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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>thus </b><a NAME="E1:15"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="trees_3.html#V3" target="_self">constituted-DTrees</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="trees_3.html#V3" target="_self">constituted-DTrees</a> implies <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font> is <a href="trees_3.html#V3" target="_self">constituted-DTrees</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:15_1"/><i><font color="Green">E10</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is   <a href="trees_2.html#NM4">DecoratedTree</a>
 <b>and </b><br/><a NAME="E2:15_1"/><i><font color="Green">E11</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is   <a href="trees_2.html#NM4">DecoratedTree</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="trees_3.html#D5" target="_self">TREES_3:def 5</a><br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="trees_3.html#D5" target="_self">TREES_3:def 5</a><br/>

<b>assume </b><a NAME="E3:15_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E4:15_1"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></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="E5:15_1"/>
<font color="Maroon">c<sub>3</sub></font> is   <a href="trees_2.html#NM4">DecoratedTree</a>
 <b>by </b><i><a class="txt" href="trees_3.html#E1:15_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="trees_3.html#E2:15_1"><i><font color="Green">E11</font></i></a></i>;<br/>


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

<b>assume </b><a NAME="E2:15"/><i><font color="Green">E10</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is   <a href="trees_2.html#NM4">DecoratedTree</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="trees_3.html#D5" target="_self">TREES_3:def 5</a><br/>

<b>thus </b><a NAME="E3:15"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="trees_3.html#V3" target="_self">constituted-DTrees</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="trees_3.html#D5" target="_self">TREES_3:def 5</a><br/>

<b>assume </b><a NAME="E1:15_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:15_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></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="E3:15_2"/>
<font color="Maroon">c<sub>3</sub></font> is   <a href="trees_2.html#NM4">DecoratedTree</a>
 <b>by </b><i><a class="txt" href="trees_3.html#E2:15"><i><font color="Green">E10</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="trees_3.html#D5" target="_self">TREES_3:def 5</a><br/>

<b>assume </b><a NAME="E4:15"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E5:15"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></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="E6:15"/>
<font color="Maroon">c<sub>3</sub></font> is   <a href="trees_2.html#NM4">DecoratedTree</a>
 <b>by </b><i><a class="txt" href="trees_3.html#E2:15"><i><font color="Green">E10</font></i></a></i>;<br/>


</div>
