<?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/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:5_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="finsub_1.html#V4" target="_self">preBoolean</a>
 ;<br/><a NAME="E2:5_1"/><b>then </b><i><font color="Green">E5</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="finsub_1.html#V1" target="_self">cup-closed</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="finsub_1.html#V3" target="_self">diff-closed</a> )
 <b>by </b><i><a class="ref" href="finsub_1.html#D4" target="_self">Def4</a></i>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E3:5_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><b>hence </b><a NAME="E4:5_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="finsub_1.html#E2:5_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="finsub_1.html#D1" target="_self">Def1</a>, <a class="ref" href="finsub_1.html#D3" target="_self">Def3</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:5"/><i><font color="Green">E5</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</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> &amp; <font color="Olive">b<sub>2</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> <a href="xboole_0.html#K2">\/</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<a NAME="E3:5"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="finsub_1.html#V1" target="_self">cup-closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <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="finsub_1.html#D1" target="_self">FINSUB_1:def 1</a><br/>

<b>assume </b><a NAME="E1:5_2"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<b>hence </b><a NAME="E2:5_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="finsub_1.html#E2:5"><i><font color="Green">E5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:5"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="finsub_1.html#V3" target="_self">diff-closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <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="finsub_1.html#D3" target="_self">FINSUB_1:def 3</a><br/>

<b>assume </b><a NAME="E1:5_3"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<b>hence </b><a NAME="E2:5_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="finsub_1.html#E2:5"><i><font color="Green">E5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:5"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="finsub_1.html#V4" target="_self">preBoolean</a>
 <b>by </b><i><a class="txt" href="finsub_1.html#E3:5"><i><font color="Green">E6</font></i></a>, <a class="ref" href="finsub_1.html#D4" target="_self">Def4</a></i>;<br/>


</div>
