<?xml version="1.0"?>
<div class="add">

<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>2</sub></font>,<font color="Maroon">a<sub>1</sub></font>] &amp; ( 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="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">i'</font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="waybel18.html#K4">.</a> <font color="Olive">i'</font></span>)</span> ) );<br/>
<a NAME="E1:9_1"/><i><font color="Green">E22</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/> ex <font color="Olive">x</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">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_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="E1:9_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<b>then reconsider </b><font color="Maroon">i1</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<b>consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="waybel18.html#K4">.</a> <font color="Maroon">i1</font></span>)</span><b> such that </b><br/><a NAME="E4:9_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">i1</font>]
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">x</font>
;<br/>

<b>thus </b><a NAME="E5:9_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">i</font>]
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">i'</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<b>assume </b><a NAME="E6:9_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i'</font>
 ;<br/>

<b>hence </b><a NAME="E7:9_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="waybel18.html#K4">.</a> <font color="Maroon">i'</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:9_1"/><i><font color="Green">E26</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>and </b><br/><a NAME="E4:9_1"/><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">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="zfrefle1.html#S1">ZFREFLE1:sch 1</a>();<br/></i>
<a NAME="E5:9_1"/><i><font color="Green">E30</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E6:9_1"/>
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>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:9_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:9_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="waybel18.html#K4">.</a> <font color="Maroon">x'</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:9_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#T9">YELLOW_6:9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:9_1"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
 
<b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3">product</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <b>by </b><i><a class="ref" href="waybel18.html#D3">WAYBEL18:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<b>thus </b><a NAME="E9:9_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">f</font> <a href="waybel18.html#K5">.</a> <font color="Maroon">i</font>,<font color="Maroon">i</font>]
 <b>by </b><i/>;<br/>


</div>
