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<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> ex <font color="Olive">f1</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> st <br/>( <font color="Olive">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">f1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">x</font>,<font color="Maroon">a<sub>1</sub></font>] ) );<br/>
<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1"><b>now </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"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_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_1"/>
(  not <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is <a href="xboole_0.html#V1">empty</a> or <font color="Maroon">F<sub>3</sub></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_1_1"><b>suppose </b></a><a NAME="E1:2_1_1_1_1"/>
not <font color="Maroon">F<sub>3</sub></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>then reconsider </b><font color="Maroon">bb</font> = <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">i</font>];<br/><a NAME="E3:2_1_1_1_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> holds <br/> ex <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">bb</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">e</font>,<font color="Olive">u</font>] )
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">ff</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">bb</font><b> such that </b><br/><a NAME="E5:2_1_1_1_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">e</font>,<font color="Maroon">ff</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>();<br/></i><b>reconsider </b><font color="Maroon">fff</font> = <font color="Maroon">ff</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> ;<br/><b>reconsider </b><font color="Maroon">j</font> = <font color="Maroon">fff</font> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">j</font> = <font color="Maroon">j</font>;<br/><b>thus </b><a NAME="E8:2_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">j</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1_2"><b>suppose </b></a><a NAME="E1:2_1_1_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">F<sub>3</sub></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"><div><i><font color="Green">E52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:2_1_1_1_2_1"/>
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> or  not <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Maroon">x</font>,<font color="Maroon">i</font>] )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E2:2_1_1_1_2_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><a NAME="E3:2_1_1_1_2"/><b>then </b>
<font color="Maroon">F<sub>2</sub></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="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">fff</font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">F<sub>3</sub></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/><b>reconsider </b><font color="Maroon">j</font> = <font color="Maroon">fff</font> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">j</font> = <font color="Maroon">j</font>;<br/><b>thus </b><a NAME="E6:2_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">j</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">fff</font>
;<br/>

<b>thus </b><a NAME="E1:2_1_1_1_2_2"/>
<font color="Maroon">fff</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 ;<br/>

<b>thus </b><a NAME="E2:2_1_1_1_2_2"/>
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">fff</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font>,<font color="Olive">e</font>,<font color="Maroon">i</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>consider </b><font color="Maroon">F</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E3:2_1"/><i><font color="Green">E53</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">F<sub>1</sub></font>() holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3">PBOOLE:sch 3</a>();<br/></i>
<a NAME="E4:2_1"/>
<font color="Maroon">F</font> is    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a><br/>

<b>assume </b><a NAME="E1:2_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<b>then consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font><b> such that </b><br/><a NAME="E3:2_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E4:2_1_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">x</font>,<font color="Maroon">i</font>]
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E5:2_1_2"/>
<font color="Maroon">F</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">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>() ;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>thus </b><a NAME="E6:2_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">F<sub>1</sub></font>() holds <br/> ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>,<font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> st <br/>( <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> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">x</font>,<font color="Olive">i</font>] ) )
 <b>by </b><i/>;<br/>


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