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

<b>reconsider </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Olive">A</font></span>)</span> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">U0</font> : not <font color="Olive">A</font> is <a href="xboole_0.html#V1">empty</a> </span>}</span>  as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon">X</font>
;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E2:45_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> implies <font color="Maroon">x</font> is  <a href="unialg_1.html#V3">strict</a>  <a href="unialg_2.html#M3" target="_self">SubAlgebra</a> of <font color="Maroon">U0</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:45_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">U0</font><b> such that </b><br/><a NAME="E3:45_1_1_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a> )
 ;<br/>
<b>thus </b><a NAME="E4:45_1_1_1"/>
<font color="Maroon">x</font> is  <a href="unialg_1.html#V3">strict</a>  <a href="unialg_2.html#M3" target="_self">SubAlgebra</a> of <font color="Maroon">U0</font>
 <b>by </b><i/>;<br/>


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

<b>assume </b><a NAME="E3:45_1_1"/>
<font color="Maroon">x</font> is  <a href="unialg_1.html#V3">strict</a>  <a href="unialg_2.html#M3" target="_self">SubAlgebra</a> of <font color="Maroon">U0</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font> as  <a href="unialg_1.html#V3">strict</a>  <a href="unialg_2.html#M3" target="_self">SubAlgebra</a> of <font color="Maroon">U0</font> ;<br/>
<b>reconsider </b><font color="Maroon">A</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">a</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">U0</font> <b>by </b><i/>;<br/>
<a NAME="E6:45_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:45_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>


</div>
