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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>thus </b><a NAME="E1:6"/>
( <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="rfinseq.html#R1" target="_self">are_fiberwise_equipotent</a>  implies for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="relat_1.html#K10">"</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K10">"</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_1"/>
<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="rfinseq.html#R1" target="_self">are_fiberwise_equipotent</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:6_1"/><i><font color="Green">E6</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="rfinseq.html#T3" target="_self">Th3</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>
<a NAME="E4:6_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="rfinseq.html#E3:6_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="funct_1.html#T84">FUNCT_1:84</a></i>;<br/>
<a NAME="E5:6_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T167">RELAT_1:167</a></i>;<br/>
<a NAME="E6:6_1"/><b>then </b><i><font color="Green">E8</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="rfinseq.html#E3:6_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E7:6_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:6_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>4</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></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="E1:6_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6_1_1_1"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:6_1_1_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="rfinseq.html#E6:6_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="rfinseq.html#E3:6_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E5:6_1_1_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="rfinseq.html#E3:6_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E6:6_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E7:6_1_1_1"/><b>then </b><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="rfinseq.html#E3:6_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:6_1_1_1"/><b>then </b><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="rfinseq.html#E3:6_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="rfinseq.html#E5:6_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E9:6_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="rfinseq.html#E3:6_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="rfinseq.html#E4:6_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="rfinseq.html#E5:6_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="rfinseq.html#E7:6_1_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<b>hence </b><a NAME="E10:6_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="rfinseq.html#E8:6_1_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>6</sub></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="E2:6_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E3:6_1_1"/><b>then </b><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:6_1_1"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="rfinseq.html#E3:6_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:6_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="rfinseq.html#E3:6_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="rfinseq.html#E3:6_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="rfinseq.html#E5:6_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<a NAME="E7:6_1_1"/><b>then </b><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rfinseq.html#E3:6_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="rfinseq.html#E6:6_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="rfinseq.html#E5:6_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E8:6_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E9:6_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rfinseq.html#E5:6_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="rfinseq.html#E7:6_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:6_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="txt" href="rfinseq.html#E4:6_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="rfinseq.html#E6:6_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/>
<b>hence </b><a NAME="E9:6_1"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/>


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

<b>assume </b><a NAME="E2:6"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="relat_1.html#K10">"</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K10">"</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 ;<br/>

<b>hence </b><a NAME="E3:6"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="rfinseq.html#D1" target="_self">RFINSEQ:def 1</a><br/>


</div>
