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<b>deffunc </b><font color="Maroon">H<sub>3</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>  = <font color="Maroon">a<sub>2</sub></font> <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">o</font>,0</span><a href="domain_1.html#K1">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">p</font></span>)</span> where <font color="Olive">o</font> is   <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive">p</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">a<sub>2</sub></font> <a href="finseq_2.html#K3">*</a>  :  ex <font color="Olive">q</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>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> &amp; <font color="Olive">o</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">q</font></span>)</span> ) </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E2:22_1"/><i><font color="Green">E34</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>and </b><br/><a NAME="E3:22_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">t</font>,<font color="Olive">d</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> where <font color="Olive">t</font> is   <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive">d</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( ( <font color="Olive">t</font> <a href="hidden.html#R2">in</a>  <a href="lang1.html#K5">Terminals</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Olive">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">t</font>) ) or ( <font color="Olive">t</font> <a href="lang1.html#R1">==&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">t</font>, <a href="xboole_0.html#K1">{}</a> , <a href="xboole_0.html#K1">{}</a> ) ) ) </span>}</span> 
 <b>and </b><br/><a NAME="E4:22_1"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">n</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="recdef_1.html#S3">RECDEF_1:sch 3</a>();<br/></i>
<a NAME="E5:22_1"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</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</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</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">o</font>,<font color="Maroon">H<sub>2</sub></font>(<font color="Olive">o</font>, <a href="mcart_1.html#K15">pr1</a> <span class="p1">(<span class="default"><a href="trees_3.html#K16">roots</a> <font color="Olive">p</font></span>)</span>, <a href="mcart_1.html#K16">pr2</a> <span class="p1">(<span class="default"><a href="trees_3.html#K16">roots</a> <font color="Olive">p</font></span>)</span>)</span><a href="domain_1.html#K1">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">p</font></span>)</span> where <font color="Olive">o</font> is   <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive">p</font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="finseq_2.html#K3">*</a>  :  ex <font color="Olive">q</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>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> &amp; <font color="Olive">o</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">q</font></span>)</span> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E6:22_1"/><i><font color="Green">E63</font></i>: 
 ex <font color="Olive">X</font> being  <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>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">f</font> &amp; ( for <font color="Olive">d</font> being  <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive">d</font> <a href="hidden.html#R2">in</a>  <a href="lang1.html#K5">Terminals</a> <font color="Maroon">F<sub>1</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">d</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Olive">d</font>)</span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> ) &amp; ( for <font color="Olive">o</font> being  <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive">p</font> being   <a href="trees_4.html#M1">FinSequence</a> of <font color="Olive">X</font>  st <font color="Olive">o</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">p</font></span>)</span> holds <br/><span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">o</font>,<font color="Maroon">H<sub>2</sub></font>(<font color="Olive">o</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="Olive">p</font></span>)</span>, <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="Olive">p</font></span>)</span>)</span><a href="domain_1.html#K1">]</a></span> <a href="trees_4.html#K12">-tree</a> <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> ) &amp; ( for <font color="Olive">F</font> being  <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>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>  st ( for <font color="Olive">d</font> being  <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive">d</font> <a href="hidden.html#R2">in</a>  <a href="lang1.html#K5">Terminals</a> <font color="Maroon">F<sub>1</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">d</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Olive">d</font>)</span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> ) &amp; ( for <font color="Olive">o</font> being  <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive">p</font> being   <a href="trees_4.html#M1">FinSequence</a> of <font color="Olive">F</font>  st <font color="Olive">o</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">p</font></span>)</span> holds <br/><span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">o</font>,<font color="Maroon">H<sub>2</sub></font>(<font color="Olive">o</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="Olive">p</font></span>)</span>, <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="Olive">p</font></span>)</span>)</span><a href="domain_1.html#K1">]</a></span> <a href="trees_4.html#K12">-tree</a> <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> ) holds <br/><font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">F</font> ) )
 
<b>from </b><i><a class="ref" href="dtconstr.html#S3" target="_self">DTCONSTR:sch 3</a>(, , );<br/></i>
<b>consider </b><font color="Maroon">TSG</font> being   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="trees_3.html#K5">FinTrees</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span>)</span><b> such that </b><br/><a NAME="E8:22_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">TSG</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">t</font> <a href="trees_3.html#K10">`1</a> </span>)</span> where <font color="Olive">t</font> is    <a href="trees_3.html#M5">Element</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>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span> : <font color="Olive">t</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">f</font> </span>}</span> 
 <b>and </b><br/><a NAME="E9:22_1"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">d</font> being  <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive">d</font> <a href="hidden.html#R2">in</a>  <a href="lang1.html#K5">Terminals</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/> <a href="trees_4.html#K2">root-tree</a> <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">TSG</font>
 <b>and </b><br/><a NAME="E10:22_1"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">o</font> being  <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive">p</font> being   <a href="trees_4.html#M1">FinSequence</a> of <font color="Maroon">TSG</font>  st <font color="Olive">o</font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1" target="_self">roots</a> <font color="Olive">p</font> holds <br/><font color="Olive">o</font> <a href="trees_4.html#K12">-tree</a> <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">TSG</font>
 <b>and </b><br/><a NAME="E11:22_1"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">F</font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="trees_3.html#K5">FinTrees</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span>)</span>  st ( for <font color="Olive">d</font> being  <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive">d</font> <a href="hidden.html#R2">in</a>  <a href="lang1.html#K5">Terminals</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/> <a href="trees_4.html#K2">root-tree</a> <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> ) &amp; ( for <font color="Olive">o</font> being  <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive">p</font> being   <a href="trees_4.html#M1">FinSequence</a> of <font color="Olive">F</font>  st <font color="Olive">o</font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1" target="_self">roots</a> <font color="Olive">p</font> holds <br/><font color="Olive">o</font> <a href="trees_4.html#K12">-tree</a> <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> ) holds <br/><font color="Maroon">TSG</font> <a href="tarski.html#R1">c=</a> <font color="Olive">F</font>
 <b>from </b><i><a class="ref" href="dtconstr.html#S4" target="_self">DTCONSTR:sch 4</a>(, , );<br/></i>
<a NAME="E12:22_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">TSG</font> <a href="hidden.html#R1">=</a>  <a href="dtconstr.html#K4" target="_self">TS</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i>, , , </i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">t</font> being  <a href="trees_2.html#NM6">DecoratedTree</a> of <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>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span>  st <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">t</font> <a href="trees_3.html#K10">`1</a> ];<br/>
<a NAME="E13:22_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">tt</font> be   <a href="trees_2.html#NM6">DecoratedTree</a> of <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>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>

<b>reconsider </b><font color="Maroon">p</font> =  <a href="xboole_0.html#K1">{}</a>  as    <a href="trees_4.html#M1">FinSequence</a> of  <a href="dtconstr.html#K4" target="_self">TS</a> <font color="Maroon">F<sub>1</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="E2:22_1_1"/>
<font color="Maroon">tt</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0
 ;<br/>

<b>then consider </b><font color="Maroon">t</font> being   <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">d</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:22_1_1"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">tt</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">t</font>,<font color="Maroon">d</font></span><a href="domain_1.html#K1">]</a></span> &amp; ( ( <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="lang1.html#K5">Terminals</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> 0 ) or ( <font color="Maroon">t</font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1" target="_self">roots</a> <font color="Maroon">p</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> 0 ) ) )
 <b>by </b><i>, </i>;<br/>
<a NAME="E5:22_1_1"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">tt</font> <a href="trees_3.html#K10">`1</a>  <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K2">root-tree</a> <font color="Maroon">t</font> &amp; <font color="Maroon">t</font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K2">root-tree</a> <font color="Maroon">t</font> )
 
<b>by </b><i>, <a class="ref" href="trees_4.html#T20">TREES_4:20</a>, <a class="ref" href="trees_4.html#T25">TREES_4:25</a></i>;<br/>
<a NAME="E6:22_1_1"/>
for <font color="Olive">T</font> being  <a href="trees_2.html#NM6">DecoratedTree</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive">T</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">T</font>]
 
<b>by </b><i><a class="ref" href="finseq_1.html#T27">FINSEQ_1:27</a></i>;<br/>
<b>hence </b><a NAME="E7:22_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">tt</font> <a href="trees_3.html#K10">`1</a> ]
 <b>by </b><i>, , , <a class="ref" href="dtconstr.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E85</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:22_1_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/><b>thus </b><a NAME="E2:22_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="trees_2.html#NM6">DecoratedTree</a> of <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>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>

<b>assume </b><a NAME="E1:22_1_2_1"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font> <a href="trees_3.html#K10">`1</a> ] )
 ;<br/>

<a NAME="E2:22_1_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</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">o</font>,0</span><a href="domain_1.html#K1">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">p</font></span>)</span> where <font color="Olive">o</font> is   <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive">p</font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_2.html#K3">*</a>  :  ex <font color="Olive">q</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>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> &amp; <font color="Olive">o</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">q</font></span>)</span> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:22_1_2_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> or <font color="Maroon">x</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">o</font>,0</span><a href="domain_1.html#K1">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">p</font></span>)</span> where <font color="Olive">o</font> is   <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive">p</font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_2.html#K3">*</a>  :  ex <font color="Olive">q</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>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> &amp; <font color="Olive">o</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">q</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">o</font> being   <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">p</font> being    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E5:22_1_2_1"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">o</font>,0</span><a href="domain_1.html#K1">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">p</font> &amp;  ex <font color="Olive">q</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>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span> st <br/>( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> &amp; <font color="Maroon">o</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">q</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="dtconstr.html#T2" target="_self">Th2</a>, </i>;<br/>
<a NAME="E6:22_1_2_1"/>
(  <a href="card_3.html#K3">Union</a> <font color="Maroon">f</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</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, <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="E7:22_1_2_1"/><b>then </b><i><font color="Green">E90</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">f</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E8:22_1_2_1"/><i><font color="Green">E91</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p</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">p</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>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">dt</font> being   <a href="trees_3.html#M5">Element</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>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span> st <br/>( <font color="Olive">dt</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">dt</font> <a href="trees_3.html#K10">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">dt</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">f</font> );<br/>
<a NAME="E9:22_1_2_1"/><i><font color="Green">E93</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</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">p</font></span>)</span> holds <br/> ex <font color="Olive">x</font> being   <a href="trees_3.html#M5">Element</a> of  <a href="trees_3.html#K5">FinTrees</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>,<font color="Olive">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:22_1_2_1_1"/>
<font color="Maroon">k</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">p</font></span>)</span>
 ;<br/>

<a NAME="E2:22_1_2_1_1"/><b>then </b><i><font color="Green">E94</font></i>: 
( <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E3:22_1_2_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">f</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">dt</font> = <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as    <a href="trees_3.html#M5">Element</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>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <b>by </b><i/>;<br/>
<a NAME="E5:22_1_2_1_1"/>
( <font color="Maroon">dt</font> <a href="trees_3.html#K10">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="trees_3.html#K8">pr1</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span>)</span> <a href="trees_3.html#K7">*</a> <font color="Maroon">dt</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">dt</font> <a href="tarski.html#R1">c=</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>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="trees_3.html#K8">pr1</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span>)</span> <a href="hidden.html#R1">=</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>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="trees_2.html#D9">TREES_2:def 9</a>, <a class="ref" href="trees_3.html#D12">TREES_3:def 12</a></i>;<br/>
<a NAME="E6:22_1_2_1_1"/><b>then </b>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">dt</font> <a href="trees_3.html#K10">`1</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">dt</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">dt</font> is <a href="finset_1.html#V1">finite</a> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">dt</font> <a href="trees_3.html#K10">`1</a>  as    <a href="trees_3.html#M5">Element</a> of  <a href="trees_3.html#K5">FinTrees</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="trees_3.html#D8">TREES_3:def 8</a></i>;<br/>
<b>take </b>
<font color="Maroon">x</font>
;<br/>

<b>take </b>
<font color="Maroon">dt</font>
;<br/>

<b>thus </b><a NAME="E8:22_1_2_1_1"/>
( <font color="Maroon">dt</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">dt</font> <a href="trees_3.html#K10">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">dt</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">f</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">p1</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="trees_3.html#K5">FinTrees</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E11:22_1_2_1"/><i><font color="Green">E98</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p1</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">p</font></span>)</span>
 <b>and </b><br/><a NAME="E12:22_1_2_1"/><i><font color="Green">E99</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</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">p</font></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>,<font color="Maroon">p1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="finseq_1.html#S5">FINSEQ_1:sch 5</a>();<br/></i>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="trees_3.html#K4">Trees</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>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <b>by </b><i>, </i>;<br/>
<a NAME="E14:22_1_2_1"/><i><font color="Green">E100</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1" target="_self">roots</a> <font color="Maroon">p1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p1</font>
 
<b>by </b><i><a class="ref" href="trees_3.html#D18">TREES_3:def 18</a></i>;<br/>
<a NAME="E15:22_1_2_1"/><i><font color="Green">E101</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <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">p</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1" target="_self">roots</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#D12">MCART_1:def 12</a></i>;<br/>
<a NAME="E16:22_1_2_1"/><b>then </b><i><font color="Green">E106</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <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">p</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p1</font>
 
<b>by </b><i>, , <a class="ref" href="trees_3.html#D18">TREES_3:def 18</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:22_1_2_1_2"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p1</font>
 ;<br/><b>then consider </b><font color="Maroon">dt</font> being    <a href="trees_3.html#M5">Element</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>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span><b> such that </b><br/><a NAME="E3:22_1_2_1_2"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">dt</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">dt</font> <a href="trees_3.html#K10">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">dt</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">f</font> )
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">r</font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">dt</font> <b>by </b><i><a class="ref" href="trees_1.html#T47">TREES_1:47</a></i>;<br/><a NAME="E5:22_1_2_1_2"/>
 ex <font color="Olive">T</font> being  <a href="trees_2.html#NM4">DecoratedTree</a> st <br/>( <font color="Olive">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <span class="p1">(<span class="default"><a href="dtconstr.html#K1" target="_self">roots</a> <font color="Maroon">p1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">T</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="txt" href="dtconstr.html#E2:22_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="trees_3.html#D18">TREES_3:def 18</a></i>;<br/><a NAME="E6:22_1_2_1_2"/><b>then </b><i><font color="Green">E109</font></i>: 
<span class="p1">(<span class="default"><a href="dtconstr.html#K1" target="_self">roots</a> <font color="Maroon">p1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">dt</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">r</font></span>)</span> <a href="domain_1.html#K2">`1</a> 
 <b>by </b><i>, <a class="ref" href="trees_3.html#T41">TREES_3:41</a></i>;<br/><a NAME="E7:22_1_2_1_2"/>
 ex <font color="Olive">T</font> being  <a href="trees_2.html#NM4">DecoratedTree</a> st <br/>( <font color="Olive">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <span class="p1">(<span class="default"><a href="dtconstr.html#K1" target="_self">roots</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">T</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i>, , <a class="txt" href="dtconstr.html#E2:22_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="trees_3.html#D18">TREES_3:def 18</a></i>;<br/><b>hence </b><a NAME="E8:22_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="dtconstr.html#K1" target="_self">roots</a> <font color="Maroon">p1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <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">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="dtconstr.html#E2:22"><i><font color="Green">E32</font></i></a>, , <a class="txt" href="dtconstr.html#E2:22_1"><i><font color="Green">E34</font></i></a>, , <a class="txt" href="dtconstr.html#E3"><i><font color="Green">Lemma36</font></i></a>, <a class="ref" href="mcart_1.html#D12">MCART_1:def 12</a></i>;<br/></div><b>end;</b></div>
<a NAME="E18:22_1_2_1"/><b>then </b><i><font color="Green">E110</font></i>: 
 <a href="dtconstr.html#K1" target="_self">roots</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</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">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="dtconstr.html#E1:22"><i><font color="Green">E31</font></i></a>, , <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>
<a NAME="E19:22_1_2_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p1</font> <a href="tarski.html#R1">c=</a>  <a href="dtconstr.html#K4" target="_self">TS</a> <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2_1_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:22_1_2_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:22_1_2_1_3"/><i><font color="Green">E111</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</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">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/>
<a NAME="E5:22_1_2_1_3"/>
 ex <font color="Olive">dt</font> being   <a href="trees_3.html#M5">Element</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>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span> st <br/>( <font color="Olive">dt</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Olive">dt</font> <a href="trees_3.html#K10">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Olive">dt</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E6:22_1_2_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K4" target="_self">TS</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i>, <a class="ref" href="dtconstr.html#T1" target="_self">Th1</a>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">p1</font> as    <a href="trees_4.html#M1">FinSequence</a> of  <a href="dtconstr.html#K4" target="_self">TS</a> <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">t</font> be   <a href="trees_2.html#NM6">DecoratedTree</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>assume </b><a NAME="E1:22_1_2_1_4"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p1</font>
 ;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:22_1_2_1_4"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</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">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/><b>consider </b><font color="Maroon">dt</font> being    <a href="trees_3.html#M5">Element</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>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span><b> such that </b><br/><a NAME="E6:22_1_2_1_4"/><i><font color="Green">E113</font></i>: 
( <font color="Maroon">dt</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">dt</font> <a href="trees_3.html#K10">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">dt</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">f</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E7:22_1_2_1_4"/>
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E8:22_1_2_1_4"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">t</font>]
 <b>by </b><i><a class="ref" href="dtconstr.html#T2" target="_self">Th2</a>, , , <a class="txt" href="dtconstr.html#E3"><i><font color="Green">Lemma36</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E22:22_1_2_1"/><b>then </b><i><font color="Green">E114</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">o</font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">p1</font>]
 
<b>by </b><i>, , <a class="txt" href="dtconstr.html#E2:22_1"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">p1</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="trees_3.html#K4">Trees</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:22_1_2_1_5"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p</font>
 ;<br/><a NAME="E2:22_1_2_1_5"/><b>then </b>
 ex <font color="Olive">dt</font> being   <a href="trees_3.html#M5">Element</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>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span> st <br/>( <font color="Olive">dt</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Olive">dt</font> <a href="trees_3.html#K10">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Olive">dt</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">f</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:22_1_2_1_5"/>
for <font color="Olive">T</font> being  <a href="trees_2.html#NM6">DecoratedTree</a> of <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>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span>  st <font color="Olive">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> holds <br/><font color="Maroon">p1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">T</font> <a href="trees_3.html#K10">`1</a> 
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E25:22_1_2_1"/>
contradiction
 <b>by </b><i>, , , , , <a class="ref" href="trees_4.html#T27">TREES_4:27</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E15:22_1"/><i><font color="Green">E126</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>let </b><font color="Maroon">t</font> be   <a href="trees_2.html#NM6">DecoratedTree</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<b>assume </b><a NAME="E16:22_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K4" target="_self">TS</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<b>then consider </b><font color="Maroon">tt</font> being    <a href="trees_3.html#M5">Element</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>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span><b> such that </b><br/><a NAME="E18:22_1"/><i><font color="Green">E127</font></i>: 
( <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">tt</font> <a href="trees_3.html#K10">`1</a>  &amp; <font color="Maroon">tt</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">f</font> )
 <b>by </b><i>, <a class="ref" href="dtconstr.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E19:22_1"/>
 ex <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">tt</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> )
 
<b>by </b><i>, , <a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/>
<b>hence </b><a NAME="E20:22_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">t</font>]
 <b>by </b><i><a class="ref" href="dtconstr.html#T2" target="_self">Th2</a>, </i>;<br/>


</div>
