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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>







<b>assume </b><a NAME="E1:20"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is    <a href="unialg_2.html#M3" target="_self">SubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</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> &amp; <font color="Maroon">c<sub>3</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>5</sub></font> &amp; <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>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</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>2</sub></font> <b>by </b><i><a class="ref" href="unialg_2.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E3:20"/>
<font color="Maroon">c<sub>6</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#E1:20"><i><font color="Green">E13</font></i></a>, <a class="ref" href="unialg_2.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E4:20"/><b>then </b><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="unialg_2.html#R2" target="_self">is_closed_on</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E5:20"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</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>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="unialg_2.html#E1:20"><i><font color="Green">E13</font></i></a>, <a class="ref" href="unialg_2.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E6:20"/>
<font color="Maroon">c<sub>3</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>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="unialg_2.html#E1:20"><i><font color="Green">E13</font></i></a>, <a class="ref" href="unialg_2.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E7:20"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="unialg_2.html#K2" target="_self">/.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="unialg_2.html#E1:20"><i><font color="Green">E13</font></i></a>, <a class="txt" href="unialg_2.html#E5:20"><i><font color="Green">E15</font></i></a>, <a class="ref" href="unialg_2.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E8:20"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="unialg_2.html#E4:20"><i><font color="Green">E14</font></i></a>, <a class="ref" href="unialg_2.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E9:20"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<b>then </b><i><font color="Green">E16</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="unialg_2.html#T2" target="_self">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="unialg_2.html#T1" target="_self">Th1</a></i>
;<br/>
<a NAME="E11:20"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#T2" target="_self">Th2</a></i>;<br/>
<b>hence </b><a NAME="E12:20"/>
 <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="unialg_2.html#E10:20"><i><font color="Green">E16</font></i></a>, <a class="ref" href="finseq_2.html#T130">FINSEQ_2:130</a></i>;<br/>


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