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

<b>let </b><font color="Maroon">D</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">Z</font> be   <a href="trees_2.html#NM6">DecoratedTree</a> of <font color="Maroon">D</font>, <font color="Maroon">Z1</font> be   <a href="trees_2.html#NM6">DecoratedTree</a> of <font color="Maroon">D</font>, <font color="Maroon">Z2</font> be   <a href="trees_2.html#NM6">DecoratedTree</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">z</font> be    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Z</font>;<br/>





<b>assume </b><a NAME="E1:15"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">Z</font> <a href="trees_2.html#K8">with-replacement</a> <font color="Maroon">z</font>,<font color="Maroon">Z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Z</font> <a href="trees_2.html#K8">with-replacement</a> <font color="Maroon">z</font>,<font color="Maroon">Z2</font>
 ;<br/>

<b>set </b><font color="Maroon">T1</font> = <font color="Maroon">Z</font> <a href="trees_2.html#K8">with-replacement</a> <font color="Maroon">z</font>,<font color="Maroon">Z1</font>;<br/>
<b>set </b><font color="Maroon">T2</font> = <font color="Maroon">Z</font> <a href="trees_2.html#K8">with-replacement</a> <font color="Maroon">z</font>,<font color="Maroon">Z2</font>;<br/>
<a NAME="E2:15"/><i><font color="Green">E38</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Z</font> <a href="trees_2.html#K8">with-replacement</a> <font color="Maroon">z</font>,<font color="Maroon">Z1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">Z</font></span>)</span> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">z</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">Z1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="trees_2.html#D12">TREES_2:def 12</a></i>;<br/>
<a NAME="E3:15"/><i><font color="Green">E39</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Z</font> <a href="trees_2.html#K8">with-replacement</a> <font color="Maroon">z</font>,<font color="Maroon">Z2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">Z</font></span>)</span> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">z</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">Z2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="trees_2.html#D12">TREES_2:def 12</a></i>;<br/>
<a NAME="E4:15"/><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">Z</font></span>)</span> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">z</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">Z1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">Z</font></span>)</span> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">z</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">Z2</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="trees_2.html#D12">TREES_2:def 12</a></i>;<br/>
<a NAME="E5:15"/><i><font color="Green">E44</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">Z1</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Z2</font>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E6:15"/>
for <font color="Olive">s</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Z1</font> holds <br/><font color="Maroon">Z1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Z2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:15_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Z1</font>
 ;<br/>

<a NAME="E2:15_1"/><b>then </b><i><font color="Green">E61</font></i>: 
( <font color="Maroon">z</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">Z</font></span>)</span> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">z</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">Z2</font></span>)</span> &amp; <font color="Maroon">z</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">Z</font></span>)</span> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">z</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">Z1</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/>
<a NAME="E3:15_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">z</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">z</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="trees_1.html#T8">TREES_1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon">w</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="E5:15_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Z2</font> &amp; <font color="Maroon">z</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">w</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">Z</font> <a href="trees_2.html#K8">with-replacement</a> <font color="Maroon">z</font>,<font color="Maroon">Z2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Z2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> )
 <b>by </b><i>, <a class="ref" href="trees_2.html#D12">TREES_2:def 12</a></i>;<br/>
<a NAME="E6:15_1"/><i><font color="Green">E64</font></i>: 
 ex <font color="Olive">u</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Z1</font> &amp; <font color="Maroon">z</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">u</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">Z</font> <a href="trees_2.html#K8">with-replacement</a> <font color="Maroon">z</font>,<font color="Maroon">Z1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Z1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font> )
 
<b>by </b><i>, , <a class="ref" href="trees_2.html#D12">TREES_2:def 12</a></i>;<br/>
<a NAME="E7:15_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/>
<b>hence </b><a NAME="E8:15_1"/>
<font color="Maroon">Z1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Z2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:15"/>
<font color="Maroon">Z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Z2</font>
 <b>by </b><i>, <a class="ref" href="trees_2.html#T33">TREES_2:33</a></i>;<br/>


</div>
