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<b>let </b><font color="Maroon">U0</font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/><b>let </b><font color="Maroon">H</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">U0</font>;<br/><b>let </b><font color="Maroon">o</font> be   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U0</font>;<br/>





<b>assume </b><a NAME="E1:8"/><i><font color="Green">E26</font></i>: 
 <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>thus </b><a NAME="E2:8"/>
( <font color="Maroon">H</font> <a href="unialg_2.html#R2">is_closed_on</a> <font color="Maroon">o</font> implies <font color="Maroon">o</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">H</font> <a href="unialg_2.html#R2">is_closed_on</a> <font color="Maroon">o</font>
 ;<br/>

<b>consider </b><font color="Maroon">s</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">H</font><b> such that </b><br/><a NAME="E3:8_1"/><i><font color="Green">E37</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T24">FINSEQ_1:24</a></i>;<br/>
<a NAME="E4:8_1"/>
<font color="Maroon">o</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 
<b>by </b><i>, , <a class="ref" href="unialg_2.html#D4">UNIALG_2:def 4</a></i>;<br/>
<b>hence </b><a NAME="E5:8_1"/>
<font color="Maroon">o</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:8"/>
( <font color="Maroon">o</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> implies <font color="Maroon">H</font> <a href="unialg_2.html#R2">is_closed_on</a> <font color="Maroon">o</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_2"/>
<font color="Maroon">o</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 ;<br/>

<a NAME="E2:8_2"/><b>then </b>
for <font color="Olive">s</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">H</font>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o</font> holds <br/><font color="Maroon">o</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<b>hence </b><a NAME="E3:8_2"/>
<font color="Maroon">H</font> <a href="unialg_2.html#R2">is_closed_on</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="unialg_2.html#D4">UNIALG_2:def 4</a></i>;<br/>


</div><b>end;</b></div>


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