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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">C</font> being <a href="msualg_1.html#V4">strict</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">A</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">C</font> &amp; <font color="Maroon">B</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Olive">C</font> );<br/>
<b>consider </b><font color="Maroon">IT</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:22_1_1"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">A</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<b>take </b>
<font color="Maroon">IT</font>
;<br/>

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

<b>thus </b><a NAME="E3:22_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> implies  ex <font color="Olive">C</font> being <a href="msualg_1.html#V4">strict</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">A</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">C</font> &amp; <font color="Maroon">B</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Olive">C</font> ) )
 <b>by </b><i/>;<br/>

<b>given </b><font color="Maroon">C</font> being  <a href="msualg_1.html#V4">strict</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">A</font><b> such that </b><a NAME="E4:22_1_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> &amp; <font color="Maroon">B</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">C</font> )
 ;<br/>

<a NAME="E5:22_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="msualg_2.html#D20">MSUALG_2:def 20</a></i>;<br/>
<b>hence </b><a NAME="E6:22_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font>
 <b>by </b><i>, </i>;<br/>


</div>
