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<b>let </b><font color="Maroon" title="c1">U0</font> be   <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a>;<br/><b>let </b><font color="Maroon" title="c2">H</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">U0</font>;<br/><b>let </b><font color="Maroon" title="c3">o</font> be   <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <font color="Maroon" title="c1">U0</font>;<br/>





<b>assume </b><a NAME="E1:8"/><i><font color="Green" title="E5">A1</font></i>: 
 <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<b>thus </b><a NAME="E2:8"/>
( <font color="Maroon" title="c2">H</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c3">o</font> implies <font color="Maroon" title="c3">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">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" title="E6">A2</font></i>: 
<font color="Maroon" title="c2">H</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c3">o</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c4">s</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">H</font><b> such that </b><br/><a NAME="E3:8_1"/><i><font color="Green" title="E7">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c3">o</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T24" title="FINSEQ_1:th.24">FINSEQ_1:24</a></i>;<br/>
<a NAME="E4:8_1"/>
<font color="Maroon" title="c3">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">H</font>
 
<b>by </b><i><a class="txt" href="unialg_3.html#E1:8_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="unialg_3.html#E3:8_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="unialg_2.html#D4" title="UNIALG_2:def.4">UNIALG_2:def 4</a></i>;<br/>
<b>hence </b><a NAME="E5:8_1"/>
<font color="Maroon" title="c3">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">H</font>
 <b>by </b><i><a class="txt" href="unialg_3.html#E1:8"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="unialg_3.html#E3:8_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/>


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

<b>thus </b><a NAME="E3:8"/>
( <font color="Maroon" title="c3">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">H</font> implies <font color="Maroon" title="c2">H</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c3">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" title="c3">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">H</font>
 ;<br/>

<a NAME="E2:8_2"/><b>then </b>
for <font color="Olive" title="b1">s</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">H</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c3">o</font> holds <br/><font color="Maroon" title="c3">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">H</font>
 
<b>by </b><i><a class="txt" href="unialg_3.html#E1:8"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/>
<b>hence </b><a NAME="E3:8_2"/>
<font color="Maroon" title="c2">H</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c3">o</font>
 <b>by </b><i><a class="ref" href="unialg_2.html#D4" title="UNIALG_2:def.4">UNIALG_2:def 4</a></i>;<br/>


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


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