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<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">S</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/>



<b>thus </b><a NAME="E1:27"/>
( <font color="Maroon">S</font> is <a href="waybel23.html#V3" target="_self">infs-closed</a> implies for <font color="Olive">X</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">S</font>  st  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Olive">X</font>,<font color="Maroon">L</font> holds <br/> <a href="yellow_0.html#K2">"/\"</a> <font color="Olive">X</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:27_1"/>
<font color="Maroon">S</font> is <a href="waybel23.html#V3" target="_self">infs-closed</a>
 ;<br/>

<a NAME="E2:27_1"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">S</font> is <a href="yellow_0.html#V7">infs-inheriting</a>
 
<b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">X</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/>

<a NAME="E3:27_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<b>assume </b><a NAME="E4:27_1"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>
 ;<br/>

<a NAME="E5:27_1"/><b>then </b>
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="yellow_0.html#D18">YELLOW_0:def 18</a></i>;<br/>
<b>hence </b><a NAME="E6:27_1"/>
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>


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

<b>assume </b><a NAME="E2:27"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">X</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">S</font>  st  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Olive">X</font>,<font color="Maroon">L</font> holds <br/> <a href="yellow_0.html#K2">"/\"</a> <font color="Olive">X</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">S</font></span>)</span>;<br/><a NAME="E1:27_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">X</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><b>assume </b><a NAME="E2:27_2"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>
 ;<br/><a NAME="E3:27_2"/><b>then </b>
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:27_2"/>
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">S</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:27"/><b>then </b>
 <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">S</font> is <a href="yellow_0.html#V7">infs-inheriting</a>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D18">YELLOW_0:def 18</a></i>;<br/>
<b>hence </b><a NAME="E5:27"/>
<font color="Maroon">S</font> is <a href="waybel23.html#V3" target="_self">infs-closed</a>
 <b>by </b><i/>;<br/>


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