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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>2</sub></font>, <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="rearran1.html#NM1">RearrangmentGen</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>









<b>assume </b><a NAME="E1:47"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> is <a href="partfun1.html#V1">total</a> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<a NAME="E2:47"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>, <a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="ref" href="rearran1.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E3:47"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="finset_1.html#V1">finite</a> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E1:47"><i><font color="Green">E35</font></i></a>, <a class="ref" href="rearran1.html#T21" target="_self">Th21</a>, <a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>
<a NAME="E4:47"/><b>then </b>
( <span class="p1">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rfunct_3.html#K23">|</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="rfunct_3.html#K23">|</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>;<br/>
<a NAME="E5:47"/><b>then </b>
 <a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="txt" href="rearran1.html#E3:47"><i><font color="Green">E37</font></i></a>, <a class="ref" href="rfunct_3.html#D14">RFUNCT_3:def 14</a></i>;<br/>
<a NAME="E6:47"/><b>then </b>
 <a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="txt" href="rearran1.html#E2:47"><i><font color="Green">E36</font></i></a>, <a class="ref" href="rfinseq.html#T2">RFINSEQ:2</a></i>;<br/>
<a NAME="E7:47"/><b>then </b><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">c<sub>1</sub></font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="ref" href="rfunct_3.html#T54">RFUNCT_3:54</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="rfunct_3.html#K20">max-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="rfunct_3.html#K20">max-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E8:47"/><i><font color="Green">E39</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="rfunct_3.html#D11">RFUNCT_3:def 11</a></i>;<br/>
<b>thus </b><a NAME="E9:47"/><i><font color="Green">E40</font></i>: 
 <a href="rfunct_3.html#K20">max-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>, <a href="rfunct_3.html#K20">max-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 <b>by </b><i><a class="txt" href="rearran1.html#E7:47"><i><font color="Green">E38</font></i></a>, <a class="ref" href="rfunct_3.html#T45">RFUNCT_3:45</a></i>;<br/>

<a NAME="E10:47"/><i><font color="Green">E41</font></i>: 
( <span class="p1">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="rfunct_3.html#K23">|</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="rfunct_3.html#K20">max-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="rfunct_3.html#K23">|</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="rfunct_3.html#K20">max-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E8:47"><i><font color="Green">E39</font></i></a>, <a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>;<br/>
<a NAME="E11:47"/><b>then </b>
 <a href="rfunct_3.html#K22">FinS</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font>, <a href="rfunct_3.html#K20">max-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="txt" href="rearran1.html#E8:47"><i><font color="Green">E39</font></i></a>, <a class="ref" href="rfunct_3.html#D14">RFUNCT_3:def 14</a></i>;<br/>
<a NAME="E12:47"/><b>then </b>
 <a href="rfunct_3.html#K22">FinS</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font>, <a href="rfunct_3.html#K20">max-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="txt" href="rearran1.html#E9:47"><i><font color="Green">E40</font></i></a>, <a class="ref" href="rfinseq.html#T2">RFINSEQ:2</a></i>;<br/>
<b>hence </b><a NAME="E13:47"/>
 <a href="rfunct_3.html#K22">FinS</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="rfunct_3.html#K22">FinS</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="rearran1.html#E8:47"><i><font color="Green">E39</font></i></a>, <a class="txt" href="rearran1.html#E10:47"><i><font color="Green">E41</font></i></a>, <a class="ref" href="rfunct_3.html#D14">RFUNCT_3:def 14</a></i>;<br/>

<b>hence </b> <a href="rfunct_3.html#K24">Sum</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font> = 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_3.html#D15">RFUNCT_3:def 15</a></i>
<br/>.= 
 <a href="rfunct_3.html#K24">Sum</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="rfunct_3.html#D15">RFUNCT_3:def 15</a></i>
;<br/><br/>


</div>
