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

<b>thus </b><a NAME="E1:9_1_1"/>
not  <a href="closure2.html#K1" target="_self">Bool</a> <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:9_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is    <a href="pboole.html#M4">ManySortedSubset</a> of <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:9_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>2</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:9_1_1_1"/>
not  <a href="closure2.html#K1" target="_self">Bool</a> <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="closure2.html#D1" target="_self">Def1</a></i>;<br/>


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

<b>thus </b><a NAME="E2:9_1_1"/>
 <a href="closure2.html#K1" target="_self">Bool</a> <font color="Maroon">c<sub>2</sub></font> is <a href="fraenkel.html#V1">functional</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_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="fraenkel.html#D1">FRAENKEL:def 1</a><br/>

<b>assume </b><a NAME="E1:9_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="closure2.html#K1" target="_self">Bool</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>hence </b><a NAME="E2:9_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> is   <a href="funct_1.html#NM1">Function</a>
 <b>by </b><i><a class="ref" href="closure2.html#D1" target="_self">Def1</a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="funct_1.html#NM1">Function</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="pralg_2.html#D1">PRALG_2:def 1</a><br/>

<b>assume </b><a NAME="E3:9_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="closure2.html#K1" target="_self">Bool</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="closure2.html#K1" target="_self">Bool</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<a NAME="E4:9_1_1"/><b>then </b><i><font color="Green">E3</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is    <a href="pboole.html#M4">ManySortedSubset</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> is    <a href="pboole.html#M4">ManySortedSubset</a> of <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="closure2.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b> <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> = 
<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="closure2.html#E4:9_1_1"><i><font color="Green">E3</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/><br/>


</div>
