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

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">T</font> be   <a href="trees_2.html#NM4">DecoratedTree</a>;<br/>



<b>set </b><font color="Maroon">D</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="trees_2.html#K8">with-replacement</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>,<font color="Maroon">T</font>;<br/>
<b>set </b><font color="Maroon">W</font> = <span class="p1">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="trees_1.html#K5">with-replacement</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>,<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font></span>)</span>;<br/>
<i><font color="Green">E33</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="trees_4.html#K4" target="_self">-tree</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">T</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> = 
 <a href="trees_3.html#K13">tree</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">T</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="trees_3.html#K13">tree</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
<b>by </b><i><a class="ref" href="funct_6.html#T33">FUNCT_6:33</a></i>
<br/>.= 
 <a href="trees_3.html#K14">^</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font></span>)</span>
<b>by </b><i><a class="ref" href="trees_3.html#D16">TREES_3:def 16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="trees_1.html#K5">with-replacement</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>,<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font></span>)</span>
<b>by </b><i><a class="ref" href="trees_3.html#T61">TREES_3:61</a></i>
;<br/>
<a NAME="E2:33"/><i><font color="Green">E34</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="trees_1.html#K2">elementary_tree</a> 1
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<b>reconsider </b><font color="Maroon">t1</font> =  <a href="xboole_0.html#K1">{}</a> , <font color="Maroon">t2</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span> as    <a href="trees_1.html#M1">Element</a> of  <a href="trees_1.html#K2">elementary_tree</a> 1 <b>by </b><i><a class="ref" href="modal_1.html#T9">MODAL_1:9</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E4:33"/>
<font color="Maroon">t2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font>
 
;<br/>
<a NAME="E5:33"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="trees_2.html#K8">with-replacement</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>,<font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="trees_1.html#K5">with-replacement</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>,<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="trees_2.html#D12">TREES_2:def 12</a></i>;<br/>
<a NAME="E6:33"/><i><font color="Green">E48</font></i>: 
(  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font> &amp; <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span> &amp; not <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="subset_1.html#K1">{}</a> <a href="numbers.html#K5">NAT</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a>, <a class="ref" href="trees_1.html#T47">TREES_1:47</a>, <a class="ref" href="xboole_1.html#T115">XBOOLE_1:115</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:33_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="trees_1.html#K5">with-replacement</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>,<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">y</font> as    <a href="trees_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="trees_1.html#K5">with-replacement</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>,<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font></span>)</span> ;<br/><a NAME="E3:33_2"/>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K2">elementary_tree</a> 1 or  ex <font color="Olive">v</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">v</font> ) )
 <b>by </b><i><a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/><a NAME="E4:33_2"/><b>then </b><i><font color="Green">E49</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font> &amp; <font color="Maroon">t2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">t1</font> ) or  ex <font color="Olive">v</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Olive">v</font> ) )
 <b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a>, <a class="ref" href="modal_1.html#T9">MODAL_1:9</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E5:33_2"/>
not <font color="Maroon">t2</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">t1</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><b>then </b><i><font color="Green">E51</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="trees_2.html#K8">with-replacement</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>,<font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">t1</font>
<b>by </b><i>, <a class="ref" href="trees_3.html#T48">TREES_3:48</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="trees_4.html#K4" target="_self">-tree</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">T</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i/>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">r</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:33_2_2"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font> )
 ;<br/><b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</font> as   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/><a NAME="E3:33_2_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/><a NAME="E4:33_2_2"/><b>then </b><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="trees_2.html#K8">with-replacement</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>,<font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="trees_3.html#T49">TREES_3:49</a></i>;<br/><a NAME="E5:33_2_2"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">T</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">T</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 <b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><a NAME="E6:33_2_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="trees_4.html#K4" target="_self">-tree</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">T</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="trees_2.html#K5">|</a> <font color="Maroon">t2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="trees_1.html#K5">with-replacement</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>,<span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">t2</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font> )
 <b>by </b><i>, <a class="ref" href="trees_1.html#T66">TREES_1:66</a></i>;<br/><b>hence </b><a NAME="E7:33_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="trees_2.html#K8">with-replacement</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>,<font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="trees_4.html#K4" target="_self">-tree</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">T</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font>
 <b>by </b><i>, , , <a class="ref" href="trees_2.html#D11">TREES_2:def 11</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:33_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="trees_2.html#K8">with-replacement</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>,<font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="trees_4.html#K4" target="_self">-tree</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">T</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:33"/>
<font color="Maroon">x</font> <a href="trees_4.html#K4" target="_self">-tree</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">T</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="trees_2.html#K8">with-replacement</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>,<font color="Maroon">T</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div>
