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<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">SD</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">D</font>,<font color="Maroon">D</font>;<br/>





<b>thus </b><a NAME="E1:7"/>
( <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="partfun2.html#K1" target="_self">id</a> <font color="Maroon">SD</font> implies (  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">SD</font> &amp; ( for <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>  st <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SD</font> holds <br/><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">d</font> <a href="hidden.html#R1">=</a> <font color="Olive">d</font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="partfun2.html#K1" target="_self">id</a> <font color="Maroon">SD</font>
 ;<br/>

<b>hence </b><a NAME="E2:7_1"/><i><font color="Green">E30</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">SD</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>;<br/>

<b>let </b><font color="Maroon">d</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/>

<b>assume </b><a NAME="E3:7_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SD</font>
 ;<br/>

<a NAME="E4:7_1"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>
<b>hence </b><a NAME="E5:7_1"/>
<font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:7"/><i><font color="Green">E34</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">SD</font> &amp; ( for <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>  st <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SD</font> holds <br/><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">d</font> <a href="hidden.html#R1">=</a> <font color="Olive">d</font> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2"><b>now </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="E1:7_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SD</font>
 ;<br/><b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> ;<br/><a NAME="E3:7_2"/>
<font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:7_2"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:7"/>
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="partfun2.html#K1" target="_self">id</a> <font color="Maroon">SD</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T34">FUNCT_1:34</a></i>;<br/>


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