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

<b>let </b><font color="Maroon">T1</font> be   <a href="trees_1.html#NM1">Tree</a>, <font color="Maroon">T2</font> be   <a href="trees_1.html#NM1">Tree</a>;<br/><b>let </b><font color="Maroon">p</font> be    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">T2</font>;<br/>



<b>thus </b><a NAME="E1:110"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T2</font> implies <span class="p1">(<span class="default"><font color="Maroon">T1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">T2</font></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">T1</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">T2</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:110_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T2</font> )
 ;<br/>

<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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="trees_2.html#D1">TREES_2:def 1</a><br/>

<b>thus </b><a NAME="E2:110_1"/>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">T1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">T2</font></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font> implies <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">T1</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">T2</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:110_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">T1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">T2</font></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E2:110_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">T2</font>
 
<b>by </b><i><a class="ref" href="trees_1.html#D9">TREES_1:def 9</a></i>;<br/>
<a NAME="E3:110_1_1"/><b>then </b>
( <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font> or <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T2</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E4:110_1_1"/><b>then </b>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font> or <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T2</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font> )
 
<b>by </b><i><a class="ref" href="trees_3.html#D5" target="_self">Def5</a>, <a class="ref" href="trees_1.html#D9">TREES_1:def 9</a></i>;<br/>
<b>hence </b><a NAME="E5:110_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">T1</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">T2</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:110_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">T1</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">T2</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font></span>)</span>
 ;<br/>

<a NAME="E4:110_1"/><b>then </b>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font> or <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T2</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:110_1"/><b>then </b>
( <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font> or <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T2</font> )
 
<b>by </b><i><a class="ref" href="trees_3.html#D5" target="_self">Def5</a>, <a class="ref" href="trees_1.html#D9">TREES_1:def 9</a></i>;<br/>
<a NAME="E6:110_1"/><b>then </b>
<font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">T2</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E7:110_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">T1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">T2</font></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="trees_1.html#D9">TREES_1:def 9</a></i>;<br/>


</div><b>end;</b></div>

<a NAME="E2:110"/>
for <font color="Olive">T1</font>, <font color="Olive">T2</font> being  <a href="trees_1.html#NM1">Tree</a><br/> for <font color="Olive">p</font> being   <a href="trees_1.html#M1">Element</a> of <font color="Olive">T1</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">T2</font>  st not <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">T1</font> holds <br/><span class="p1">(<span class="default"><font color="Olive">T1</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">T2</font></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">T2</font> <a href="trees_1.html#K4">|</a> <font color="Olive">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">T1</font> be   <a href="trees_1.html#NM1">Tree</a>, <font color="Maroon">T2</font> be   <a href="trees_1.html#NM1">Tree</a>;<br/><b>let </b><font color="Maroon">p</font> be    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">T2</font>;<br/>



<b>assume </b><a NAME="E1:110_2"/><i><font color="Green">E40</font></i>: 
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font>
 ;<br/>

<a NAME="E2:110_2"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T2</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="trees_2.html#D1">TREES_2:def 1</a><br/>

<b>thus </b><a NAME="E3:110_2"/>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">T1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">T2</font></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font> implies <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T2</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:110_2_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">T1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">T2</font></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E2:110_2_1"/><b>then </b>
<font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">T2</font>
 
<b>by </b><i><a class="ref" href="trees_1.html#D9">TREES_1:def 9</a></i>;<br/>
<a NAME="E3:110_2_1"/><b>then </b>
( <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font> or <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T2</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:110_2_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T2</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="trees_3.html#D5" target="_self">Def5</a>, <a class="ref" href="trees_3.html#T3" target="_self">Th3</a>, <a class="ref" href="trees_1.html#T46">TREES_1:46</a>, <a class="ref" href="trees_1.html#D9">TREES_1:def 9</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E4:110_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T2</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E5:110_2"/><b>then </b>
<font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T2</font>
 
<b>by </b><i><a class="ref" href="trees_3.html#T3" target="_self">Th3</a>, <a class="ref" href="trees_1.html#D9">TREES_1:def 9</a></i>;<br/>
<a NAME="E6:110_2"/><b>then </b>
<font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">T2</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E7:110_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">T1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">T2</font></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="trees_1.html#D9">TREES_1:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:110"/>
( ( not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font> implies <span class="p1">(<span class="default"><font color="Maroon">T1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">T2</font></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T2</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font> ) &amp; ( not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T2</font> implies <span class="p1">(<span class="default"><font color="Maroon">T1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">T2</font></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T1</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font> ) )
 ;<br/>


</div>
