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

<b>let </b><font color="Maroon">F</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">G</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>assume </b><a NAME="E1:8"/><i><font color="Green">E31</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> <font color="Maroon">G</font>
 ;<br/>

<b>thus </b><a NAME="E2:8"/>
( <font color="Maroon">F</font>,<font color="Maroon">G</font> <a href="rfinseq.html#R1" target="_self">are_fiberwise_equipotent</a>  implies  ex <font color="Olive">P</font> being  <a href="funct_2.html#NM2">Permutation</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> st <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">P</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_1"/>
<font color="Maroon">F</font>,<font color="Maroon">G</font> <a href="rfinseq.html#R1" target="_self">are_fiberwise_equipotent</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">I</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:8_1"/><i><font color="Green">E32</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">I</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">I</font> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">I</font> = <font color="Maroon">I</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>, <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <b>by </b><i>, , <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">I</font> = <font color="Maroon">I</font> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <b>by </b><i>, , <a class="ref" href="funct_2.html#T83">FUNCT_2:83</a></i>;<br/>
<b>take </b>
<font color="Maroon">I</font>
;<br/>

<b>thus </b><a NAME="E6:8_1"/>
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">I</font>
 <b>by </b><i/>;<br/>


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

<b>given </b><font color="Maroon">P</font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font><b> such that </b><a NAME="E3:8"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">P</font>
 ;<br/>

<a NAME="E4:8"/><i><font color="Green">E35</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">P</font> is <a href="funct_1.html#V2">one-to-one</a> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E5:8"/>
( <font color="Maroon">P</font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>, <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
;<br/>
<a NAME="E6:8"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:8"/>
<font color="Maroon">F</font>,<font color="Maroon">G</font> <a href="rfinseq.html#R1" target="_self">are_fiberwise_equipotent</a> 
 <b>by </b><i>, , , </i>;<br/>


</div>
