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<b>let </b><font color="Maroon">I</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X</font> be    <a href="pboole.html#M1" target="_self">ManySortedSet</a> of <font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">Y</font> be    <a href="pboole.html#M1" target="_self">ManySortedSet</a> of <font color="Maroon">I</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:141"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">X</font> is <a href="relat_1.html#V2">non-empty</a>
 <b>and </b><br/><a NAME="E2:141"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">X</font> <a href="pboole.html#R7" target="_self">[=</a> <font color="Maroon">Y</font>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E3:141"/><i><font color="Green">E26</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/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</font>) <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="pboole.html#T22" target="_self">Th22</a>, <a class="ref" href="pboole.html#T18" target="_self">Th18</a></i>;<br/>
<b>consider </b><font color="Maroon">f</font> being    <a href="pboole.html#M1" target="_self">ManySortedSet</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E5:141"/><i><font color="Green">E27</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/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</font>)
 <b>from </b><i><a class="ref" href="pboole.html#S1" target="_self">PBOOLE:sch 1</a>(<a class="ref" href="pboole.html#T25" target="_self">Th25</a>);<br/></i>
<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D5" target="_self">PBOOLE:def 5</a><br/>

<b>assume </b><a NAME="E6:141"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<b>let </b><font color="Maroon">e</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>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">i</font>,<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">e</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:141"/><i><font color="Green">E29</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 <b>and </b><br/><a NAME="E9:141"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">u</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as    <a href="pboole.html#M1" target="_self">ManySortedSet</a> of <font color="Maroon">I</font> <b>by </b><i><a class="ref" href="pboole.html#T36" target="_self">Th36</a>, </i>;<br/>
<i><font color="Green">E51</font></i>: <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
 <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">i</font>,<font color="Maroon">i</font>,<font color="Maroon">e</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#T35" target="_self">Th35</a>, <a class="ref" href="pboole.html#T37" target="_self">Th37</a></i>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/>
<b>assume </b><a NAME="E12:141"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E13:141"/>
<font color="Maroon">g</font> <a href="pboole.html#R1" target="_self">in</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D4" target="_self">PBOOLE:def 4</a><br/>

<b>assume </b><a NAME="E1:141_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:141_2_1"/>
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> or <font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_2_1_1"><b>suppose </b></a><a NAME="E1:141_2_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:141_2_1_1"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="pboole.html#T38" target="_self">Th38</a>, <a class="ref" href="pboole.html#T39" target="_self">Th39</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_2_1_2"><b>suppose </b></a><a NAME="E1:141_2_1_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> = 
 <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">i</font>,<font color="Maroon">u</font>,<font color="Maroon">e</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#T37" target="_self">Th37</a>, <a class="ref" href="pboole.html#T49" target="_self">Th49</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
<b>by </b><i><a class="ref" href="pboole.html#T50" target="_self">Th50</a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/><b>hence </b><a NAME="E3:141_2_1_2"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="pboole.html#T34" target="_self">Th34</a>, <a class="ref" href="pboole.html#T49" target="_self">Th49</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E14:141"/><b>then </b>
<font color="Maroon">g</font> <a href="pboole.html#R1" target="_self">in</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="pboole.html#T24" target="_self">Th24</a>, <a class="ref" href="pboole.html#T11" target="_self">Th11</a></i>;<br/>
<b>hence </b><a NAME="E15:141"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="pboole.html#T35" target="_self">Th35</a>, <a class="ref" href="pboole.html#T38" target="_self">Th38</a>, </i>;<br/>


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