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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>,<font color="Maroon">a<sub>2</sub></font>];<br/>
<a NAME="E1:25_1"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive">f</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">c</font>,<font color="Olive">f</font>] holds <br/> not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">c</font>,<font color="Olive">f</font>]
 
;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">F<sub>3</sub></font>()<b> such that </b><br/><a NAME="E3:25_1"/><i><font color="Green">E25</font></i>: 
( ( for <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive">e</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">c</font>,<font color="Olive">e</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> iff ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">c</font>,<font color="Olive">e</font>] or <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">c</font>,<font color="Olive">e</font>] ) ) ) &amp; ( for <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive">g</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">c</font>,<font color="Olive">g</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/>( ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">c</font>,<font color="Olive">g</font>] implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">c</font>,<font color="Olive">g</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">c</font>,<font color="Olive">g</font>) ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">c</font>,<font color="Olive">g</font>] implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">c</font>,<font color="Olive">g</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive">c</font>,<font color="Olive">g</font>) ) ) ) )
 <b>from </b><i><a class="ref" href="scheme1.html#S15" target="_self">SCHEME1:sch 15</a>();<br/></i>
<a NAME="E4:25_1"/><i><font color="Green">E28</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:25_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:25_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:25_1_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() <b>by </b><i/>;<br/>
<a NAME="E7:25_1_1"/>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">y</font>,<font color="Maroon">z</font>] or  not <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">y</font>,<font color="Maroon">z</font>] )
 
;<br/>
<b>hence </b><a NAME="E8:25_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:25_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 ;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">F<sub>3</sub></font>() <b>by </b><i>, , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon">g</font>
;<br/>

<b>thus </b><a NAME="E8:25_1"/>
for <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">c</font>,<font color="Olive">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> holds <br/>( ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">c</font>,<font color="Olive">d</font>] implies <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">c</font>,<font color="Olive">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">c</font>,<font color="Olive">d</font>) ) &amp; ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">c</font>,<font color="Olive">d</font>] implies <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">c</font>,<font color="Olive">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive">c</font>,<font color="Olive">d</font>) ) )
 <b>by </b><i>, </i>;<br/>


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