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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  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/>



<b>assume </b><a NAME="E1:16"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>2</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>
 ;<br/>

<a NAME="E2:16"/><i><font color="Green">E12</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#R2" target="_self">is_closed_on</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<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="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="unialg_2.html#D4" target="_self">UNIALG_2:def 4</a><br/>



<b>assume </b><a NAME="E1:16_1"/><i><font color="Green">E13</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:16_1"/><i><font color="Green">E14</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="unialg_2.html#E1:16"><i><font color="Green">E11</font></i></a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E3:16_1"/><i><font color="Green">E15</font></i>: 
 <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>2</sub></font>
 
<b>by </b><i><a class="txt" href="unialg_2.html#E1:16"><i><font color="Green">E11</font></i></a>, <a class="ref" href="unialg_2.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E4:16_1"/>
<font color="Maroon">c<sub>4</sub></font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</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>2</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E5:16_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="unialg_2.html#E1:16_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="unialg_2.html#E3:16_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:16_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="unialg_2.html#E2:16_1"><i><font color="Green">E14</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:16"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="unialg_2.html#K2" target="_self">/.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>proof </b></a><div class="add">

<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/>

<a NAME="E1:16_2"/><i><font color="Green">E13</font></i>: 
 <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>2</sub></font>
 
<b>by </b><i><a class="txt" href="unialg_2.html#E1:16"><i><font color="Green">E11</font></i></a>, <a class="ref" href="unialg_2.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E2:16_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#R2" target="_self">is_closed_on</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="unialg_2.html#E2:16"><i><font color="Green">E12</font></i></a></i>;<br/>
<a NAME="E3:16_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="unialg_2.html#K2" target="_self">/.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</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>3</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D6" target="_self">Def6</a></i>;<br/>
<b>hence </b><a NAME="E4:16_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="unialg_2.html#K2" target="_self">/.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="unialg_2.html#E1:16_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:16"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="unialg_2.html#V1" target="_self">opers_closed</a> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="unialg_2.html#K2" target="_self">/.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="unialg_2.html#E2:16"><i><font color="Green">E12</font></i></a>, <a class="ref" href="unialg_2.html#D5" target="_self">Def5</a></i>;<br/>


</div>
