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<b>reconsider </b><font color="Maroon">f</font> =  <a href="lopclset.html#K8" target="_self">UFilter</a> <font color="Maroon">BL</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<a NAME="E2:23_1_1"/><i><font color="Green">E25</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">BL</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:23_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:23_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:23_1_1_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:23_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">F</font> where <font color="Olive">F</font> is    <a href="filter_0.html#M1">Filter</a> of <font color="Maroon">BL</font> : ( <font color="Olive">F</font> <a href="filter_0.html#NR1" target="_self">is_ultrafilter</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> ) </span>}</span> 
 
<b>by </b><i>, <a class="txt" href="lopclset.html#E2:23_1_1"><i><font color="Green">E25</font></i></a>, </i>;<br/>
<a NAME="E5:23_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a>  <a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font>
 
<b>by </b><i>, <a class="txt" href="lopclset.html#E2:23_1_1"><i><font color="Green">E25</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E6:23_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">BL</font>, <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span> <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E5:23_1_1"/>
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">BL</font> holds  <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">F</font> where <font color="Olive">F</font> is    <a href="filter_0.html#M1">Filter</a> of <font color="Maroon">BL</font> : ( <font color="Olive">F</font> <a href="filter_0.html#NR1" target="_self">is_ultrafilter</a>  &amp; <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:23_1_1"/>
 <a href="lopclset.html#K8" target="_self">UFilter</a> <font color="Maroon">BL</font> is   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">BL</font>, <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">BL</font></span>)</span>
 ;<br/>


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