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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> = <font color="Maroon">R</font> <a href="waybel_4.html#K5">-below</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:17_1_1"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="lattice7.html#K4">LOWER</a> <font color="Maroon">L</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>reconsider </b><font color="Maroon">I</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>) as  <a href="waybel_0.html#V12">lower</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E2:17_1_1_1"/>
 <a href="lattice7.html#K4">LOWER</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : <font color="Olive">X</font> is <a href="waybel_0.html#V12">lower</a> </span>}</span> 
 
<b>by </b><i><a class="ref" href="lattice7.html#D7">LATTICE7:def 7</a></i>;<br/>
<a NAME="E3:17_1_1_1"/><b>then </b>
<font color="Maroon">I</font> <a href="hidden.html#R2">in</a>  <a href="lattice7.html#K4">LOWER</a> <font color="Maroon">L</font>
 
;<br/>
<b>hence </b><a NAME="E4:17_1_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>) is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="lattice7.html#K4">LOWER</a> <font color="Maroon">L</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="lattice7.html#K4">LOWER</a> <font color="Maroon">L</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:17_1_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S9">FUNCT_2:sch 9</a>();<br/></i>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>thus </b><a NAME="E4:17_1_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="waybel_4.html#K5">-below</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


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