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

<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">F</font> be   <a href="pboole.html#NM2">ManySortedFunction</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">C</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="fraenkel.html#V1">functional</a> <a href="pralg_2.html#V1">with_common_domain</a>  <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:8"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 ;<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>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E2:8"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="pralg_2.html#K1">DOM</a> <font color="Maroon">C</font> )
 ;<br/>

<b>set </b><font color="Maroon">E</font> =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">d'</font></span>)</span></span>)</span> where <font color="Olive">d'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : verum </span>}</span> ;<br/>
<b>reconsider </b><font color="Maroon">F'</font> = <font color="Maroon">F</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<a NAME="E4:8"/><i><font color="Green">E43</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:8"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F'</font> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K1">DOM</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">d'</font></span>)</span></span>)</span> where <font color="Olive">d'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : verum </span>}</span> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<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="E1:8_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F'</font>
 ;<br/>

<b>then consider </b><font color="Maroon">d'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:8_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">d'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp; <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">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">d'</font> = <font color="Maroon">d'</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">Fd</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:8_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">Fd</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d'</font>
 ;<br/>
<a NAME="E7:8_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:8_1"/><b>then </b><i><font color="Green">E68</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">Fd</font> <a href="hidden.html#R1">=</a>  <a href="pralg_2.html#K1">DOM</a> <font color="Maroon">C</font>
 
<b>by </b><i>, , <a class="ref" href="pralg_2.html#D2">PRALG_2:def 2</a></i>;<br/>
<a NAME="E9:8_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">Fd</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">d'</font></span>)</span></span>)</span> where <font color="Olive">d'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : verum </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</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="E1:8_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Fd</font>
 ;<br/>

<a NAME="E2:8_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">Fd</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">d''</font></span>)</span></span>)</span> where <font color="Olive">d''</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : verum </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:8_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">d'</font></span>)</span></span>)</span> where <font color="Olive">d'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : verum </span>}</span> 
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:8_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K1">DOM</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">d'</font></span>)</span></span>)</span> where <font color="Olive">d'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : verum </span>}</span> </span>)</span>
 <b>by </b><i>, , , <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:8"/><b>then </b>
<font color="Maroon">F</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">D</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K1">DOM</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p5">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">d'</font></span>)</span></span>)</span> where <font color="Olive">d'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : verum </span>}</span> </span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>hence </b><a NAME="E7:8"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font>
 <b>by </b><i>, <a class="ref" href="funct_6.html#T86">FUNCT_6:86</a></i>;<br/>


</div>
