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

<b>let </b><font color="Maroon">n1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:61_1_2"/><i><font color="Green">E203</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">X</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">P</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Olive">X</font>,<font color="Maroon">k</font>,<font color="Olive">x</font>,<font color="Olive">y</font></span>)</span>, <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Olive">X</font>,<font color="Maroon">k</font>,<font color="Olive">x</font>,<font color="Olive">y</font>  st  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Olive">X</font> &amp; <font color="Olive">P</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> holds <br/> ex <font color="Olive">XFS</font> being  <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>(  <a href="afinsq_1.html#K2">dom</a> <font color="Olive">XFS</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Olive">P</font> &amp; ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Olive">XFS</font> &amp; <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> holds <br/><font color="Olive">XFS</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K2" target="_self">Intersection</a> <font color="Maroon">F</font>,<font color="Olive">f</font>,<font color="Olive">x</font></span>)</span> ) &amp; <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a>  <a href="stirl2_1.html#K7">Sum</a> <font color="Olive">XFS</font> )
 <b>and </b><br/><a NAME="E2:61_1_2"/><i><font color="Green">E204</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">X</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">P</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Olive">X</font>,<font color="Maroon">k</font>,<font color="Olive">x</font>,<font color="Olive">y</font></span>)</span>, <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Olive">X</font>,<font color="Maroon">k</font>,<font color="Olive">x</font>,<font color="Olive">y</font>  st  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Olive">X</font> &amp; <font color="Olive">P</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> holds <br/> ex <font color="Olive">XFS</font> being  <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>(  <a href="afinsq_1.html#K2">dom</a> <font color="Olive">XFS</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Olive">P</font> &amp; ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Olive">XFS</font> &amp; <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> holds <br/><font color="Olive">XFS</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K2" target="_self">Intersection</a> <font color="Maroon">F</font>,<font color="Olive">f</font>,<font color="Olive">x</font></span>)</span> ) &amp; <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a>  <a href="stirl2_1.html#K7">Sum</a> <font color="Olive">XFS</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">D</font> =  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="card_fin.html#T2" target="_self">Th2</a></i>;<br/>
<b>set </b><font color="Maroon">Ch1</font> =  <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>,0,1;<br/>
<a NAME="E4:61_1_2"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>,0,1</span>)</span>, <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>,0,1 <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">P</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:61_1_2"/><i><font color="Green">E205</font></i>: 
( <font color="Maroon">P</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>,0,1</span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>,0,1 )
 <b>by </b><i><a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>,0,1</span>)</span>, <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>,0,1 <b>by </b><i>, <a class="ref" href="funct_2.html#T3">FUNCT_2:3</a></i>;<br/>
<b>consider </b><font color="Maroon">XFS1</font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:61_1_2"/><i><font color="Green">E206</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS1</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E10:61_1_2"/><i><font color="Green">E207</font></i>: 
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS1</font> &amp; <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> holds <br/><font color="Maroon">XFS1</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K2" target="_self">Intersection</a> <font color="Maroon">F</font>,<font color="Olive">f</font>,0</span>)</span>
 <b>and </b><br/><a NAME="E11:61_1_2"/><i><font color="Green">E208</font></i>: 
<font color="Maroon">n1</font> <a href="hidden.html#R1">=</a>  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">XFS1</font>
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">XFS2</font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E13:61_1_2"/><i><font color="Green">E210</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS2</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E14:61_1_2"/><i><font color="Green">E211</font></i>: 
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS2</font> &amp; <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> holds <br/><font color="Maroon">XFS2</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K2" target="_self">Intersection</a> <font color="Maroon">F</font>,<font color="Olive">f</font>,0</span>)</span>
 <b>and </b><br/><a NAME="E15:61_1_2"/><i><font color="Green">E212</font></i>: 
<font color="Maroon">n2</font> <a href="hidden.html#R1">=</a>  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">XFS2</font>
 <b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:61_1_2_1"/><i><font color="Green">E213</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS1</font>
 ;<br/><a NAME="E2:61_1_2_1"/>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then consider </b><font color="Maroon">Pz</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">D</font>,<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span><b> such that </b><br/><a NAME="E4:61_1_2_1"/><i><font color="Green">E215</font></i>: 
( <font color="Maroon">Pz</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">Pz</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 <b>by </b><i/>;<br/><a NAME="E5:61_1_2_1"/>
( <font color="Maroon">XFS2</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K2" target="_self">Intersection</a> <font color="Maroon">F</font>,<font color="Maroon">Pz</font>,0</span>)</span> &amp; <font color="Maroon">XFS1</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K2" target="_self">Intersection</a> <font color="Maroon">F</font>,<font color="Maroon">Pz</font>,0</span>)</span> )
 <b>by </b><i>, , , , , </i>;<br/><b>hence </b><a NAME="E6:61_1_2_1"/>
<font color="Maroon">XFS2</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XFS1</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">z</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E17:61_1_2"/>
<font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font>
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div>
