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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">R</font> being   <a href="struct_0.html#L1">1-sorted</a>  st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R</font> );<br/>
<a NAME="E1:31_1_1"/><i><font color="Green">E23</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">J</font> holds <br/> ex <font color="Olive">j</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">j</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_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="E1:31_1_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">J</font> = <font color="Maroon">J</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">A</font> as  <a href="pralg_1.html#V2" target="_self">1-sorted-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">J</font> ;<br/>
<b>reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> <b>by </b><i><a class="ref" href="pralg_1.html#T1" target="_self">Th1</a></i>;<br/>
<b>take </b><font color="Maroon">j</font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="pralg_1.html#K10" target="_self">.</a> <font color="Maroon">i'</font></span>)</span>;<br/>

<b>take </b><font color="Maroon">R</font> = <font color="Maroon">B</font> <a href="pralg_1.html#K10" target="_self">.</a> <font color="Maroon">i'</font>;<br/>

<b>thus </b><a NAME="E5:31_1_1_1"/>
( <font color="Maroon">R</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; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">M</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:31_1_1"/><i><font color="Green">E38</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font>
 <b>and </b><br/><a NAME="E4:31_1_1"/><i><font color="Green">E39</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">J</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Maroon">M</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">M</font> = <font color="Maroon">M</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">J</font> <b>by </b><i><a class="ref" href="pralg_1.html#T1" target="_self">Th1</a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon">M</font>
;<br/>

<b>thus </b><a NAME="E6:31_1_1"/>
for <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> holds <br/> ex <font color="Olive">R</font> being   <a href="struct_0.html#L1">1-sorted</a>  st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> &amp; <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R</font> )
 <b>by </b><i/>;<br/>


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