href="domain_1.html#K1">]</a></span> <a href="trees_4.html#K12">-tree</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> ) )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>;<br/>
<a NAME="E7:17_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_4"><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>reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="xboole_0.html#K1">{}</a>  as    <a href="trees_4.html#M1">FinSequence</a> of <font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="ref" href="finseq_1.html#T29">FINSEQ_1:29</a></i>;<br/>
<b>assume </b><a NAME="E2:17_1_4"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> 0
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">F<sub>2</sub></font>(), <font color="Maroon">c<sub>9</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>3</sub></font>()<b> such that </b><br/><a NAME="E4:17_1_4"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K2">root-tree</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; ( ( <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">F<sub>2</sub></font>() &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c<sub>8</sub></font>) ) or ( <font color="Maroon">c<sub>8</sub></font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K2" target="_self">pr1</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1" target="_self">roots</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font>) ) ) )
 <b>by </b><i><a class="txt" href="dtconstr.html#E2:17"><i><font color="Green">E9</font></i></a>, <a class="ref" href="dtconstr.html#T3" target="_self">Th3</a>, <a class="ref" href="dtconstr.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E5:17_1_4"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K2">root-tree</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="ref" href="trees_4.html#T20">TREES_4:20</a></i>;<br/>
<b>hence </b><a NAME="E6:17_1_4"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="dtconstr.html#E6:17_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="dtconstr.html#E4:17_1_4"><i><font color="Green">E13</font></i></a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:17_1_5"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>6</sub></font>]
 ;<br/><b>thus </b><a NAME="E2:17_1_5"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_5_1"><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="E1:17_1_5_1"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; 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="E2:17_1_5_1"/><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="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>)</span><a href="domain_1.html#K1">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">F<sub>2</sub></font>(), B is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_2.html#K3">*</a>  : ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="trees_3.html#K5">FinTrees</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K2" target="_self">pr1</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1" target="_self">roots</a> <font color="Olive">b<sub>3</sub></font></span>)</span> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="dtconstr.html#E3:17"><i><font color="Green">E10</font></i></a></i>;<br/>
<a NAME="E3:17_1_5_1"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>)</span><a href="domain_1.html#K1">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">F<sub>2</sub></font>(), B is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_2.html#K3">*</a>  : ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="trees_3.html#K5">FinTrees</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K2" target="_self">pr1</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1" target="_self">roots</a> <font color="Olive">b<sub>3</sub></font></span>)</span> ) </span>}</span>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">F<sub>2</sub></font>(), <font color="Maroon">c<sub>9</sub></font> being    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E5:17_1_5_1"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>)</span><a href="domain_1.html#K1">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">c<sub>9</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="trees_3.html#K5">FinTrees</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span> 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; <font color="Maroon">c<sub>8</sub></font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K2" target="_self">pr1</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1" target="_self">roots</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="dtconstr.html#E1:17_1_5"><i><font color="Green">E14</font></i></a>, <a class="txt" href="dtconstr.html#E1:17_1_5_1"><i><font color="Green">E15</font></i></a></i>;<br/>
<a NAME="E6:17_1_5_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E7:17_1_5_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="dtconstr.html#E1:17_1_5"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as    <a href="trees_4.html#M1">FinSequence</a> of <font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E9:17_1_5_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K2" target="_self">pr1</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1" target="_self">roots</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="dtconstr.html#E5:17_1_5_1"><i><font color="Green">E16</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:17_1_5_1"/>
contradiction
 <b>by </b><i><a class="txt" href="dtconstr.html#E6:17_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="dtconstr.html#E1:17_1_5_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="dtconstr.html#E5:17_1_5_1"><i><font color="Green">E16</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E9:17_1"/><i><font color="Green">E14</font></i>: 
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="dtconstr.html#E7:17_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="dtconstr.html#E8:17_1"><i><font color="Green">E13</font></i></a>);<br/></i>
<b>thus </b><a NAME="E10:17_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_6"><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="E1:17_1_6"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<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="E3:17_1_6"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="dtconstr.html#E1:17"><i><font color="Green">E8</font></i></a>, <a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/>
<a NAME="E4:17_1_6"/>
<font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="dtconstr.html#E9:17_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="dtconstr.html#E3:17_1_6"><i><font color="Green">E15</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:17_1_6"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="dtconstr.html#E3:17_1_6"><i><font color="Green">E15</font></i></a></i>;<br/>


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


</div>
