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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="unialg_1.html#V3">strict</a> <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/>

<a NAME="E1:38"/><i><font color="Green">E24</font></i>: 
 <a href="subset_1.html#K2">[#]</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="subset_1.html#D4">SUBSET_1:def 4</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <span class="p1">(<span class="default"><a href="subset_1.html#K2">[#]</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E2:38"/><i><font color="Green">E25</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <span class="p2">(<span class="default"><a href="subset_1.html#K2">[#]</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E3:38"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <span class="p2">(<span class="default"><a href="subset_1.html#K2">[#]</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="unialg_2.html#E1:38"><i><font color="Green">E24</font></i></a>, <a class="ref" href="unialg_2.html#D13" target="_self">Def13</a></i>;<br/>
<a NAME="E4:38"/><b>then </b><i><font color="Green">E26</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <span class="p2">(<span class="default"><a href="subset_1.html#K2">[#]</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="unialg_2.html#E2:38"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <span class="p2">(<span class="default"><a href="subset_1.html#K2">[#]</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="unialg_2.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E6:38"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="unialg_2.html#V1" target="_self">opers_closed</a> &amp; the <a href="unialg_1.html#U1">charact</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <span class="p2">(<span class="default"><a href="subset_1.html#K2">[#]</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="unialg_2.html#K3" target="_self">Opers</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="unialg_2.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E7:38"/><i><font color="Green">E28</font></i>: 
 <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E8:38"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font> holds <br/>the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:38_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="unialg_2.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E3:38_1"/>
<span class="p1">(<span class="default"><a href="unialg_2.html#K3" target="_self">Opers</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="unialg_2.html#K2" target="_self">/.</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="unialg_2.html#E7:38"><i><font color="Green">E28</font></i></a>, <a class="txt" href="unialg_2.html#E1:38_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="unialg_2.html#D7" target="_self">Def7</a></i>;<br/>
<b>hence </b><a NAME="E4:38_1"/>
<span class="p1">(<span class="default"><a href="unialg_2.html#K3" target="_self">Opers</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="unialg_2.html#E4:38"><i><font color="Green">E26</font></i></a>, <a class="ref" href="unialg_2.html#T7" target="_self">Th7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:38"/>
 <a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <span class="p1">(<span class="default"><a href="subset_1.html#K2">[#]</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="unialg_2.html#E4:38"><i><font color="Green">E26</font></i></a>, <a class="txt" href="unialg_2.html#E6:38"><i><font color="Green">E27</font></i></a>, <a class="txt" href="unialg_2.html#E7:38"><i><font color="Green">E28</font></i></a>, <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>


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