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<b>let </b><font color="Maroon" title="c1">U0</font> be  <a href="unialg_2.html#V2" title="UNIALG_2:attr.2">with_const_op</a> <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="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="unialg_3.html#K1" title="UNIALG_3:func.1">Sub</a> <font color="Maroon" title="c1">U0</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c3">S</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>assume </b><a NAME="E1:21"/><i><font color="Green" title="E12">A1</font></i>: 
<font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_3.html#K6" title="UNIALG_3:func.6">Carr</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">H</font></span>)</span>
 ;<br/>

<a NAME="E2:21"/><i><font color="Green" title="E13">A2</font></i>: 
<span class="p1">(<span class="default"><a href="unialg_3.html#K6" title="UNIALG_3:func.6">Carr</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="unialg_3.html#T9" target="_self" title="UNIALG_3:th.9">Th9</a></i>;<br/>
<a NAME="E3:21"/>
for <font color="Olive" title="b1">o</font> being  <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <font color="Maroon" title="c1">U0</font> holds  <font color="Maroon" title="c3">S</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Olive" title="b1">o</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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>let </b><font color="Maroon" title="c5">s</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c3">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="unialg_2.html#D4" title="UNIALG_2:def.4">UNIALG_2:def 4</a><br/>



<b>assume </b><a NAME="E1:21_1"/><i><font color="Green" title="E14">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">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="c4">o</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:21_1_1"/><i><font color="Green" title="E15">A4</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="unialg_3.html#K6" title="UNIALG_3:func.6">Carr</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">H</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c7">H1</font> = <font color="Maroon" title="c6">a</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">U0</font> ;<br/><b>consider </b><font color="Maroon" title="c8">H2</font> being    <a href="unialg_3.html#M2" title="UNIALG_3:mode.2">Element</a> of  <a href="unialg_3.html#K1" title="UNIALG_3:func.1">Sub</a> <font color="Maroon" title="c1">U0</font><b> such that </b><br/><a NAME="E4:21_1_1"/><i><font color="Green" title="E16">A5</font></i>: 
( <font color="Maroon" title="c8">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">H</font> &amp; <font color="Maroon" title="c7">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="unialg_3.html#K6" title="UNIALG_3:func.6">Carr</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">H2</font> )
 <b>by </b><i><a class="txt" href="unialg_3.html#E1:21_1_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/><a NAME="E5:21_1_1"/><i><font color="Green" title="E17">A6</font></i>: 
<font color="Maroon" title="c7">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">H2</font>
 <b>by </b><i><a class="txt" href="unialg_3.html#E4:21_1_1"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="unialg_3.html#D4" target="_self" title="UNIALG_3:def.4">Def4</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c9">H3</font> = <font color="Maroon" title="c7">H1</font> as  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/><a NAME="E7:21_1_1"/>
<font color="Maroon" title="c3">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">H1</font>
 <b>by </b><i><a class="txt" href="unialg_3.html#E1:21"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="unialg_3.html#E1:21_1_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="setfam_1.html#T4" title="SETFAM_1:th.4">SETFAM_1:4</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">s1</font> = <font color="Maroon" title="c5">s</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c9">H3</font> <b>by </b><i><a class="ref" href="finseq_2.html#T27" title="FINSEQ_2:th.27">FINSEQ_2:27</a></i>;<br/><a NAME="E9:21_1_1"/>
<font color="Maroon" title="c9">H3</font> is <a href="unialg_2.html#V1" title="UNIALG_2:attr.1">opers_closed</a>
 <b>by </b><i><a class="txt" href="unialg_3.html#E5:21_1_1"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="unialg_2.html#D8" title="UNIALG_2:def.8">UNIALG_2:def 8</a></i>;<br/><a NAME="E10:21_1_1"/><b>then </b>
<font color="Maroon" title="c9">H3</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c4">o</font>
 <b>by </b><i><a class="ref" href="unialg_2.html#D5" title="UNIALG_2:def.5">UNIALG_2:def 5</a></i>;<br/><a NAME="E11:21_1_1"/><b>then </b>
<font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">H3</font>
 <b>by </b><i><a class="txt" href="unialg_3.html#E1:21_1"><i><font color="Green" title="E14">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="E12:21_1_1"/>
<font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">a</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:21_1"/>
<font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font>
 <b>by </b><i><a class="txt" href="unialg_3.html#E1:21"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="unialg_3.html#E2:21"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:21"/>
<font color="Maroon" title="c3">S</font> is <a href="unialg_2.html#V1" title="UNIALG_2:attr.1">opers_closed</a>
 <b>by </b><i><a class="ref" href="unialg_2.html#D5" title="UNIALG_2:def.5">UNIALG_2:def 5</a></i>;<br/>


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