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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>( ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>] implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">a<sub>1</sub></font>) ) &amp; ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>] implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">a<sub>1</sub></font>) ) );<br/>
<div><i><font color="Green">E29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_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="E1:10_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/><b>thus </b><a NAME="E2:10_1_1"/>
 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="Maroon">i</font>,<font color="Olive">j</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_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="E1:10_1_1_1_1"/>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">i</font>] or  not <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">i</font>] )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:10_1_1_1_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">i</font>]
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">i</font>)
;<br/><b>thus </b><a NAME="E2:10_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">i</font>)]
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:10_1_1_1_1_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">i</font>]
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">i</font>)
;<br/><b>thus </b><a NAME="E2:10_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">F<sub>3</sub></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="E3:10_1"/><i><font color="Green">E40</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>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E4:10_1"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  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">P<sub>1</sub></font>[<font color="Olive">i</font>] implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">i</font>) ) &amp; ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">i</font>] implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">i</font>) ) )
 <b>by </b><i/>;<br/>


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