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<a NAME="E1:30_1_1"/><i><font color="Green">E20</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> 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="E2:30_1_1"/><i><font color="Green">E21</font></i>: 
( 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 <font color="Maroon">c<sub>3</sub></font> <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> &amp; 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 <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>then 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> <a href="xboole_0.html#K3">/\</a> 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="txt" href="unialg_2.html#E1:30"><i><font color="Green">E19</font></i></a>, <a class="txt" href="unialg_2.html#E1:30_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<div><i><font color="Green">E22</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><b>assume </b><a NAME="E1:30_1_1_1_1"/><i><font color="Green">E23</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>set </b><font color="Maroon">c<sub>7</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>8</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of 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#E2:30_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="finseq_2.html#T27">FINSEQ_2:27</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</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/><a NAME="E4:30_1_1_1_1"/>
<font color="Maroon">c<sub>10</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="E5:30_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="unialg_2.html#R2" target="_self">is_closed_on</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="unialg_2.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E6:30_1_1_1_1"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="unialg_2.html#E1:30_1_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="unialg_2.html#D4" target="_self">Def4</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="unialg_2.html#E2:30_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="finseq_2.html#T27">FINSEQ_2:27</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</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/><a NAME="E9:30_1_1_1_1"/>
<font color="Maroon">c<sub>12</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="E10:30_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="unialg_2.html#R2" target="_self">is_closed_on</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="unialg_2.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E11:30_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="unialg_2.html#E1:30_1_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="unialg_2.html#D4" target="_self">Def4</a></i>;<br/><b>hence </b><a NAME="E12:30_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="unialg_2.html#E6:30_1_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:30_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="unialg_2.html#R2" target="_self">is_closed_on</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="unialg_2.html#D4" target="_self">Def4</a></i>;<br/></div><b>end;</b></div>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="unialg_1.html#G1">UAStr</a>(# <font color="Maroon">c<sub>4</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>4</sub></font></span>)</span> #);<br/>
<a NAME="E5:30_1_1"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being 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> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="unialg_1.html#G1">UAStr</a>(# <font color="Maroon">c<sub>4</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>4</sub></font></span>)</span> #) holds <br/>( the <a href="unialg_1.html#U1">charact</a> of <a href="unialg_1.html#G1">UAStr</a>(# <font color="Maroon">c<sub>4</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>4</sub></font></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="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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#E4:30_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="unialg_2.html#D5" target="_self">Def5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="unialg_1.html#G1">UAStr</a>(# <font color="Maroon">c<sub>4</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>4</sub></font></span>)</span> #) as   <a href="unialg_1.html#NM2">Universal_Algebra</a> <b>by </b><i><a class="ref" href="unialg_2.html#T17" target="_self">Th17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as  <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> <b>by </b><i><a class="txt" href="unialg_2.html#E5:30_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="unialg_2.html#D8" target="_self">Def8</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>thus </b><a NAME="E8:30_1_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> <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 <font color="Maroon">c<sub>3</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being 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> st <font color="Olive">b<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>7</sub></font> holds <br/>( the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>7</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="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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#E4:30_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="unialg_2.html#D5" target="_self">Def5</a></i>;<br/>


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