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

<b>let </b><font color="Maroon">U1</font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>, <font color="Maroon">U2</font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/>

<b>assume </b><a NAME="E1:13"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">U1</font> is    <a href="unialg_2.html#M3">SubAlgebra</a> of <font color="Maroon">U2</font>
 ;<br/>

<a NAME="E2:13"/><b>then </b>
 <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">U1</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">U2</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">A</font> =  <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U1</font> as  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of  <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">U2</font> ;<br/>
<a NAME="E4:13"/>
<font color="Maroon">A</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of  <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:13_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U2</font></span>)</span>
 <b>by </b><i>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a><br/>

<b>let </b><font color="Maroon">B</font> be   <a href="msualg_2.html#NM1">MSSubset</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U2</font></span>)</span>;<br/>

<b>assume </b><a NAME="E2:13_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">B</font> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>
 ;<br/>

<b>hence </b><a NAME="E3:13_1"/>
<font color="Maroon">B</font> is <a href="msualg_2.html#V3">opers_closed</a>
 <b>by </b><i>, </i>;<br/>

<b>thus </b><a NAME="E4:13_1"/>
the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">A</font> <a href="pboole.html#R6">=</a>  <a href="msualg_2.html#K4">Opers</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U2</font></span>)</span>,<font color="Maroon">B</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:13"/>
 <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U1</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of  <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U2</font>
 ;<br/>


</div>
