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<b>let </b><font color="Maroon">D1</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">D2</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">d1</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D1</font>;<br/><b>let </b><font color="Maroon">d2</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D2</font>;<br/><b>let </b><font color="Maroon">F</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="trees_3.html#NM8">DTree-set</a> of <font color="Maroon">D1</font>,<font color="Maroon">D2</font>;<br/><b>let </b><font color="Maroon">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">F</font>;<br/>











<a NAME="E1:52"/><i><font color="Green">E33</font></i>: 
 <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> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 
<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="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">T</font> being   <a href="trees_3.html#M5">Element</a> of <font color="Maroon">F</font> 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">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">T</font> <a href="trees_3.html#K11">`2</a>  );<br/>
<a NAME="E2:52"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</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#K4">Trees</a> <font color="Maroon">D2</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Olive">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:52_1"/>
<font color="Maroon">i</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/>

<b>then reconsider </b><font color="Maroon">T</font> = <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as    <a href="trees_3.html#M5">Element</a> of <font color="Maroon">F</font> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">T</font> <a href="trees_3.html#K11">`2</a>  as    <a href="trees_3.html#M5">Element</a> of  <a href="trees_3.html#K4">Trees</a> <font color="Maroon">D2</font> <b>by </b><i><a class="ref" href="trees_3.html#D7">TREES_3:def 7</a></i>;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

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

<b>thus </b><a NAME="E4:52_1"/>
( <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> <a href="trees_3.html#K11">`2</a>  )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">p2</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="trees_3.html#K4">Trees</a> <font color="Maroon">D2</font><b> such that </b><br/><a NAME="E4:52"/><i><font color="Green">E47</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p2</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> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</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>1</sub></font>[<font color="Olive">i</font>,<font color="Maroon">p2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>] ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S5">FINSEQ_1:sch 5</a>();<br/></i>
<b>take </b>
<font color="Maroon">p2</font>
;<br/>

<b>thus </b><a NAME="E5:52"/><i><font color="Green">E48</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>

<b>hence </b><a NAME="E6:52"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> holds <br/> ex <font color="Olive">T</font> being   <a href="trees_3.html#M5">Element</a> of <font color="Maroon">F</font> 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="Olive">i</font> &amp; <font color="Maroon">p2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">T</font> <a href="trees_3.html#K11">`2</a>  )
 <b>by </b><i/>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:52_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 ;<br/><a NAME="E2:52_2"/><b>then </b>
 ex <font color="Olive">T</font> being   <a href="trees_3.html#M5">Element</a> of <font color="Maroon">F</font> 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">i</font> &amp; <font color="Maroon">p2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">T</font> <a href="trees_3.html#K11">`2</a>  )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:52_2"/>
for <font color="Olive">T</font> being  <a href="trees_3.html#NM7">DecoratedTree</a> of <font color="Maroon">D1</font>,<font color="Maroon">D2</font>  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">i</font> holds <br/><font color="Maroon">p2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">T</font> <a href="trees_3.html#K11">`2</a> 
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:52"/>
<span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">d1</font>,<font color="Maroon">d2</font></span><a href="domain_1.html#K1">]</a></span> <a href="trees_4.html#K8" target="_self">-tree</a> <font color="Maroon">p</font></span>)</span> <a href="trees_3.html#K11">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">d2</font> <a href="trees_4.html#K8" target="_self">-tree</a> <font color="Maroon">p2</font>
 <b>by </b><i>, </i>;<br/>


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