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

<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="hidden.html#M1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:69"/><i><font color="Green">E47</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">Fy</font> is <a href="finset_1.html#V1">finite</a>
 <b>and </b><br/><a NAME="E2:69"/><i><font color="Green">E50</font></i>: 
<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>
 ;<br/>

<b>reconsider </b><font color="Maroon">X</font> =  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Fy</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i/>;<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>consider </b><font color="Maroon">Px</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><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">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> such that </b><br/><a NAME="E5:69"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">Px</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="card_fin.html#E1:69"><i><font color="Green">E47</font></i></a></i>;<br/>
<a NAME="E6:69"/>
<font color="Maroon">X</font> <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>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E7:69"/><b>then </b>
 <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"><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>
 
<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">XFSx</font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:69"/><i><font color="Green">E53</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFSx</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</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>and </b><br/><a NAME="E10:69"/><i><font color="Green">E57</font></i>: 
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="afinsq_1.html#K2">dom</a> <font color="Maroon">XFSx</font> holds <br/><font color="Maroon">XFSx</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">Px</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font></span>)</span>
 <b>and </b><br/><a NAME="E11:69"/><i><font color="Green">E58</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"><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="stirl2_1.html#K7">Sum</a> <font color="Maroon">XFSx</font>
 <b>by </b><i>, <a class="txt" href="card_fin.html#E2:69"><i><font color="Green">E50</font></i></a></i>;<br/>
<a NAME="E12:69"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/>
<a NAME="E13:69"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<b>then reconsider </b><font color="Maroon">k</font> = <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">X</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="nat_1.html#T60">NAT_1:60</a></i>;<br/>
<a NAME="E15:69"/><i><font color="Green">E59</font></i>: 
( not  <a href="card_1.html#K4">card</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#R2">in</a>  <a href="card_1.html#K4">card</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">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="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"><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>  <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/>
<b>then consider </b><font color="Maroon">P</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><a href="card_1.html#K4">card</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="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</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><a href="domain_1.html#K6">}</a></span>,<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> such that </b><br/><a NAME="E17:69"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">P</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</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">Px</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</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> )
 <b>by </b><i><a class="ref" href="stirl2_1.html#T67">STIRL2_1:67</a></i>;<br/>
<a NAME="E18:69"/>
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="E19:69"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">P</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, <a class="txt" href="card_fin.html#E17:69"><i><font color="Green">E60</font></i></a>, <a class="txt" href="card_fin.html#E19:69"><i><font color="Green">E61</font></i></a>, <a class="ref" href="stirl2_1.html#T68">STIRL2_1:68</a></i>;<br/>
<a NAME="E20:69"/><i><font color="Green">E62</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 
;<br/>
<a NAME="E21:69"/><b>then </b><i><font color="Green">E63</font></i>: 
 <a href="card_1.html#K4">card</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">k</font>
 
<b>by </b><i>, <a class="ref" href="stirl2_1.html#T65">STIRL2_1:65</a></i>;<br/>
<a NAME="E22:69"/><b>then </b>
(  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> <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">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="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</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><a href="domain_1.html#K6">}</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="txt" href="card_fin.html#E21:69"><i><font color="Green">E63</font></i></a>, <a class="ref" href="afinsq_1.html#T4">AFINSQ_1:4</a>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font>,<font color="Maroon">X</font> ;<br/>
<b>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="E25:69"/><i><font color="Green">E64</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E26:69"/><i><font color="Green">E68</font></i>: 
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="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS</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#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font></span>)</span>
 <b>and </b><br/><a NAME="E27:69"/><i><font color="Green">E70</font></i>: 
 <a href="card_fin.html#K5" target="_self">Card_Intersection</a> <font color="Maroon">Fy</font>,1 <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="txt" href="card_fin.html#E20:69"><i><font color="Green">E62</font></i></a>, <a class="txt" href="card_fin.html#E2:69"><i><font color="Green">E50</font></i></a></i>;<br/>
<a NAME="E28:69"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">XFS</font> <a href="card_fin.html#K4" target="_self">|</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XFSx</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>proof </b></a><div class="add">

<a NAME="E1:69_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E2:69_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">k</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="card_1.html#T56">CARD_1:56</a></i>;<br/>
<a NAME="E3:69_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFSx</font>
 
<b>by </b><i><a class="txt" href="card_fin.html#E25:69"><i><font color="Green">E64</font></i></a>, <a class="ref" href="card_fin.html#T3" target="_self">Th3</a>, <a class="txt" href="card_fin.html#E10:69"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E4:69_1"/>
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFSx</font> holds <br/><font color="Maroon">XFS</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XFSx</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:69_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFSx</font>
 ;<br/>

<a NAME="E2:69_1_1"/><i><font color="Green">E77</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">Px</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i><a class="txt" href="card_fin.html#E17:69"><i><font color="Green">E60</font></i></a>, <a class="txt" href="card_fin.html#E25:69"><i><font color="Green">E64</font></i></a>, <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="E3:69_1_1"/><b>then </b><i><font color="Green">E79</font></i>: 
( <font color="Maroon">Px</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS</font> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E19:69"><i><font color="Green">E61</font></i></a>, <a class="txt" href="card_fin.html#E10:69"><i><font color="Green">E57</font></i></a>, <a class="txt" href="card_fin.html#E25:69"><i><font color="Green">E64</font></i></a>, <a class="ref" href="card_fin.html#T3" target="_self">Th3</a>, , <a class="txt" href="card_fin.html#E28:69"><i><font color="Green">E71</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<a NAME="E4:69_1_1"/>
( <font color="Maroon">Px</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">Px</font> &amp; <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> <font color="Maroon">X</font> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="txt" href="card_fin.html#E10:69"><i><font color="Green">E57</font></i></a>, <a class="txt" href="card_fin.html#E25:69"><i><font color="Green">E64</font></i></a>, <a class="txt" href="card_fin.html#E28:69"><i><font color="Green">E71</font></i></a></i>;<br/>
<a NAME="E5:69_1_1"/><b>then </b>
( <font color="Maroon">Px</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</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="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#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">X</font> <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>  <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"><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="xboole_1.html#T28">XBOOLE_1:28</a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E6:69_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><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> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">Px</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</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> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Px</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span> &amp; <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> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Px</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Px</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a>, <a class="txt" href="card_fin.html#E10:69"><i><font color="Green">E57</font></i></a>, <a class="txt" href="card_fin.html#E25:69"><i><font color="Green">E64</font></i></a>, <a class="txt" href="card_fin.html#E28:69"><i><font color="Green">E71</font></i></a></i>;<br/>
<a NAME="E7:69_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><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> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">Px</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> &amp; <font color="Maroon">XFS</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="card_fin.html#T4" target="_self">Th4</a>, <a class="txt" href="card_fin.html#E2:69_1"><i><font color="Green">E73</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:69_1_1"/>
<font color="Maroon">XFS</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XFSx</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="card_fin.html#E11:69"><i><font color="Green">E58</font></i></a>, <a class="txt" href="card_fin.html#E28:69"><i><font color="Green">E71</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:69_1"/>
<font color="Maroon">XFS</font> <a href="card_fin.html#K4" target="_self">|</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XFSx</font>
 <b>by </b><i><a class="txt" href="card_fin.html#E27:69"><i><font color="Green">E70</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E29:69"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E30:69"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<a NAME="E31:69"/><b>then </b><i><font color="Green">E81</font></i>: 
( <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">XFSx</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">XFS</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">XFS</font> <a href="card_fin.html#K4" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">XFS</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</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>, <a class="txt" href="card_fin.html#E26:69"><i><font color="Green">E68</font></i></a>, </i>;<br/>
<a NAME="E32:69"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</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="E33:69"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">XFS</font> <a href="card_fin.html#K4" target="_self">|</a> <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> <font color="Maroon">XFS</font> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E19:69"><i><font color="Green">E61</font></i></a>, <a class="txt" href="card_fin.html#E21:69"><i><font color="Green">E63</font></i></a>, <a class="ref" href="card_fin.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a>, <a class="ref" href="relat_1.html#T98">RELAT_1:98</a>, , <a class="ref" href="stirl2_1.html#T65">STIRL2_1:65</a></i>;<br/>
<b>hence </b><a NAME="E34:69"/>
 <a href="card_fin.html#K5" target="_self">Card_Intersection</a> <font color="Maroon">Fy</font>,1 <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>,1</span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</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>
 <b>by </b><i><a class="txt" href="card_fin.html#E15:69"><i><font color="Green">E59</font></i></a>, , <a class="txt" href="card_fin.html#E27:69"><i><font color="Green">E70</font></i></a></i>;<br/>


</div>
