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

<b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:10"/><i><font color="Green">E13</font></i>: 
for <font color="Olive">X</font>, <font color="Olive">Y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds <br/> ( <font color="Olive">X</font> <a href="xboole_0.html#K5">\+\</a> <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:10_1"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/><b>then reconsider </b><font color="Maroon">Z1</font> = <font color="Maroon">X</font> <a href="xboole_0.html#K5">\+\</a> <font color="Maroon">Y</font>, <font color="Maroon">Z2</font> = <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/><a NAME="E3:10_1"/>
<font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Z1</font> <a href="xboole_0.html#K5">\+\</a> <font color="Maroon">Z2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T94">XBOOLE_1:94</a></i>;<br/><b>hence </b><a NAME="E4:10_1"/>
<font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/><a NAME="E5:10_1"/>
<font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="xboole_0.html#K5">\+\</a> <font color="Maroon">Z2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T100">XBOOLE_1:100</a></i>;<br/><b>hence </b><a NAME="E6:10_1"/>
<font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="finsub_1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:10"/>
<font color="Maroon">A</font> is <a href="finsub_1.html#V4" target="_self">preBoolean</a>
 <b>by </b><i/>;<br/>


</div>
