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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="unialg_2.html#V2" target="_self">with_const_op</a> <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  <a href="unialg_1.html#V3">strict</a>  <a href="unialg_2.html#M3" target="_self">SubAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:43"/><i><font color="Green">E29</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#R2">meets</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="unialg_2.html#T20" target="_self">Th20</a></i>;<br/>
<a NAME="E2:43"/><b>then </b><i><font color="Green">E30</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K5" target="_self">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D10" target="_self">Def10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>5</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> 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/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font> 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> ;<br/>
<a NAME="E5:43"/>
<font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D14" target="_self">Def14</a></i>;<br/>
<a NAME="E6:43"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>6</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"><font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D13" target="_self">Def13</a></i>;<br/>
<a NAME="E7:43"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E8:43"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</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"><font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="unialg_2.html#E6:43"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E9:43"/><b>then </b><i><font color="Green">E32</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</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"><font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K5" target="_self">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="unialg_2.html#E2:43"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E10:43"/><i><font color="Green">E33</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K5" target="_self">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="unialg_2.html#E2:43"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K5" target="_self">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>3</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="E12:43"/><i><font color="Green">E34</font></i>: 
( the <a href="unialg_1.html#U1">charact</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K5" target="_self">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>3</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>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> is <a href="unialg_2.html#V1" target="_self">opers_closed</a> )
 
<b>by </b><i><a class="txt" href="unialg_2.html#E1:43"><i><font color="Green">E29</font></i></a>, <a class="ref" href="unialg_2.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E13:43"/><i><font color="Green">E35</font></i>: 
( the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>2</sub></font> <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>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="unialg_2.html#V1" target="_self">opers_closed</a> )
 
<b>by </b><i><a class="ref" href="unialg_2.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E14:43"/><i><font color="Green">E36</font></i>: 
(  <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>7</sub></font></span>)</span> <a href="hidden.html#R1">=</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> &amp;  <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>4</sub></font></span>)</span> <a href="hidden.html#R1">=</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> )
 
<b>by </b><i><a class="ref" href="unialg_2.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E15:43"/>
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 <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K5" target="_self">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>2</sub></font> <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="43_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:43_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>8</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>9</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>8</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/>
<b>thus </b>the <a href="unialg_1.html#U1">charact</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K5" target="_self">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> = 
<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>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="unialg_2.html#E1:43"><i><font color="Green">E29</font></i></a>, <a class="ref" href="unialg_2.html#D10" target="_self">Def10</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="unialg_2.html#K2" target="_self">/.</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="unialg_2.html#E14:43"><i><font color="Green">E36</font></i></a>, <a class="txt" href="unialg_2.html#E1:43_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="unialg_2.html#D7" target="_self">Def7</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="unialg_2.html#K2" target="_self">/.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="unialg_2.html#E9:43"><i><font color="Green">E32</font></i></a>, <a class="txt" href="unialg_2.html#E10:43"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>
<br/>.= 
<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>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="unialg_2.html#E14:43"><i><font color="Green">E36</font></i></a>, <a class="txt" href="unialg_2.html#E1:43_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="unialg_2.html#D7" target="_self">Def7</a></i>
<br/>.= 
the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="unialg_2.html#D8" target="_self">Def8</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E16:43"/><b>then </b>
the <a href="unialg_1.html#U1">charact</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K5" target="_self">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="unialg_2.html#E12:43"><i><font color="Green">E34</font></i></a>, <a class="txt" href="unialg_2.html#E13:43"><i><font color="Green">E35</font></i></a>, <a class="txt" href="unialg_2.html#E14:43"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>
<b>hence </b><a NAME="E17:43"/>
<font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K5" target="_self">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="unialg_2.html#E9:43"><i><font color="Green">E32</font></i></a>, <a class="txt" href="unialg_2.html#E10:43"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
