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

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">Fy</font> be  <a href="finset_1.html#V2">finite-yielding</a> <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">X</font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">n</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="E1:74"/><i><font color="Green">E47</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Fy</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Fy</font> )
 <b>and </b><br/><a NAME="E2:74"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>set </b><font color="Maroon">Xx</font> = <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>;<br/>
<b>set </b><font color="Maroon">X0</font> = <span class="p1">{<span class="default"> <font color="Olive">f</font> where <font color="Olive">f</font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,0</span><a href="domain_1.html#K7">}</a></span> : (  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">X1</font> = <span class="p1">{<span class="default"> <font color="Olive">f</font> where <font color="Olive">f</font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,0</span><a href="domain_1.html#K7">}</a></span> : (  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span> ;<br/>
<a NAME="E3:74"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/>
<a NAME="E4:74"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive">f</font> where <font color="Olive">f</font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,0</span><a href="domain_1.html#K7">}</a></span> : (  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">f</font> where <font color="Olive">f</font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,0</span><a href="domain_1.html#K7">}</a></span> : (  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="card_fin.html#K1" target="_self">Choose</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,1,0
 
<b>by </b><i/>;<br/>
<a NAME="E5:74"/><b>then </b>
( <span class="p1">{<span class="default"> <font color="Olive">f</font> where <font color="Olive">f</font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,0</span><a href="domain_1.html#K7">}</a></span> : (  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="card_fin.html#K1" target="_self">Choose</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,1,0 &amp; <span class="p1">{<span class="default"> <font color="Olive">f</font> where <font color="Olive">f</font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,0</span><a href="domain_1.html#K7">}</a></span> : (  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="card_fin.html#K1" target="_self">Choose</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,1,0 )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">X0</font> = <span class="p1">{<span class="default"> <font color="Olive">f</font> where <font color="Olive">f</font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,0</span><a href="domain_1.html#K7">}</a></span> : (  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 ) </span>}</span> , <font color="Maroon">X1</font> = <span class="p1">{<span class="default"> <font color="Olive">f</font> where <font color="Olive">f</font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,0</span><a href="domain_1.html#K7">}</a></span> : (  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<b>consider </b><font color="Maroon">P0</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="card_1.html#K4">card</a> <font color="Maroon">X0</font>,<font color="Maroon">X0</font><b> such that </b><br/><a NAME="E8:74"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">P0</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="card_fin.html#E1:74"><i><font color="Green">E47</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X0</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> );<br/>
<a NAME="E9:74"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">x0</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x0</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X0</font> holds <br/> ex <font color="Olive">P0x0</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">P0x0</font> <a href="hidden.html#R2">in</a>  <a href="card_fin.html#K1" target="_self">Choose</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,1,0 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x0</font>,<font color="Olive">P0x0</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x0</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:74_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X0</font>
 ;<br/>

<a NAME="E2:74_1"/>
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P0</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="card_1.html#T47">CARD_1:47</a></i>;<br/>
<a NAME="E3:74_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">P0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">P0</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:74_1"/><b>then </b>
<font color="Maroon">P0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X0</font>
 
;<br/>
<b>then consider </b><font color="Maroon">P0x0</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,0</span><a href="domain_1.html#K7">}</a></span><b> such that </b><br/><a NAME="E6:74_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">P0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P0x0</font> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">P0x0</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">P0x0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>
<a NAME="E7:74_1"/>
<font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E8:74_1"/><b>then </b><i><font color="Green">E60</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P0x0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">P0x0</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">P0x0</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relat_1.html#T99">RELAT_1:99</a></i>;<br/>
<a NAME="E9:74_1"/><b>then </b>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">P0x0</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">P0x0</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,0</span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Px</font> = <font color="Maroon">P0x0</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,0</span><a href="domain_1.html#K7">}</a></span> <b>by </b><i><a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E11:74_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/>
<a NAME="E12:74_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E13:74_1"/><b>then </b>
<font color="Maroon">P0x0</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Px</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="card_fin.html#T3" target="_self">Th3</a>, <a class="ref" href="card_fin.html#T4" target="_self">Th4</a>, </i>;<br/>
<a NAME="E14:74_1"/><b>then </b>
<font color="Maroon">Px</font> <a href="hidden.html#R2">in</a>  <a href="card_fin.html#K1" target="_self">Choose</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,1,0
 
<b>by </b><i><a class="ref" href="card_fin.html#T3" target="_self">Th3</a>, </i>;<br/>
<b>hence </b><a NAME="E15:74_1"/>
 ex <font color="Olive">P0x0</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">P0x0</font> <a href="hidden.html#R2">in</a>  <a href="card_fin.html#K1" target="_self">Choose</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,1,0 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x0</font>,<font color="Olive">P0x0</font>] )
 <b>by </b><i>, <a class="ref" href="card_fin.html#T3" target="_self">Th3</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">P0x</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="card_1.html#K4">card</a> <font color="Maroon">X0</font>, <a href="card_fin.html#K1" target="_self">Choose</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,1,0<b> such that </b><br/><a NAME="E11:74"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">x1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X0</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x1</font>,<font color="Maroon">P0x</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x1</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>(<a class="txt" href="card_fin.html#E13:74"><i><font color="Green">E62</font></i></a>);<br/></i>
<a NAME="E12:74"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/>
<a NAME="E13:74"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,1,0</span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X0</font>
 
<b>by </b><i/>;<br/>
<a NAME="E14:74"/>
for <font color="Olive">x1</font>, <font color="Olive">x2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P0x</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P0x</font> &amp; <font color="Maroon">P0x</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P0x</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x2</font> holds <br/><font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Olive">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:74_2"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P0x</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P0x</font> &amp; <font color="Maroon">P0x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P0x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> )
 ;<br/>

<a NAME="E2:74_2"/>
( (  <a href="card_fin.html#K1" target="_self">Choose</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,1,0 <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,1,0</span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) &amp; ( <font color="Maroon">X0</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="card_1.html#K4">card</a> <font color="Maroon">X0</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 
<b>by </b><i><a class="ref" href="card_1.html#T47">CARD_1:47</a></i>;<br/>
<a NAME="E3:74_2"/><b>then </b><i><font color="Green">E64</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P0x</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X0</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P0</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X0</font> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E3:74_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">f1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:74_2"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">f1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X0</font> &amp; <font color="Maroon">P0x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> )
 <b>by </b><i><a class="txt" href="card_fin.html#E1:74_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="card_fin.html#T3" target="_self">Th3</a></i>;<br/>
<b>consider </b><font color="Maroon">f2</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:74_2"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">f2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X0</font> &amp; <font color="Maroon">P0x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> )
 <b>by </b><i><a class="ref" href="card_fin.html#T3" target="_self">Th3</a>, <a class="txt" href="card_fin.html#E1:74_2"><i><font color="Green">E63</font></i></a></i>;<br/>
<a NAME="E8:74_2"/><i><font color="Green">E71</font></i>: 
(  ex <font color="Olive">F</font> being  <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,0</span><a href="domain_1.html#K7">}</a></span> st <br/>( <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Olive">F</font> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Olive">F</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Olive">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 ) &amp;  ex <font color="Olive">F</font> being  <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,0</span><a href="domain_1.html#K7">}</a></span> st <br/>( <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Olive">F</font> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Olive">F</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Olive">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 ) )
 
<b>by </b><i>, <a class="txt" href="card_fin.html#E5:74_2"><i><font color="Green">E68</font></i></a></i>;<br/>
<a NAME="E9:74_2"/><b>then </b><i><font color="Green">E73</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E10:74_2"/>
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> holds <br/><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_2_1"><b>proof </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="E1:74_2_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:74_2_1_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="card_fin.html#E7:74_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="card_fin.html#E8:74_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:74_2_1_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:74_2_1_1_1_1"/><b>then </b>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> )
 <b>by </b><i><a class="txt" href="card_fin.html#E7:74_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="card_fin.html#E8:74_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:74_2_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T71">FUNCT_1:71</a></i>;<br/><b>hence </b><a NAME="E4:74_2_1_1_1_1"/>
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="card_fin.html#T3" target="_self">Th3</a>, , <a class="txt" href="card_fin.html#E5:74_2"><i><font color="Green">E68</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:74_2_1_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:74_2_1_1_1_2"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:74_2_1_1_1_2"/>
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:74_2_1"/>
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E11:74_2"/><b>then </b>
( <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P0</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P0</font> )
 
<b>by </b><i><a class="ref" href="card_fin.html#T3" target="_self">Th3</a>, <a class="ref" href="card_fin.html#T4" target="_self">Th4</a>, <a class="txt" href="card_fin.html#E7:74_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E12:74_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i><a class="txt" href="card_fin.html#E11:74"><i><font color="Green">E61</font></i></a>, , <a class="txt" href="card_fin.html#E5:74_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:74"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">P0x</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E16:74"/>
<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E17:74"/><b>then </b><i><font color="Green">E77</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">Fy</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<b>then consider </b><font color="Maroon">XFS0</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="E19:74"/><i><font color="Green">E79</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS0</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P0x</font>
 <b>and </b><br/><a NAME="E20:74"/><i><font color="Green">E80</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">XFS0</font> &amp; <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P0x</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> holds <br/><font color="Maroon">XFS0</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> <span class="p2">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">Fy</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<font color="Olive">f</font>,1</span>)</span>
 <b>and </b><br/><a NAME="E21:74"/><i><font color="Green">E81</font></i>: 
 <a href="card_fin.html#K5" target="_self">Card_Intersection</a> <span class="p1">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">Fy</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">XFS0</font>
 <b>by </b><i>, <a class="ref" href="card_fin.html#T3" target="_self">Th3</a>, <a class="ref" href="card_fin.html#T2" target="_self">Th2</a></i>;<br/>
<b>consider </b><font color="Maroon">P1</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="card_1.html#K4">card</a> <font color="Maroon">X1</font>,<font color="Maroon">X1</font><b> such that </b><br/><a NAME="E23:74"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">P1</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="card_fin.html#E1:74"><i><font color="Green">E47</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> );<br/>
<a NAME="E24:74"/><i><font color="Green">E86</font></i>: 
for <font color="Olive">x1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X1</font> holds <br/> ex <font color="Olive">P1x1</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">P1x1</font> <a href="hidden.html#R2">in</a>  <a href="card_fin.html#K1" target="_self">Choose</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<font color="Maroon">k</font>,1,0 &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x1</font>,<font color="Olive">P1x1</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:74_3"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X1</font>
 ;<br/>

<a NAME="E2:74_3"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P1</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="card_1.html#T47">CARD_1:47</a></i>;<br/>
<a NAME="E3:74_3"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">P1</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:74_3"/><b>then </b>
<font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 
;<br/>
<b>then consider </b><font color="Maroon">P1x1</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,0</span><a href="domain_1.html#K7">}</a></span><b> such that </b><br/><a NAME="E6:74_3"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1x1</font> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">P1x1</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">P1x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/>
<a NAME="E7:74_3"/>
<font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E8:74_3"/><b>then </b><i><font color="Green">E92</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P1x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">P1x1</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">P1x1</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relat_1.html#T99">RELAT_1:99</a></i>;<br/>
<a NAME="E9:74_3"/><b>then </b>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">P1x1</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">P1x1</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,0</span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Px</font> = <font color="Maroon">P1x1</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,0</span><a href="domain_1.html#K7">}</a></span> <b>by </b><i><a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E11:74_3"/>
( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P1x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E12:74_3"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P1x1</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E13:74_3"/><b>then </b><i><font color="Green">E183</font></i>: 
<font color="Maroon">P1x1</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Px</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="txt" href="card_fin.html#E15:74"><i><font color="Green">E75</font></i></a>, </i>;<br/>
<a NAME="E14:74_3"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Px</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/>
<a NAME="E15:74_3"/><b>then </b>
( <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">Px</font> <a href="relset_1.html#K11">"</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 
<b>by </b><i>, , <a class="ref" href="card_2.html#T54">CARD_2:54</a></i>;<br/>
<a NAME="E16:74_3"/><b>then </b>
<font color="Maroon">Px</font> <a href="hidden.html#R2">in</a>  <a href="card_fin.html#K1" target="_self">Choose</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<font color="Maroon">k</font>,1,0
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E17:74_3"/>
 ex <font color="Olive">P1x1</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">P1x1</font> <a href="hidden.html#R2">in</a>  <a href="card_fin.html#K1" target="_self">Choose</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<font color="Maroon">k</font>,1,0 &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x1</font>,<font color="Olive">P1x1</font>] )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">P1x</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="card_1.html#K4">card</a> <font color="Maroon">X1</font>, <a href="card_fin.html#K1" target="_self">Choose</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<font color="Maroon">k</font>,1,0<b> such that </b><br/><a NAME="E26:74"/><i><font color="Green">E184</font></i>: 
for <font color="Olive">x1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X1</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x1</font>,<font color="Maroon">P1x</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x1</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>();<br/></i>
<a NAME="E27:74"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/>
<a NAME="E28:74"/><b>then </b><i><font color="Green">E187</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<font color="Maroon">k</font>,1,0</span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E29:74"/>
for <font color="Olive">x1</font>, <font color="Olive">x2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P1x</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P1x</font> &amp; <font color="Maroon">P1x</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1x</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x2</font> holds <br/><font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Olive">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:74_4"/><i><font color="Green">E189</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P1x</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P1x</font> &amp; <font color="Maroon">P1x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> )
 ;<br/>

<a NAME="E2:74_4"/>
( (  <a href="card_fin.html#K1" target="_self">Choose</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<font color="Maroon">k</font>,1,0 <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<font color="Maroon">k</font>,1,0</span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) &amp; ( <font color="Maroon">X1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="card_1.html#K4">card</a> <font color="Maroon">X1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 
<b>by </b><i><a class="ref" href="card_1.html#T47">CARD_1:47</a></i>;<br/>
<a NAME="E3:74_4"/><b>then </b><i><font color="Green">E191</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P1x</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X1</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X1</font> )
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">f1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:74_4"/><i><font color="Green">E193</font></i>: 
( <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">f1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> &amp; <font color="Maroon">P1x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">f2</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:74_4"/><i><font color="Green">E194</font></i>: 
( <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">f2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> &amp; <font color="Maroon">P1x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E8:74_4"/><i><font color="Green">E195</font></i>: 
(  ex <font color="Olive">F</font> being  <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,0</span><a href="domain_1.html#K7">}</a></span> st <br/>( <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Olive">F</font> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Olive">F</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Olive">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 ) &amp;  ex <font color="Olive">F</font> being  <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,0</span><a href="domain_1.html#K7">}</a></span> st <br/>( <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Olive">F</font> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Olive">F</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Olive">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 ) )
 
<b>by </b><i>, <a class="txt" href="card_fin.html#E17:74"><i><font color="Green">E77</font></i></a></i>;<br/>
<a NAME="E9:74_4"/><b>then </b><i><font color="Green">E196</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E10:74_4"/>
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> holds <br/><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_4_1"><b>proof </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="E1:74_4_1"/><i><font color="Green">E197</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_4_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_4_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:74_4_1_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="card_fin.html#E19:74"><i><font color="Green">E79</font></i></a>, <a class="txt" href="card_fin.html#E20:74"><i><font color="Green">E80</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_4_1_1_1_1"><b>suppose </b></a><a NAME="E1:74_4_1_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:74_4_1_1_1_1"/><b>then </b>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> )
 <b>by </b><i><a class="txt" href="card_fin.html#E19:74"><i><font color="Green">E79</font></i></a>, <a class="txt" href="card_fin.html#E20:74"><i><font color="Green">E80</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:74_4_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T71">FUNCT_1:71</a></i>;<br/><b>hence </b><a NAME="E4:74_4_1_1_1_1"/>
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, , <a class="txt" href="card_fin.html#E17:74"><i><font color="Green">E77</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_4_1_1_1_2"><b>suppose </b></a><a NAME="E1:74_4_1_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:74_4_1_1_1_2"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:74_4_1_1_1_2"/>
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:74_4_1"/>
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E11:74_4"/><b>then </b>
( <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P1</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P1</font> )
 
<b>by </b><i>, <a class="txt" href="card_fin.html#E15:74"><i><font color="Green">E75</font></i></a>, <a class="txt" href="card_fin.html#E19:74"><i><font color="Green">E79</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E12:74_4"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i>, , <a class="txt" href="card_fin.html#E17:74"><i><font color="Green">E77</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E30:74"/><b>then </b><i><font color="Green">E198</font></i>: 
<font color="Maroon">P1x</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<b>set </b><font color="Maroon">I</font> =  <a href="yellow20.html#K1">Intersect</a> <span class="p1">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">Fy</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">Fy</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>;<br/>
<a NAME="E31:74"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="yellow20.html#K1">Intersect</a> <span class="p2">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">Fy</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">Fy</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="card_fin.html#T4" target="_self">Th4</a>, </i>;<br/>
<b>then 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="E33:74"/><i><font color="Green">E199</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">P1x</font>
 <b>and </b><br/><a NAME="E34:74"/><i><font color="Green">E200</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">P1x</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> <span class="p2">(<span class="default"><a href="yellow20.html#K1">Intersect</a> <span class="p3">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">Fy</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">Fy</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>,<font color="Olive">f</font>,1</span>)</span>
 <b>and </b><br/><a NAME="E35:74"/><i><font color="Green">E202</font></i>: 
 <a href="card_fin.html#K5" target="_self">Card_Intersection</a> <span class="p1">(<span class="default"><a href="yellow20.html#K1">Intersect</a> <span class="p2">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">Fy</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">Fy</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">XFS1</font>
 <b>by </b><i>, , <a class="ref" href="card_fin.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E36:74"/>
( ( <font color="Maroon">X0</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="card_1.html#K4">card</a> <font color="Maroon">X0</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) &amp; ( <font color="Maroon">X1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="card_1.html#K4">card</a> <font color="Maroon">X1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 
<b>by </b><i><a class="ref" href="card_1.html#T47">CARD_1:47</a></i>;<br/>
<a NAME="E37:74"/><b>then </b><i><font color="Green">E203</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P0</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X0</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X1</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">XP0</font> = <font color="Maroon">P0</font>, <font color="Maroon">XP1</font> = <font color="Maroon">P1</font> as   <a href="afinsq_1.html#NM1">XFinSequence</a> <b>by </b><i><a class="ref" href="afinsq_1.html#T7">AFINSQ_1:7</a></i>;<br/>
<b>set </b><font color="Maroon">Ch</font> =  <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,1,0;<br/>
<a NAME="E39:74"/>
( <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">P0</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">P1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X0</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">X1</font> )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a>, <a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a></i>;<br/>
<a NAME="E40:74"/><b>then </b><i><font color="Green">E204</font></i>: 
( <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">X0</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">X1</font></span>)</span> <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">X</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,1,0</span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">X0</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">XP0</font> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">X1</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">XP1</font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">XP0</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">XP1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X0</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">X1</font> &amp; <font color="Maroon">X0</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">X1</font> <a href="hidden.html#R1">=</a>  <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,1,0 )
 
<b>by </b><i>, , , , <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a>, <a class="ref" href="afinsq_1.html#T29">AFINSQ_1:29</a>, </i>;<br/>
<a NAME="E41:74"/><b>then </b>
(  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">XP0</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">XP1</font></span>)</span> <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">X</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,1,0</span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">XP0</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">XP1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,1,0 )
 
<b>by </b><i><a class="ref" href="afinsq_1.html#D4">AFINSQ_1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">XP01</font> = <font color="Maroon">XP0</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">XP1</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">X</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,1,0</span>)</span>, <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,1,0 <b>by </b><i><a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E43:74"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/>
<a NAME="E44:74"/><b>then </b>
<font color="Maroon">X0</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">X1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E45:74"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">P0</font> <a href="xboole_0.html#R1">misses</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">P1</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T64">XBOOLE_1:64</a></i>;<br/>
<a NAME="E46:74"/><b>then </b>
<font color="Maroon">XP01</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="card_fin.html#E11:74"><i><font color="Green">E61</font></i></a>, , </i>;<br/>
<b>then consider </b><font color="Maroon">XFS</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="E48:74"/><i><font color="Green">E205</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">XP01</font>
 <b>and </b><br/><a NAME="E49:74"/><i><font color="Green">E206</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">XFS</font> &amp; <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XP01</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> holds <br/><font color="Maroon">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">Fy</font>,<font color="Olive">f</font>,1</span>)</span>
 <b>and </b><br/><a NAME="E50:74"/><i><font color="Green">E207</font></i>: 
 <a href="card_fin.html#K5" target="_self">Card_Intersection</a> <font color="Maroon">Fy</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">XFS</font>
 <b>by </b><i>, <a class="ref" href="card_fin.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E51:74"/>
<font color="Maroon">XFS</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XFS0</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">XFS1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_5"><b>proof </b></a><div class="add">

<a NAME="E1:74_5"/>
( (  <a href="card_fin.html#K1" target="_self">Choose</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,1,0 <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,1,0</span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) &amp; (  <a href="card_fin.html#K1" target="_self">Choose</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<font color="Maroon">k</font>,1,0 <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<font color="Maroon">k</font>,1,0</span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) &amp; ( <font color="Maroon">X0</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="card_1.html#K4">card</a> <font color="Maroon">X0</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) &amp; ( <font color="Maroon">X1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="card_1.html#K4">card</a> <font color="Maroon">X1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 
<b>by </b><i><a class="ref" href="card_1.html#T47">CARD_1:47</a></i>;<br/>
<a NAME="E2:74_5"/><b>then </b><i><font color="Green">E208</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P0x</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X0</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P0</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X0</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P1x</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X1</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X1</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:74_5"/><b>then </b>
(  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">XP0</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">XP1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS1</font></span>)</span> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS0</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">XFS0</font> )
 
<b>by </b><i>, <a class="txt" href="card_fin.html#E15:74"><i><font color="Green">E75</font></i></a>, <a class="txt" href="card_fin.html#E19:74"><i><font color="Green">E79</font></i></a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a>, <a class="ref" href="afinsq_1.html#D4">AFINSQ_1:def 4</a></i>;<br/>
<a NAME="E4:74_5"/><b>then </b><i><font color="Green">E210</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">XFS0</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">XFS1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="card_fin.html#E20:74"><i><font color="Green">E80</font></i></a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E5:74_5"/><i><font color="Green">E211</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS0</font> holds <br/><font color="Maroon">XFS</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XFS0</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_5_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:74_5_1"/><i><font color="Green">E212</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS0</font>
 ;<br/>

<b>consider </b><font color="Maroon">fx</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:74_5_1"/><i><font color="Green">E213</font></i>: 
( <font color="Maroon">fx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">fx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X0</font> &amp; <font color="Maroon">P0x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fx</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> )
 <b>by </b><i>, , <a class="txt" href="card_fin.html#E24:74"><i><font color="Green">E86</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">fx'</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,0</span><a href="domain_1.html#K7">}</a></span><b> such that </b><br/><a NAME="E5:74_5_1"/><i><font color="Green">E215</font></i>: 
( <font color="Maroon">fx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fx'</font> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">fx'</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">fx'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i/>;<br/>
<a NAME="E6:74_5_1"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fx</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">fx</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">fx</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">fx</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, , <a class="ref" href="relat_1.html#T87">RELAT_1:87</a>, <a class="ref" href="relat_1.html#T167">RELAT_1:167</a></i>;<br/>
<a NAME="E7:74_5_1"/><b>then </b><i><font color="Green">E250</font></i>: 
( <font color="Maroon">fx</font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">fx</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">fx</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:74_5_1"/><i><font color="Green">E251</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fx</font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a>, <a class="ref" href="card_1.html#T47">CARD_1:47</a></i>;<br/>
<a NAME="E10:74_5_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS0</font> &amp; <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E24:74"><i><font color="Green">E86</font></i></a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E11:74_5_1"/><b>then </b>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS1</font></span>)</span> &amp; <font color="Maroon">fx</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E12:74_5_1"/><b>then </b>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS</font> &amp; <font color="Maroon">XP01</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XP0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">fx</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fx</font> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E20:74"><i><font color="Green">E80</font></i></a>, <a class="ref" href="card_fin.html#T5" target="_self">Th5</a>, , <a class="txt" href="card_fin.html#E15:74"><i><font color="Green">E75</font></i></a>, <a class="txt" href="card_fin.html#E19:74"><i><font color="Green">E79</font></i></a>, <a class="txt" href="card_fin.html#E24:74"><i><font color="Green">E86</font></i></a>, , <a class="ref" href="afinsq_1.html#D4">AFINSQ_1:def 4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E13:74_5_1"/><b>then </b>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS</font> &amp; <font color="Maroon">XP01</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fx</font> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">fx</font> )
 
<b>by </b><i>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E14:74_5_1"/><b>then </b>
( <font color="Maroon">XFS</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">n</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">Fy</font>,<font color="Maroon">fx</font>,1</span>)</span> &amp; <font color="Maroon">XFS0</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">n</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> <span class="p2">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">fx</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,1</span>)</span> &amp;  <a href="card_fin.html#K2" target="_self">Intersection</a> <span class="p1">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">fx</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,1 <a href="hidden.html#R1">=</a>  <a href="card_fin.html#K2" target="_self">Intersection</a> <span class="p1">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<font color="Maroon">fx</font>,1 &amp;  <a href="card_fin.html#K2" target="_self">Intersection</a> <span class="p1">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<font color="Maroon">fx</font>,1 <a href="hidden.html#R1">=</a>  <a href="card_fin.html#K2" target="_self">Intersection</a> <font color="Maroon">Fy</font>,<font color="Maroon">fx</font>,1 )
 
<b>by </b><i>, , <a class="txt" href="card_fin.html#E21:74"><i><font color="Green">E81</font></i></a>, <a class="txt" href="card_fin.html#E24:74"><i><font color="Green">E86</font></i></a>, , , , </i>;<br/>
<b>hence </b><a NAME="E15:74_5_1"/>
<font color="Maroon">XFS</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XFS0</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">n</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:74_5"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS1</font> holds <br/><font color="Maroon">XFS</font> <a href="stirl2_1.html#K8">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">XFS0</font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">XFS1</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_5_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:74_5_2"/><i><font color="Green">E252</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS1</font>
 ;<br/>

<b>consider </b><font color="Maroon">fx</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:74_5_2"/><i><font color="Green">E253</font></i>: 
( <font color="Maroon">fx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">fx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> &amp; <font color="Maroon">P1x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fx</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> )
 <b>by </b><i>, <a class="txt" href="card_fin.html#E15:74"><i><font color="Green">E75</font></i></a>, <a class="txt" href="card_fin.html#E24:74"><i><font color="Green">E86</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">fx'</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,0</span><a href="domain_1.html#K7">}</a></span><b> such that </b><br/><a NAME="E5:74_5_2"/><i><font color="Green">E254</font></i>: 
( <font color="Maroon">fx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fx'</font> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">fx'</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">fx'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i/>;<br/>
<a NAME="E6:74_5_2"/><i><font color="Green">E255</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fx</font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_5_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:74_5_2_1"/><i><font color="Green">E256</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">fx'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:74_5_2_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fx</font> &amp; <font color="Maroon">fx</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:74_5_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fx</font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:74_5_2"/><i><font color="Green">E257</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">fx</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">fx</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_5_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:74_5_2_2"/><i><font color="Green">E258</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">fx</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">fx</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">fx</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></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">fx</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fx</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T167">RELAT_1:167</a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">fx1</font> = <span class="p1">(<span class="default"><font color="Maroon">fx</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<a NAME="E3:74_5_2_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/>
<a NAME="E4:74_5_2_2"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">fx</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:74_5_2_2"/><b>then </b>
not <font color="Maroon">x</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">fx</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E6:74_5_2_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">fx1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fx</font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fx1</font> )
 
<b>by </b><i>, , , , <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E7:74_5_2_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">fx1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">fx1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">fx1</font> )
 
<b>by </b><i>, <a class="ref" href="card_2.html#T54">CARD_2:54</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:74_5_2_2"/><i><font color="Green">E260</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fx1</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a>, , <a class="ref" href="card_1.html#T47">CARD_1:47</a></i>;<br/>
<a NAME="E10:74_5_2_2"/>
( <font color="Maroon">y</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">fx</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">fx</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E11:74_5_2_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">fx</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</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">fx</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">fx</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E12:74_5_2_2"/>
( 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">fx</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">fx</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:74_5_2"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS1</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XP1</font> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E24:74"><i><font color="Green">E86</font></i></a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a>, <a class="txt" href="card_fin.html#E15:74"><i><font color="Green">E75</font></i></a>, <a class="ref" href="card_fin.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E9:74_5_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS1</font></span>)</span> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XP1</font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E10:74_5_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS</font> &amp; <font color="Maroon">XP1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XP01</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">XP0</font></span>)</span></span>)</span> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS0</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XP0</font> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E20:74"><i><font color="Green">E80</font></i></a>, <a class="ref" href="card_fin.html#T5" target="_self">Th5</a>, , <a class="txt" href="card_fin.html#E15:74"><i><font color="Green">E75</font></i></a>, <a class="txt" href="card_fin.html#E19:74"><i><font color="Green">E79</font></i></a>, <a class="ref" href="afinsq_1.html#D4">AFINSQ_1:def 4</a></i>;<br/>
<a NAME="E11:74_5_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS</font> &amp; <font color="Maroon">XP01</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">XP0</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">fx</font> &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">XP0</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS0</font> )
 
<b>by </b><i>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E12:74_5_2"/><b>then </b>
( <font color="Maroon">XFS</font> <a href="stirl2_1.html#K8">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font></span>)</span> <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">Fy</font>,<font color="Maroon">fx</font>,1</span>)</span> &amp; <font color="Maroon">XFS1</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">n</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> <span class="p2">(<span class="default"><a href="yellow20.html#K1">Intersect</a> <span class="p3">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p4">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">fx</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,1</span>)</span> &amp;  <a href="card_fin.html#K2" target="_self">Intersection</a> <span class="p1">(<span class="default"><a href="yellow20.html#K1">Intersect</a> <span class="p2">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">fx</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_fin.html#K2" target="_self">Intersection</a> <span class="p2">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">fx</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,1</span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> &amp;  <a href="card_fin.html#K2" target="_self">Intersection</a> <span class="p1">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">fx</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,1 <a href="hidden.html#R1">=</a>  <a href="card_fin.html#K2" target="_self">Intersection</a> <font color="Maroon">Fy</font>,<span class="p1">(<span class="default"><font color="Maroon">fx</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,1 )
 
<b>by </b><i>, <a class="ref" href="card_fin.html#T4" target="_self">Th4</a>, , , <a class="txt" href="card_fin.html#E21:74"><i><font color="Green">E81</font></i></a>, <a class="txt" href="card_fin.html#E24:74"><i><font color="Green">E86</font></i></a>, , , </i>;<br/>
<a NAME="E13:74_5_2"/><b>then </b>
( <font color="Maroon">XFS</font> <a href="stirl2_1.html#K8">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font></span>)</span> <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">Fy</font>,<font color="Maroon">fx</font>,1</span>)</span> &amp; <font color="Maroon">XFS1</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_fin.html#K2" target="_self">Intersection</a> <font color="Maroon">Fy</font>,<span class="p3">(<span class="default"><font color="Maroon">fx</font> <a href="relat_1.html#K7">|</a> <span class="p4">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,1</span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p2">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="card_fin.html#K2" target="_self">Intersection</a> <font color="Maroon">Fy</font>,<span class="p2">(<span class="default"><font color="Maroon">fx</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,1</span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_fin.html#K2" target="_self">Intersection</a> <font color="Maroon">Fy</font>,<font color="Maroon">fx</font>,1 )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E14:74_5_2"/>
<font color="Maroon">XFS</font> <a href="stirl2_1.html#K8">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">XFS0</font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">XFS1</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:74_5"/>
<font color="Maroon">XFS</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XFS0</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">XFS1</font>
 <b>by </b><i>, <a class="ref" href="card_fin.html#T7" target="_self">Th7</a>, <a class="ref" href="afinsq_1.html#D4">AFINSQ_1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E52:74"/><b>then </b>
( <a href="binop_2.html#K47">addnat</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">XFS</font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K47">addnat</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K47">addnat</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">XFS0</font></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K47">addnat</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">XFS1</font></span>)</span> &amp; <a href="binop_2.html#K47">addnat</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">XFS0</font> <a href="hidden.html#R1">=</a>  <a href="card_fin.html#K5" target="_self">Card_Intersection</a> <span class="p1">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">Fy</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> &amp; <a href="binop_2.html#K47">addnat</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">XFS1</font> <a href="hidden.html#R1">=</a>  <a href="card_fin.html#K5" target="_self">Card_Intersection</a> <span class="p1">(<span class="default"><a href="yellow20.html#K1">Intersect</a> <span class="p2">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">Fy</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">Fy</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">k</font> &amp; <a href="binop_2.html#K47">addnat</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">XFS</font> <a href="hidden.html#R1">=</a>  <a href="card_fin.html#K5" target="_self">Card_Intersection</a> <font color="Maroon">Fy</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> )
 
<b>by </b><i>, <a class="txt" href="card_fin.html#E17:74"><i><font color="Green">E77</font></i></a>, <a class="txt" href="card_fin.html#E23:74"><i><font color="Green">E82</font></i></a>, <a class="ref" href="stirl2_1.html#T47">STIRL2_1:47</a>, <a class="ref" href="stirl2_1.html#D4">STIRL2_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E53:74"/>
 <a href="card_fin.html#K5" target="_self">Card_Intersection</a> <font color="Maroon">Fy</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_fin.html#K5" target="_self">Card_Intersection</a> <span class="p2">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">Fy</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="card_fin.html#K5" target="_self">Card_Intersection</a> <span class="p2">(<span class="default"><a href="yellow20.html#K1">Intersect</a> <span class="p3">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">Fy</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">Fy</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="binop_2.html#D23">BINOP_2:def 23</a></i>;<br/>


</div>
