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

<b>let </b><font color="Maroon" title="c1">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">Z</font>, <font color="Maroon" title="c3">Z1</font>, <font color="Maroon" title="c4">Z2</font> be   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of <font color="Maroon" title="c1">D</font>;<br/><b>let </b><font color="Maroon" title="c5">z</font> be    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">Z</font>;<br/>





<b>assume </b><a NAME="E1:9"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c2">Z</font> <a href="trees_2.html#K8" title="TREES_2:func.8">with-replacement</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c3">Z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Z</font> <a href="trees_2.html#K8" title="TREES_2:func.8">with-replacement</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">Z2</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c6">T1</font> = <font color="Maroon" title="c2">Z</font> <a href="trees_2.html#K8" title="TREES_2:func.8">with-replacement</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c3">Z1</font>;<br/>
<b>set </b><font color="Maroon" title="c7">T2</font> = <font color="Maroon" title="c2">Z</font> <a href="trees_2.html#K8" title="TREES_2:func.8">with-replacement</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">Z2</font>;<br/>
<a NAME="E2:9"/><i><font color="Green" title="E9">A2</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Z</font> <a href="trees_2.html#K8" title="TREES_2:func.8">with-replacement</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c3">Z1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <font color="Maroon" title="c5">z</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">Z1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="trees_2.html#D12" title="TREES_2:def.12">TREES_2:def 12</a></i>;<br/>
<a NAME="E3:9"/><i><font color="Green" title="E10">A3</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Z</font> <a href="trees_2.html#K8" title="TREES_2:func.8">with-replacement</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">Z2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <font color="Maroon" title="c5">z</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">Z2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="trees_2.html#D12" title="TREES_2:def.12">TREES_2:def 12</a></i>;<br/>
<a NAME="E4:9"/><i><font color="Green" title="E11">A4</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <font color="Maroon" title="c5">z</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">Z1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <font color="Maroon" title="c5">z</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">Z2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="modal_1.html#E1:9"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="modal_1.html#E2:9"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="trees_2.html#D12" title="TREES_2:def.12">TREES_2:def 12</a></i>;<br/>
<a NAME="E5:9"/><i><font color="Green" title="E12">A5</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">Z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">Z2</font>
 
<b>by </b><i><a class="txt" href="modal_1.html#E1:9"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="modal_1.html#E2:9"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="modal_1.html#E3:9"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="modal_1.html#T15" target="_self" title="MODAL_1:th.15">Th15</a></i>;<br/>
<a NAME="E6:9"/>
for <font color="Olive" title="b1">s</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">Z1</font> holds <br/><font color="Maroon" title="c3">Z1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Z2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">s</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:9_1"/>
<font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">Z1</font>
 ;<br/>

<a NAME="E2:9_1"/><b>then </b><i><font color="Green" title="E13">A6</font></i>: 
( <font color="Maroon" title="c5">z</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <font color="Maroon" title="c5">z</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">Z2</font></span>)</span> &amp; <font color="Maroon" title="c5">z</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <font color="Maroon" title="c5">z</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">Z1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="modal_1.html#E4:9"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="trees_1.html#D12" title="TREES_1:def.12">TREES_1:def 12</a></i>;<br/>
<a NAME="E3:9_1"/><i><font color="Green" title="E14">A7</font></i>: 
<font color="Maroon" title="c5">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c5">z</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c8">s</font>
 
<b>by </b><i><a class="ref" href="trees_1.html#T8" title="TREES_1:th.8">TREES_1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">w</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:9_1"/><i><font color="Green" title="E15">A8</font></i>: 
( <font color="Maroon" title="c9">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">Z2</font> &amp; <font color="Maroon" title="c5">z</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c8">s</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">z</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">w</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">Z</font> <a href="trees_2.html#K8" title="TREES_2:func.8">with-replacement</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">Z2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Z2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">w</font> )
 <b>by </b><i><a class="txt" href="modal_1.html#E2:9_1"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="trees_2.html#D12" title="TREES_2:def.12">TREES_2:def 12</a></i>;<br/>
<a NAME="E6:9_1"/><i><font color="Green" title="E16">A9</font></i>: 
 ex <font color="Olive" title="b1">u</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">Z1</font> &amp; <font color="Maroon" title="c5">z</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c8">s</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">z</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">u</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">Z</font> <a href="trees_2.html#K8" title="TREES_2:func.8">with-replacement</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c3">Z1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Z1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">u</font> )
 
<b>by </b><i><a class="txt" href="modal_1.html#E2:9_1"><i><font color="Green" title="E13">A6</font></i></a>, <a class="txt" href="modal_1.html#E3:9_1"><i><font color="Green" title="E14">A7</font></i></a>, <a class="ref" href="trees_2.html#D12" title="TREES_2:def.12">TREES_2:def 12</a></i>;<br/>
<a NAME="E7:9_1"/>
<font color="Maroon" title="c8">s</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c9">w</font>
 
<b>by </b><i><a class="txt" href="modal_1.html#E5:9_1"><i><font color="Green" title="E15">A8</font></i></a>, <a class="ref" href="finseq_1.html#T46" title="FINSEQ_1:th.46">FINSEQ_1:46</a></i>;<br/>
<b>hence </b><a NAME="E8:9_1"/>
<font color="Maroon" title="c3">Z1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Z2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">s</font>
 <b>by </b><i><a class="txt" href="modal_1.html#E1:9"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="modal_1.html#E5:9_1"><i><font color="Green" title="E15">A8</font></i></a>, <a class="txt" href="modal_1.html#E6:9_1"><i><font color="Green" title="E16">A9</font></i></a>, <a class="ref" href="finseq_1.html#T46" title="FINSEQ_1:th.46">FINSEQ_1:46</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:9"/>
<font color="Maroon" title="c3">Z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Z2</font>
 <b>by </b><i><a class="txt" href="modal_1.html#E5:9"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="trees_2.html#T33" title="TREES_2:th.33">TREES_2:33</a></i>;<br/>


</div>
