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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="pboole.html#M1" target="_self">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>







<b>thus </b><a NAME="E1:37"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R6" target="_self">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K3" target="_self">/\</a> <font color="Maroon">c<sub>4</sub></font> implies ( <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R2" target="_self">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R2" target="_self">c=</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="pboole.html#M1" target="_self">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R2" target="_self">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R2" target="_self">c=</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R2" target="_self">c=</a> <font color="Maroon">c<sub>2</sub></font> ) ) )
 <b>by </b><i><a class="ref" href="pboole.html#T17" target="_self">Th17</a>, <a class="ref" href="pboole.html#T19" target="_self">Th19</a></i>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:37"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R2" target="_self">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E3:37"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R2" target="_self">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E4:37"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="pboole.html#M1" target="_self">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R2" target="_self">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R2" target="_self">c=</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R2" target="_self">c=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>thus </b><a NAME="E5:37"/>
<font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R2" target="_self">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K3" target="_self">/\</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="pboole.html#E2:37"><i><font color="Green">E27</font></i></a>, <a class="txt" href="pboole.html#E3:37"><i><font color="Green">E28</font></i></a>, <a class="ref" href="pboole.html#T19" target="_self">Th19</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D13" target="_self">PBOOLE:def 13</a><br/>

<a NAME="E6:37"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K3" target="_self">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#R2" target="_self">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K3" target="_self">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#R2" target="_self">c=</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K3" target="_self">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#R2" target="_self">c=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="pboole.html#E4:37"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:37"/>
<font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K3" target="_self">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#R2" target="_self">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#T17" target="_self">Th17</a></i>;<br/>


</div>
