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<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>  st <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E1:2_1_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/> ex <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Olive">u</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:2_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <b>by </b><i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<b>take </b>
<font color="Maroon">f</font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
;<br/>

<b>thus </b><a NAME="E3:2_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">f</font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">H</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:2_1_1"/><i><font color="Green">E33</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp; ( for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>] ) )
 <b>from </b><i><a class="ref" href="zfrefle1.html#S1">ZFREFLE1:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon">H</font> = <font color="Maroon">H</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:2_1_1"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>,<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:2_1_1_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <b>by </b><i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<a NAME="E3:2_1_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:2_1_1_2"/>
<font color="Maroon">C</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E5:2_1_1_2"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1_1_2_1"/>
( <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:2_1_1_2_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>set </b><font color="Maroon">h</font> = <font color="Maroon">f</font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>;<br/><a NAME="E2:2_1_1_2_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><a NAME="E3:2_1_1_2_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><a NAME="E4:2_1_1_2_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1_1_2_1_1_1_1"/>
( <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:2_1_1_2_1_1_1_1_1"/>
<font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_1_1_2_1_1_1_1_1"/>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T55">FUNCT_2:55</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:2_1_1_2_1_1_1_1_2"/>
<font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_1_1_2_1_1_1_1_2"/>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T25">RELSET_1:25</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:2_1_1_2_1_1"/>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:2_1_1_2_1_2"/><i><font color="Green">E49</font></i>: 
not <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:2_1_1_2_1_2"/><b>then </b><i><font color="Green">E50</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><i><font color="Green">E51</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><a NAME="E4:2_1_1_2_1_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><b>hence </b><a NAME="E5:2_1_1_2_1_2"/>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <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/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">H</font> = <font color="Maroon">H</font> as    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">C</font>,<font color="Maroon">B</font> <b>by </b><i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<b>take </b>
<font color="Maroon">H</font>
;<br/>

<b>thus </b><a NAME="E7:2_1_1"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/>for <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>,<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>  st <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> holds <br/><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span>
 <b>by </b><i/>;<br/>


</div>
