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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">j</font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">I</font> <a href="finseq_2.html#K3">*</a>  st <br/>( <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">j</font></span>)</span> );<br/>
<a NAME="E1:157_1_1"/><i><font color="Green">E24</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> <a href="finseq_2.html#K3">*</a>  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="157_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="E1:157_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> <a href="finseq_2.html#K3">*</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">j</font> = <font color="Maroon">i</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">I</font> <a href="finseq_2.html#K3">*</a>  ;<br/>
<b>reconsider </b><font color="Maroon">e</font> =  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">j</font></span>)</span> as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon">e</font>
;<br/>

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

<b>thus </b><a NAME="E4:157_1_1_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">j</font></span>)</span> )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being    <a href="pboole.html#M1" target="_self">ManySortedSet</a> of <font color="Maroon">I</font> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E3:157_1_1"/><i><font color="Green">E25</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> <a href="finseq_2.html#K3">*</a>  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" target="_self">PBOOLE:sch 3</a>(<a class="ref" href="pboole.html#T34" target="_self">Th34</a>);<br/></i>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>let </b><font color="Maroon">i</font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">I</font> <a href="finseq_2.html#K3">*</a> ;<br/>

<a NAME="E4:157_1_1"/>
 ex <font color="Olive">j</font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">I</font> <a href="finseq_2.html#K3">*</a>  st <br/>( <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">j</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="pboole.html#T35" target="_self">Th35</a></i>;<br/>
<b>hence </b><a NAME="E5:157_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">i</font></span>)</span>
 ;<br/>


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