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<a NAME="E1:1_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>() holds <br/> ex <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">e</font>,<font color="Olive">u</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>()
 ;<br/>

<b>then reconsider </b><font color="Maroon">e1</font> = <font color="Maroon">e</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() ;<br/>
<b>reconsider </b><font color="Maroon">e1</font> = <font color="Maroon">e1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">u</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E5:1_1_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">e1</font>,<font color="Maroon">u</font>] )
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">u1</font> = <font color="Maroon">u</font> as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon">u1</font>
;<br/>

<b>thus </b><a NAME="E7:1_1_1"/>
( <font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">e</font>,<font color="Maroon">u1</font>] )
 <b>by </b><i/>;<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="E3:1_1"/><i><font color="Green">E52</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>2</sub></font>() &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>3</sub></font>() )
 <b>and </b><br/><a NAME="E4:1_1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>() holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">e</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">F<sub>2</sub></font>(),<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>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E6:1_1"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">e</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>() holds <br/> ex <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Olive">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">e</font>,<font color="Olive">u</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<b>assume </b><a NAME="E1:1_1_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>()
 ;<br/>

<b>then reconsider </b><font color="Maroon">e1</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">F<sub>3</sub></font>() <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">fe</font> = <font color="Maroon">e1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<b>take </b>
<font color="Maroon">fe</font>
;<br/>

<b>thus </b><a NAME="E4:1_1_2"/>
( <font color="Maroon">fe</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Maroon">fe</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">e</font>,<font color="Maroon">fe</font>] )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E7:1_1"/>
for <font color="Olive">e</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>() holds <br/> ex <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Olive">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">e</font>,<font color="Olive">u</font>] )
 <b>by </b><i/>;<br/>


</div>
