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

<b>let </b><font color="Maroon">T</font> be   <a href="trees_2.html#NM4">DecoratedTree</a>;<br/><b>let </b><font color="Maroon">T1</font> be   <a href="trees_2.html#NM4">DecoratedTree</a>;<br/><b>let </b><font color="Maroon">t</font> be    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font>;<br/>





<a NAME="E1:20"/><i><font color="Green">E28</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="trees_a.html#K3" target="_self">tree</a> <font color="Maroon">T</font>,<span class="p2"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">t</font></span><a href="trees_a.html#K2" target="_self">}</a></span>,<font color="Maroon">T1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_2.html#K8">with-replacement</a> <font color="Maroon">t</font>,<font color="Maroon">T1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="trees_a.html#K3" target="_self">tree</a> <font color="Maroon">T</font>,<span class="p2"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">t</font></span><a href="trees_a.html#K2" target="_self">}</a></span>,<font color="Maroon">T1</font></span>)</span> = 
 <a href="trees_a.html#K1" target="_self">tree</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font></span>)</span>,<span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">t</font></span><a href="trees_a.html#K2" target="_self">}</a></span>,<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font></span>)</span> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">t</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_2.html#K8">with-replacement</a> <font color="Maroon">t</font>,<font color="Maroon">T1</font></span>)</span>
<b>by </b><i><a class="ref" href="trees_2.html#D12">TREES_2:def 12</a></i>
;<br/>
<b>hence </b><a NAME="E2:20_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="trees_a.html#K3" target="_self">tree</a> <font color="Maroon">T</font>,<span class="p2"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">t</font></span><a href="trees_a.html#K2" target="_self">}</a></span>,<font color="Maroon">T1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_2.html#K8">with-replacement</a> <font color="Maroon">t</font>,<font color="Maroon">T1</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:20"/>
for <font color="Olive">q</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">q</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="trees_a.html#K3" target="_self">tree</a> <font color="Maroon">T</font>,<span class="p2"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">t</font></span><a href="trees_a.html#K2" target="_self">}</a></span>,<font color="Maroon">T1</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="trees_a.html#K3" target="_self">tree</a> <font color="Maroon">T</font>,<span class="p2"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">t</font></span><a href="trees_a.html#K2" target="_self">}</a></span>,<font color="Maroon">T1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_2.html#K8">with-replacement</a> <font color="Maroon">t</font>,<font color="Maroon">T1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:20_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="trees_a.html#K3" target="_self">tree</a> <font color="Maroon">T</font>,<span class="p2"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">t</font></span><a href="trees_a.html#K2" target="_self">}</a></span>,<font color="Maroon">T1</font></span>)</span>
 ;<br/>

<a NAME="E2:20_2"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="trees_a.html#K1" target="_self">tree</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font></span>)</span>,<span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">t</font></span><a href="trees_a.html#K2" target="_self">}</a></span>,<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:20_2"/><i><font color="Green">E35</font></i>: 
 <a href="trees_a.html#K1" target="_self">tree</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font></span>)</span>,<span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">t</font></span><a href="trees_a.html#K2" target="_self">}</a></span>,<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">t1</font> where <font color="Olive">t1</font> is    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font> : for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">t</font></span><a href="trees_a.html#K2" target="_self">}</a></span> holds <br/> not <font color="Olive">p</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">t1</font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">p</font> is    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font>, <font color="Olive">s</font> is    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font> : <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">t</font></span><a href="trees_a.html#K2" target="_self">}</a></span> </span>}</span> 
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_2_1"/>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">t1</font> where <font color="Olive">t1</font> is    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font> : for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">t</font></span><a href="trees_a.html#K2" target="_self">}</a></span> holds <br/> not <font color="Olive">p</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">t1</font> </span>}</span>  or <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p'</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">p'</font> is    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font>, <font color="Olive">s</font> is    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font> : <font color="Olive">p'</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">t</font></span><a href="trees_a.html#K2" target="_self">}</a></span> </span>}</span>  )
 </span><b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_2_1_1"><b>suppose </b></a><a NAME="E1:20_2_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">t1</font> where <font color="Olive">t1</font> is    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font> : for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">t</font></span><a href="trees_a.html#K2" target="_self">}</a></span> holds <br/> not <font color="Olive">p</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">t1</font> </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">t'</font> being    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:20_2_1_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t'</font> &amp; ( for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">t</font></span><a href="trees_a.html#K2" target="_self">}</a></span> holds <br/> not <font color="Olive">p</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">t'</font> ) )
 ;<br/><b>consider </b><font color="Maroon">p</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="E5:20_2_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 ;<br/><a NAME="E6:20_2_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">t</font></span><a href="trees_a.html#K2" target="_self">}</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E7:20_2_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
not <font color="Maroon">p</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">t'</font>
 <b>by </b><i/>;<br/><a NAME="E8:20_2_1_1"/>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_2.html#K8">with-replacement</a> <font color="Maroon">t</font>,<font color="Maroon">T1</font></span>)</span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">t1</font> where <font color="Olive">t1</font> is    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font> : not <font color="Maroon">p</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">t1</font> </span>}</span>  implies <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_2.html#K8">with-replacement</a> <font color="Maroon">t</font>,<font color="Maroon">T1</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">q</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E9:20_2_1_1"/>
<span class="p1">(<span class="default"><a href="trees_a.html#K3" target="_self">tree</a> <font color="Maroon">T</font>,<span class="p2"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">t</font></span><a href="trees_a.html#K2" target="_self">}</a></span>,<font color="Maroon">T1</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">T</font> <a href="trees_2.html#K8">with-replacement</a> <font color="Maroon">t</font>,<font color="Maroon">T1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font>
 <b>by </b><i>, , , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_2_1_2"><b>suppose </b></a><a NAME="E1:20_2_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p'</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">p'</font> is    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font>, <font color="Olive">s</font> is    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font> : <font color="Olive">p'</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">t</font></span><a href="trees_a.html#K2" target="_self">}</a></span> </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p</font> being    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font>, <font color="Maroon">r</font> being    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font><b> such that </b><br/><a NAME="E3:20_2_1_2"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">t</font></span><a href="trees_a.html#K2" target="_self">}</a></span> )
 ;<br/><a NAME="E4:20_2_1_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">s</font> is    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font> : <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> </span>}</span> 
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">p1</font> being    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font>, <font color="Maroon">r1</font> being    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font><b> such that </b><br/><a NAME="E6:20_2_1_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">t</font></span><a href="trees_a.html#K2" target="_self">}</a></span> &amp; <span class="p1">(<span class="default"><a href="trees_a.html#K3" target="_self">tree</a> <font color="Maroon">T</font>,<span class="p2"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">t</font></span><a href="trees_a.html#K2" target="_self">}</a></span>,<font color="Maroon">T1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E7:20_2_1_2"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">r2</font> being    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font><b> such that </b><br/><a NAME="E9:20_2_1_2"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_2.html#K8">with-replacement</a> <font color="Maroon">p</font>,<font color="Maroon">T1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r2</font> )
 <b>by </b><i>, , , </i>;<br/><b>thus </b><a NAME="E10:20_2_1_2"/>
<span class="p1">(<span class="default"><a href="trees_a.html#K3" target="_self">tree</a> <font color="Maroon">T</font>,<span class="p2"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">t</font></span><a href="trees_a.html#K2" target="_self">}</a></span>,<font color="Maroon">T1</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">T</font> <a href="trees_2.html#K8">with-replacement</a> <font color="Maroon">t</font>,<font color="Maroon">T1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font>
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E3:20"/>
 <a href="trees_a.html#K3" target="_self">tree</a> <font color="Maroon">T</font>,<span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">t</font></span><a href="trees_a.html#K2" target="_self">}</a></span>,<font color="Maroon">T1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> <a href="trees_2.html#K8">with-replacement</a> <font color="Maroon">t</font>,<font color="Maroon">T1</font>
 <b>by </b><i>, <a class="ref" href="trees_2.html#T33">TREES_2:33</a></i>;<br/>


</div>
