<?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="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:65"/><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>
 ;<br/>

<b>let </b><font color="Maroon">P</font> be   <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>assume </b><a NAME="E2:65"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">P</font> is <a href="funct_1.html#V2">one-to-one</a>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:65_1"/>
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1"><b>suppose </b></a><a NAME="E1:65_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">XFS</font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="afinsq_1.html#NM1">XFinSequence</a> ;<br/><a NAME="E3:65_1_1"/>
(  <a href="relat_1.html#K2">rng</a> <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R1">c=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><b>then reconsider </b><font color="Maroon">XFS</font> = <font color="Maroon">XFS</font> as   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D8">ORDINAL1:def 8</a></i>;<br/><b>take </b>
<font color="Maroon">XFS</font>
;<br/><b>thus </b><a NAME="E5:65_1_1"/>
(  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS</font> &amp; ( 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>by </b><i><a class="ref" href="relat_1.html#T60">RELAT_1:60</a>, , <a class="ref" href="card_1.html#T47">CARD_1:47</a></i>;<br/><a NAME="E6:65_1_1"/>
(  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">XFS</font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="card_fin.html#K5" target="_self">Card_Intersection</a> <font color="Maroon">Fy</font>,1 <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="card_fin.html#E1:65_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="stirl2_1.html#T53">STIRL2_1:53</a>, , , <a class="ref" href="card_1.html#T47">CARD_1:47</a></i>;<br/><b>hence </b><a NAME="E7:65_1_1"/>
 <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>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_2"><b>suppose </b></a><a NAME="E1:65_1_2"/>
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">cX</font> =  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">cX</font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</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">a<sub>1</sub></font></span>)</span>;<br/><b>consider </b><font color="Maroon">XFS</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">cX</font>, <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:65_1_2"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">cX</font> holds  <font color="Maroon">XFS</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i><a NAME="E5:65_1_2"/><i><font color="Green">E57</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">XFS</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cX</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">XFS</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  )
 <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">XFS</font> = <font color="Maroon">XFS</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>reconsider </b><font color="Maroon">XFS</font> = <font color="Maroon">XFS</font> as   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="ordinal1.html#D8">ORDINAL1:def 8</a></i>;<br/><b>take </b>
<font color="Maroon">XFS</font>
;<br/><b>thus </b><a NAME="E8:65_1_2"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>thus </b><a NAME="E9:65_1_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="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>by </b><i><a class="txt" href="card_fin.html#E4:65_1_2"><i><font color="Green">E53</font></i></a>, </i>;<br/><b>thus </b><a NAME="E10:65_1_2"/>
 <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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_2_1"><b>proof </b></a><div class="add">

<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</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"><span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> 1</span>)</span>;<br/>
<a NAME="E1:65_1_2_1"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">cX</font> holds <br/><font color="Maroon">H<sub>2</sub></font>(<font color="Olive">x</font>) <a href="hidden.html#R2">in</a>  <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,1,0,1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_2_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:65_1_2_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">cX</font>
 ;<br/>

<b>set </b><font color="Maroon">F0</font> = <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> 0;<br/>
<b>set </b><font color="Maroon">F1</font> = <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"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> 1;<br/>
<a NAME="E2:65_1_2_1_1"/>
<font color="Maroon">x</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="E3:65_1_2_1_1"/><b>then </b>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:65_1_2_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.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"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#R1">misses</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"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="card_1.html#K4">card</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a>, <a class="ref" href="card_1.html#T50">CARD_1:50</a></i>;<br/>
<b>hence </b><a NAME="E5:65_1_2_1_1"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">x</font>) <a href="hidden.html#R2">in</a>  <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,1,0,1
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">P1</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">cX</font>, <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,1,0,1<b> such that </b><br/><a NAME="E3:65_1_2_1"/><i><font color="Green">E60</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> <font color="Maroon">cX</font> holds <br/><font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">z</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S2">FUNCT_2:sch 2</a>();<br/></i>
<a NAME="E4:65_1_2_1"/>
 <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,1,0,1 <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">P1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:65_1_2_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,1,0,1
 ;<br/>

<b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span><b> such that </b><br/><a NAME="E3:65_1_2_1_2"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="card_fin.html#E5:65_1_2"><i><font color="Green">E57</font></i></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="E5:65_1_2_1_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">F</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="card_fin.html#E1:65_1_2_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="card_2.html#T60">CARD_2:60</a></i>;<br/>
<a NAME="E6:65_1_2_1_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E7:65_1_2_1_2"/><b>then </b>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">X</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="card_1.html#T66">CARD_1:66</a></i>;<br/>
<a NAME="E8:65_1_2_1_2"/><b>then </b>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">P</font> is <a href="funct_2.html#V2">onto</a> )
 
<b>by </b><i>, <a class="ref" href="stirl2_1.html#T70">STIRL2_1:70</a></i>;<br/>
<a NAME="E9:65_1_2_1_2"/><b>then </b>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:65_1_2_1_2"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E12:65_1_2_1_2"/>
<font color="Maroon">cX</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
;<br/>
<a NAME="E13:65_1_2_1_2"/><b>then </b>
(  <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>,1,0,1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">X</font></span>)</span> <a href="newton.html#K9">choose</a> 1 &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E14:65_1_2_1_2"/><b>then </b>
 <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>,1,0,1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">cX</font>
 
<b>by </b><i><a class="ref" href="newton.html#T33">NEWTON:33</a></i>;<br/>
<a NAME="E15:65_1_2_1_2"/><b>then </b><i><font color="Green">E70</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cX</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cX</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="E16:65_1_2_1_2"/>
<font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_2_1_2_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">F0</font> = <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> 0;<br/>
<b>set </b><font color="Maroon">F1</font> = <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"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> 1;<br/>
<b>set </b><font color="Maroon">P1x</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</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"><span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> 1</span>)</span>;<br/>
<a NAME="E1:65_1_2_1_2_1"/><i><font color="Green">E71</font></i>: 
( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.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"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, <a class="txt" href="card_fin.html#E3:65_1_2_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="card_fin.html#E1:65_1_2_1_2"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E2:65_1_2_1_2_1"/><b>then </b><i><font color="Green">E73</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> 0</span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <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"><span class="p5">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> 1</span>)</span></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</a> <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"><span class="p5">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> 0</span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <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"><span class="p5">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> 1</span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_2_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">d</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:65_1_2_1_2_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_2_1_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_2_1_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:65_1_2_1_2_1_1_1_1"/>
( <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">d</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"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="card_fin.html#E5:65_1_2_1_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="card_fin.html#T3" target="_self">Th3</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="65_1_2_1_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:65_1_2_1_2_1_1_1_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:65_1_2_1_2_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relat_1.html#K1">dom</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"><span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> 1</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"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#R1">misses</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"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a>, <a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:65_1_2_1_2_1_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</a> <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"><span class="p5">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i>, <a class="txt" href="card_fin.html#E1:65_1_2_1_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="card_fin.html#T4" target="_self">Th4</a>, <a class="ref" href="funct_4.html#T17">FUNCT_4:17</a></i>;<br/><a NAME="E4:65_1_2_1_2_1_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</a> <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"><span class="p5">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="ref" href="card_fin.html#T4" target="_self">Th4</a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><b>hence </b><a NAME="E5:65_1_2_1_2_1_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</a> <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"><span class="p5">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_2_1_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:65_1_2_1_2_1_1_1_1_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">d</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"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:65_1_2_1_2_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</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"><span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> 1</span>)</span> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; not <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> )
 <b>by </b><i><a class="txt" href="card_fin.html#E1:65_1_2_1_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E3:65_1_2_1_2_1_1_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</a> <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"><span class="p5">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</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"><span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> 1</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> &amp; not <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><a NAME="E4:65_1_2_1_2_1_1_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</a> <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"><span class="p5">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> &amp; not <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="card_fin.html#T4" target="_self">Th4</a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:65_1_2_1_2_1_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</a> <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"><span class="p5">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:65_1_2_1_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</a> <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"><span class="p5">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E4:65_1_2_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</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"><span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="txt" href="card_fin.html#E6:65_1_2_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E5:65_1_2_1_2_1"/>
<font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="txt" href="card_fin.html#E1:65_1_2_1_2"><i><font color="Green">E61</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E17:65_1_2_1_2"/>
<font color="Maroon">z</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="txt" href="card_fin.html#E1:65_1_2_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="card_fin.html#E1:65_1_2_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="card_fin.html#E3:65_1_2_1_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:65_1_2_1"/><b>then </b>
(  <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,1,0,1 <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">P1</font> &amp; <font color="Maroon">cX</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E6:65_1_2_1"/><b>then </b>
( <font color="Maroon">P1</font> is <a href="funct_2.html#V2">onto</a> &amp;  <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>,1,0,1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">X</font></span>)</span> <a href="newton.html#K9">choose</a> 1 &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E7:65_1_2_1"/><b>then </b><i><font color="Green">E79</font></i>: 
( <font color="Maroon">P1</font> is <a href="funct_2.html#V2">onto</a> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,1,0,1</span>)</span> &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> )
 
<b>by </b><i><a class="ref" href="card_1.html#T66">CARD_1:66</a>, <a class="ref" href="newton.html#T33">NEWTON:33</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">P1</font> = <font color="Maroon">P1</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>,1,0,1</span>)</span>, <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,1,0,1 ;<br/>
<a NAME="E9:65_1_2_1"/>
( <font color="Maroon">P1</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Fy</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, <a class="txt" href="card_fin.html#E5:65_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="stirl2_1.html#T70">STIRL2_1:70</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="E11:65_1_2_1"/><i><font color="Green">E80</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">P1</font>
 <b>and </b><br/><a NAME="E12:65_1_2_1"/><i><font color="Green">E81</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">P1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> holds <br/><font color="Maroon">XFS1</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K2" target="_self">Intersection</a> <font color="Maroon">Fy</font>,<font color="Olive">f</font>,0</span>)</span>
 <b>and </b><br/><a NAME="E13:65_1_2_1"/><i><font color="Green">E82</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">XFS1</font>
 <b>by </b><i><a class="ref" href="card_fin.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E14:65_1_2_1"/>
<font color="Maroon">XFS</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XFS1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_2_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:65_1_2_1_3"/>
(  <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,1,0,1 <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> <font color="Maroon">X</font>,1,0,1</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/>
<a NAME="E2:65_1_2_1_3"/><b>then </b><i><font color="Green">E86</font></i>: 
 <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> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,1,0,1</span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:65_1_2_1_3"/><b>then </b><i><font color="Green">E89</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS1</font> <a href="hidden.html#R1">=</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#E5:65_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="card_fin.html#E1:65_1_2_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:65_1_2_1_3"/>
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">XFS1</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> <font color="Maroon">XFS</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_2_1_3_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="E1:65_1_2_1_3_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS1</font>
 ;<br/>

<a NAME="E2:65_1_2_1_3_1"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">z</font>) &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">z</font>) <a href="hidden.html#R2">in</a>  <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,1,0,1 )
 
<b>by </b><i>, , <a class="txt" href="card_fin.html#E5:65_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="card_fin.html#E1:65_1_2_1"><i><font color="Green">E58</font></i></a>, </i>;<br/>
<b>then consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span><b> such that </b><br/><a NAME="E4:65_1_2_1_3_1"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">z</font>) &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><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="E6:65_1_2_1_3_1"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="card_fin.html#T3" target="_self">Th3</a>, <a class="ref" href="card_2.html#T60">CARD_2:60</a></i>;<br/>
<a NAME="E7:65_1_2_1_3_1"/>
( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> 0</span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">z</font>) <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> 0</span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <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"><span class="p5">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> 1</span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/>
<a NAME="E8:65_1_2_1_3_1"/><b>then </b>
( not <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <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"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> 0</span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <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"><span class="p5">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> 0</span>)</span> <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">z</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="relat_1.html#K1">dom</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"><span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> 1</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"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a>, <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E9:65_1_2_1_3_1"/><b>then </b>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">z</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">z</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">z</font>) &amp; 0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E10:65_1_2_1_3_1"/><b>then </b>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</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="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E11:65_1_2_1_3_1"/><b>then </b>
( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cX</font> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E5:65_1_2"><i><font color="Green">E57</font></i></a>, , <a class="txt" href="card_fin.html#E1:65_1_2_1"><i><font color="Green">E58</font></i></a>, , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="card_fin.html#T4" target="_self">Th4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E12:65_1_2_1_3_1"/><b>then </b>
(  <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">f</font>,0</span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span></span>)</span> &amp; <font color="Maroon">XFS1</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K2" target="_self">Intersection</a> <font color="Maroon">Fy</font>,<font color="Maroon">f</font>,0</span>)</span> &amp; <font color="Maroon">XFS</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">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="Maroon">z</font></span>)</span> &amp; <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">z</font></span>)</span> <a href="hidden.html#R1">=</a> <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">z</font> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E4:65_1_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a>, , , , , <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/>
<b>hence </b><a NAME="E13:65_1_2_1_3_1"/>
<font color="Maroon">XFS1</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XFS</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">z</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:65_1_2_1_3"/>
<font color="Maroon">XFS</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XFS1</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E15:65_1_2_1"/>
 <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#E3:65_1_2_1"><i><font color="Green">E60</font></i></a></i>;<br/>


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

</div>
