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

<a NAME="E1:41"/>
 not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T33" title="ARYTM_3:th.33">ARYTM_3:33</a></span>;<br/>
<a NAME="E2:41"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A1</font></span>: 
 <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#D3" title="NUMBERS:def.3">NUMBERS:def 3</a>, <a class="ref" href="xboole_1.html#T87" title="XBOOLE_1:th.87">XBOOLE_1:87</a>, <a class="ref" href="zfmisc_1.html#T50" title="ZFMISC_1:th.50">ZFMISC_1:50</a></span>;<br/>
<a NAME="E3:41"/><span class="kw">then </span>
 <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a>, <a class="ref" href="zfmisc_1.html#T67" title="ZFMISC_1:th.67">ZFMISC_1:67</a></span>;<br/>
<a NAME="E4:41"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A2</font></span>: 
 <a href="arytm_2.html#K1" title="ARYTM_2:func.1">DEDEKIND_CUTS</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E5:41"/>
 <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E6:41"/><span class="kw">then </span>
<a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="arytm_2.html#K1" title="ARYTM_2:func.1">DEDEKIND_CUTS</a> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="ref" href="xboole_1.html#T13" title="XBOOLE_1:th.13">XBOOLE_1:13</a></span>;<br/>
<a NAME="E7:41"/><span class="kw">then </span>
 <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_2.html#D2" title="ARYTM_2:def.2">ARYTM_2:def 2</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E8:41"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A3</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T11" title="CARD_1:th.11">CARD_1:11</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c2">INFNAT</font> = <span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span>;<br/>
<a NAME="E9:41"/><span class="lab"><font color="Green" title="E36">A4</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T14" title="CARD_1:th.14">CARD_1:14</a></span>;<br/>
<a NAME="E10:41"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a></span>)</span></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T34" title="CARD_2:th.34">CARD_2:34</a></span>;<br/>
<a NAME="E11:41"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:41"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E12:41"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:41"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="ref" href="card_2.html#T76" title="CARD_2:th.76">CARD_2:76</a></span>;<br/>
<a NAME="E13:41"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (2,<a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topgen_3.html#T17" target="_self" title="TOPGEN_3:th.17">Th17</a>, <a class="ref" href="card_2.html#T31" title="CARD_2:th.31">CARD_2:31</a></span>;<br/>
<a NAME="E14:41"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (2,<a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>)</span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> <span class="p1">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (2,<a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T83" title="CARD_2:th.83">CARD_2:83</a></span>;<br/>
<a NAME="E15:41"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A5</font></span>: 
<span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (2,<a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T76" title="CARD_2:th.76">CARD_2:76</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  =  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default">$1 <a href="topgen_3.html#K4" title="TOPGEN_3:func.4">-powers</a> <span class="p2">(<span class="default">1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2</span>)</span></span>)</span>;<br/>
<span class="lab"><font color="Green" title="E38">A6</font></span>:  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> = 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T4" title="CARD_2:th.4">CARD_2:4</a></span>
<br/>.= 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T69" title="CARD_1:th.69">CARD_1:69</a></span>
;<br/>
<a NAME="E17:41"/><span class="lab"><font color="Green" title="E39">A7</font></span>: 
 <a href="topgen_3.html#K3" title="TOPGEN_3:func.3">continuum</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T11" title="CARD_1:th.11">CARD_1:11</a>, <a class="ref" href="numbers.html#D1" title="NUMBERS:def.1">NUMBERS:def 1</a></span>;<br/>
<a NAME="E18:41"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T34" title="CARD_2:th.34">CARD_2:34</a></span>;<br/>
<a NAME="E19:41"/><span class="kw">then </span>
 <a href="topgen_3.html#K3" title="TOPGEN_3:func.3">continuum</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:41"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="txt" href="#E16:41"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:41"/>
 <a href="topgen_3.html#K3" title="TOPGEN_3:func.3">continuum</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (2,<a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:41"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (2,<a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topgen_3.html#K3" title="TOPGEN_3:func.3">continuum</a> </span><br/>

<a NAME="E21:41"/>
 <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> is  <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topgen_3.html#T28" target="_self" title="TOPGEN_3:th.28">Th28</a>, <a class="ref" href="card_4.html#T2" title="CARD_4:th.2">CARD_4:2</a></span>;<br/>
<a NAME="E22:41"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A8</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D14" title="CARD_3:def.14">CARD_3:def 14</a></span>;<br/>
<a NAME="E23:41"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T14" title="CARD_1:th.14">CARD_1:14</a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="ordinal1.html#T12" title="ORDINAL1:th.12">ORDINAL1:12</a></span>;<br/>
<a NAME="E24:41"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A9</font></span>: 
<span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p2">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T76" title="CARD_2:th.76">CARD_2:76</a></span>;<br/>
<a NAME="E25:41"/><span class="lab"><font color="Green" title="E42">A10</font></span>: 
 <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T14" title="CARD_1:th.14">CARD_1:14</a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">INFNAT</font> = <span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:41"><span class="lab"><font color="Green" title="E40">A8</font></span></a>, <a class="ref" href="card_1.html#T68" title="CARD_1:th.68">CARD_1:68</a>, <a class="ref" href="ordinal1.html#T12" title="ORDINAL1:th.12">ORDINAL1:12</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">INFNAT</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E28:41"/><span class="lab"><font color="Green" title="E43">A11</font></span>: 
 for <font color="Olive" title="b1">X</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">INFNAT</font> holds  <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">X</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></span>
<a NAME="E29:41"/><span class="lab"><font color="Green" title="E44">A12</font></span>: 
<font color="Maroon" title="c4">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c4">f</font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c4">f</font> or  not <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> or <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c4">f</font> or  not <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c4">f</font> or  not <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font> or <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:41_2"/><span class="lab"><font color="Green" title="E45">A13</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">and </span><br/><a NAME="E2:41_2"/><span class="lab"><font color="Green" title="E46">A14</font></span>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font> or <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )</span><br/>

<a NAME="E3:41_2"/><span class="lab"><font color="Green" title="E47">A15</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">b</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:41"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="txt" href="#E2:41_2"><span class="lab"><font color="Green" title="E46">A14</font></span></a></span>;<br/>
<a NAME="E4:41_2"/>
 not <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_2"><span class="lab"><font color="Green" title="E46">A14</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E5:41_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A16</font></span>: 
<font color="Maroon" title="c6">b</font> is   <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_2"><span class="lab"><font color="Green" title="E46">A14</font></span></a>, <a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E6:41_2"/>
 not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_2"><span class="lab"><font color="Green" title="E45">A13</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E7:41_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A17</font></span>: 
<font color="Maroon" title="c5">a</font> is   <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_2"><span class="lab"><font color="Green" title="E45">A13</font></span></a>, <a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E8:41_2"/>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">a</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:41"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="txt" href="#E1:41_2"><span class="lab"><font color="Green" title="E45">A13</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:41_2"/>
(  not <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font> or <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:41_2"><span class="lab"><font color="Green" title="E49">A17</font></span></a>, <a class="txt" href="#E5:41_2"><span class="lab"><font color="Green" title="E48">A16</font></span></a>, <a class="txt" href="#E3:41_2"><span class="lab"><font color="Green" title="E47">A15</font></span></a>, <a class="ref" href="topgen_3.html#T27" target="_self" title="TOPGEN_3:th.27">Th27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E30:41"/><span class="lab"><font color="Green" title="E45">A18</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<a NAME="E31:41"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">INFNAT</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E32:41"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c3">INFNAT</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="topgen_3.html#K3" title="TOPGEN_3:func.3">continuum</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:41"><span class="lab"><font color="Green" title="E44">A12</font></span></a>, <a class="txt" href="#E30:41"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="ref" href="card_1.html#T10" title="CARD_1:th.10">CARD_1:10</a></span>;<br/>
<a NAME="E33:41"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="topgen_3.html#K3" title="TOPGEN_3:func.3">continuum</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:41"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="txt" href="#E22:41"><span class="lab"><font color="Green" title="E40">A8</font></span></a>, <a class="txt" href="#E25:41"><span class="lab"><font color="Green" title="E42">A10</font></span></a>, <a class="ref" href="card_2.html#T98" title="CARD_2:th.98">CARD_2:98</a>, <a class="ref" href="ordinal1.html#T12" title="ORDINAL1:th.12">ORDINAL1:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E34:41"/>
 <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (2,<a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topgen_3.html#K3" title="TOPGEN_3:func.3">continuum</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="card_2.html#T31" title="CARD_2:th.31">CARD_2:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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