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<div class="add">

<b>let </b><font color="Maroon">U0</font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/>

<a NAME="E1:22"/><i><font color="Green">E20</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U0</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U0</font>
 
;<br/>
<a NAME="E2:22"/>
for <font color="Olive">B</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">U0</font>  st <font color="Olive">B</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U0</font> holds <br/>( the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U0</font> <a href="hidden.html#R1">=</a>  <a href="unialg_2.html#K3" target="_self">Opers</a> <font color="Maroon">U0</font>,<font color="Olive">B</font> &amp; <font color="Olive">B</font> is <a href="unialg_2.html#V1" target="_self">opers_closed</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">B</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">U0</font>;<br/>

<b>assume </b><a NAME="E1:22_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U0</font>
 ;<br/>

<a NAME="E2:22_1"/><i><font color="Green">E29</font></i>: 
 <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U0</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K3" target="_self">Opers</a> <font color="Maroon">U0</font>,<font color="Maroon">B</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:22_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U0</font> holds <br/>the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K3" target="_self">Opers</a> <font color="Maroon">U0</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:22_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U0</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">o</font> = the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> as   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U0</font> <b>by </b><i/>;<br/>
<a NAME="E3:22_1_1"/>
<span class="p1">(<span class="default"><a href="unialg_2.html#K3" target="_self">Opers</a> <font color="Maroon">U0</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> <a href="unialg_2.html#K2" target="_self">/.</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#T1" target="_self">Th1</a>, , </i>;<br/>
<b>hence </b><a NAME="E4:22_1_1"/>
<span class="p1">(<span class="default"><a href="unialg_2.html#K3" target="_self">Opers</a> <font color="Maroon">U0</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:22_1"/>
( the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U0</font> <a href="hidden.html#R1">=</a>  <a href="unialg_2.html#K3" target="_self">Opers</a> <font color="Maroon">U0</font>,<font color="Maroon">B</font> &amp; <font color="Maroon">B</font> is <a href="unialg_2.html#V1" target="_self">opers_closed</a> )
 <b>by </b><i>, <a class="ref" href="unialg_2.html#T1" target="_self">Th1</a>, , <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:22"/>
<font color="Maroon">U0</font> is    <a href="unialg_2.html#M3" target="_self">SubAlgebra</a> of <font color="Maroon">U0</font>
 <b>by </b><i>, </i>;<br/>


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