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

<a NAME="E1:43"/><i><font color="Green">E33</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K2">REAL+</a>  <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <a href="arytm_2.html#K2">REAL+</a> </span>)</span> <a href="card_2.html#K1">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_2.html#T47">CARD_2:47</a></i>;<br/>
<i><font color="Green">E34</font></i>:  <a href="card_1.html#K1">Card</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> = 
 <a href="card_1.html#K1">Card</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="arytm_2.html#K2">REAL+</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><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="card_2.html#T11">CARD_2:11</a></i>
<br/>.= 
 <a href="card_1.html#K1">Card</a> <a href="arytm_2.html#K2">REAL+</a> 
<b>by </b><i><a class="ref" href="card_2.html#T13">CARD_2:13</a></i>
;<br/>
<a NAME="E3:43"/>
not <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">0,0</span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 
<b>by </b><i><a class="ref" href="arytm_3.html#T39">ARYTM_3:39</a></i>;<br/>
<a NAME="E4:43"/><b>then </b>
(  <a href="arytm_3.html#K7">RAT+</a>  <a href="subset_1.html#R1">misses</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default">0,0</span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K6">}</a></span> &amp;  <a href="numbers.html#K3">RAT</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K7">RAT+</a>  <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<a href="arytm_3.html#K7">RAT+</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default">0,0</span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T56">ZFMISC_1:56</a>, <a class="ref" href="numbers.html#D3">NUMBERS:def 3</a></i>;<br/>
<a NAME="E5:43"/><b>then </b>
 <a href="numbers.html#K3">RAT</a>  <a href="hidden.html#R1">=</a> <a href="arytm_3.html#K7">RAT+</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<a href="arytm_3.html#K7">RAT+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default">0,0</span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T87">XBOOLE_1:87</a></i>;<br/>
<a NAME="E6:43"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="arytm_3.html#K7">RAT+</a>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K3">RAT</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E7:43"/><b>then </b>
 <a href="pcomps_1.html#K1">bool</a> <a href="arytm_3.html#K7">RAT+</a>  <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> <a href="numbers.html#K3">RAT</a> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T79">ZFMISC_1:79</a></i>;<br/>
<a NAME="E8:43"/><b>then </b>
 <a href="arytm_2.html#K1">DEDEKIND_CUTS</a>  <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> <a href="numbers.html#K3">RAT</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E9:43"/><b>then </b><i><font color="Green">E36</font></i>: 
<a href="arytm_3.html#K7">RAT+</a>  <a href="xboole_0.html#K2">\/</a> <a href="arytm_2.html#K1">DEDEKIND_CUTS</a>  <a href="tarski.html#R1">c=</a> <a href="numbers.html#K3">RAT</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <a href="numbers.html#K3">RAT</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E6:43"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a></i>;<br/>
<a NAME="E10:43"/>
 <a href="arytm_2.html#K2">REAL+</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K7">RAT+</a>  <a href="xboole_0.html#K2">\/</a> <a href="arytm_2.html#K1">DEDEKIND_CUTS</a> </span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">{<span class="default"> <span class="p2">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  : <font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  where B is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a>  </span>}</span> 
 
<b>by </b><i><a class="ref" href="arytm_2.html#D2">ARYTM_2:def 2</a></i>;<br/>
<a NAME="E11:43"/><b>then </b>
 <a href="arytm_2.html#K2">REAL+</a>  <a href="tarski.html#R1">c=</a> <a href="arytm_3.html#K7">RAT+</a>  <a href="xboole_0.html#K2">\/</a> <a href="arytm_2.html#K1">DEDEKIND_CUTS</a> 
 
;<br/>
<a NAME="E12:43"/><b>then </b>
 <a href="arytm_2.html#K2">REAL+</a>  <a href="tarski.html#R1">c=</a> <a href="numbers.html#K3">RAT</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <a href="numbers.html#K3">RAT</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E9:43"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E13:43"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <a href="arytm_2.html#K2">REAL+</a>  <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="numbers.html#K3">RAT</a>  <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <a href="numbers.html#K3">RAT</a> </span>)</span></span>)</span> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="numbers.html#K3">RAT</a>  <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <a href="numbers.html#K3">RAT</a> </span>)</span></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <a href="numbers.html#K3">RAT</a> </span>)</span> <a href="card_2.html#K1">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <a href="numbers.html#K3">RAT</a> </span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="card_2.html#T47">CARD_2:47</a>, <a class="ref" href="card_1.html#T27">CARD_1:27</a></i>;<br/>
<a NAME="E14:43"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <a href="arytm_2.html#K2">REAL+</a>  <a href="card_1.html#NR1" target="_self">&lt;=`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <a href="numbers.html#K3">RAT</a> </span>)</span> <a href="card_2.html#K1">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <a href="numbers.html#K3">RAT</a> </span>)</span></span>)</span> &amp;  <a href="card_1.html#K1">Card</a> <a href="numbers.html#K3">RAT</a>  <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <a href="numbers.html#K3">RAT</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a>, <a class="ref" href="card_1.html#T30">CARD_1:30</a></i>;<br/>
<a NAME="E15:43"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <a href="arytm_2.html#K2">REAL+</a>  <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <a href="numbers.html#K3">RAT</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="card_4.html#T34">CARD_4:34</a></i>;<br/>
<a NAME="E16:43"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <a href="arytm_2.html#K2">REAL+</a>  <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_2.html#K3">exp</a> 2,<span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span>
 
<b>by </b><i><a class="ref" href="topgen_3.html#T17" target="_self">Th17</a>, <a class="ref" href="card_2.html#T44">CARD_2:44</a></i>;<br/>
<a NAME="E17:43"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <a href="arytm_2.html#K2">REAL+</a> </span>)</span> <a href="card_2.html#K1">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <a href="arytm_2.html#K2">REAL+</a> </span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <span class="p1">(<span class="default"><a href="card_2.html#K3">exp</a> 2,<span class="p2">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span></span>)</span> <a href="card_2.html#K1">+`</a> <span class="p1">(<span class="default"><a href="card_2.html#K3">exp</a> 2,<span class="p2">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_4.html#T41">CARD_4:41</a></i>;<br/>
<a NAME="E18:43"/><b>then </b><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <a href="arytm_2.html#K2">REAL+</a> </span>)</span> <a href="card_2.html#K1">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <a href="arytm_2.html#K2">REAL+</a> </span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_2.html#K3">exp</a> 2,<span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span>
 
<b>by </b><i><a class="ref" href="card_4.html#T34">CARD_4:34</a></i>;<br/>
<a NAME="E19:43"/>
 <a href="numbers.html#K1">REAL</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K2">REAL+</a>  <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default">0,0</span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="numbers.html#D1">NUMBERS:def 1</a></i>;<br/>
<a NAME="E20:43"/><b>then </b>
 <a href="numbers.html#K1">REAL</a>  <a href="tarski.html#R1">c=</a> <a href="arytm_2.html#K2">REAL+</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
;<br/>
<a NAME="E21:43"/><b>then </b>
 <a href="topgen_3.html#K3" target="_self">continuum</a>  <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K2">REAL+</a>  <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T27">CARD_1:27</a></i>;<br/>
<a NAME="E22:43"/><b>then </b>
 <a href="topgen_3.html#K3" target="_self">continuum</a>  <a href="card_1.html#NR1" target="_self">&lt;=`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <a href="arytm_2.html#K2">REAL+</a> </span>)</span> <a href="card_2.html#K1">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <a href="arytm_2.html#K2">REAL+</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E1:43"><i><font color="Green">E33</font></i></a>, <a class="txt" href="topgen_3.html#E2:43"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E23:43"/>
 <a href="topgen_3.html#K3" target="_self">continuum</a>  <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_2.html#K3">exp</a> 2,<span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span>
 <b>by </b><i><a class="txt" href="topgen_3.html#E18:43"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  =  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="topgen_3.html#K4" target="_self">-powers</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <a href="numbers.html#K5">NAT</a> </span>)</span>;<br/>
<a NAME="E24:43"/>
 <a href="finsub_1.html#K5">Fin</a> <a href="numbers.html#K5">NAT</a>  is <a href="card_4.html#V1">countable</a>
 
<b>by </b><i><a class="ref" href="card_4.html#T44">CARD_4:44</a>, <a class="ref" href="topgen_3.html#T28" target="_self">Th28</a></i>;<br/>
<a NAME="E25:43"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K3">alef</a> 0 &amp;  <a href="card_1.html#K3">alef</a> 0 <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <a href="numbers.html#K5">NAT</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="card_1.html#T30">CARD_1:30</a>, <a class="ref" href="card_1.html#T83">CARD_1:83</a>, <a class="ref" href="card_1.html#T84">CARD_1:84</a>, <a class="ref" href="card_4.html#D1">CARD_4:def 1</a></i>;<br/>
<a NAME="E26:43"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <a href="numbers.html#K5">NAT</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <a href="numbers.html#K5">NAT</a> </span>)</span> 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#T4">CARD_2:4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>3</sub></font>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E29:43"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> holds <font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<a NAME="E30:43"/><i><font color="Green">E40</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E31:43"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:43_2"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<a NAME="E2:43_2"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <a href="numbers.html#K5">NAT</a>  &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsub_1.html#K5">Fin</a> <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <a href="numbers.html#K5">NAT</a>  &amp; not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsub_1.html#K5">Fin</a> <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:43_2"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> is  <a href="finset_1.html#V1">infinite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">c<sub>6</sub></font> is  <a href="finset_1.html#V1">infinite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>5</sub></font>) &amp; <font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>6</sub></font>) )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E29:43"><i><font color="Green">E39</font></i></a>, <a class="txt" href="topgen_3.html#E1:43_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:43_2"/>
(  not <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="topgen_3.html#T27" target="_self">Th27</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E32:43"/><b>then </b>
( <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <a href="numbers.html#K5">NAT</a> </span>)</span></span>)</span> <a href="card_2.html#K1">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <a href="numbers.html#K5">NAT</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <a href="numbers.html#K5">NAT</a> </span>)</span> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>3</sub></font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="topgen_3.html#K3" target="_self">continuum</a>  )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E26:43"><i><font color="Green">E38</font></i></a>, <a class="txt" href="topgen_3.html#E30:43"><i><font color="Green">E40</font></i></a>, <a class="ref" href="card_1.html#T26">CARD_1:26</a>, <a class="ref" href="card_4.html#T34">CARD_4:34</a></i>;<br/>
<a NAME="E33:43"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="topgen_3.html#K3" target="_self">continuum</a> 
 
<b>by </b><i><a class="txt" href="topgen_3.html#E26:43"><i><font color="Green">E38</font></i></a>, <a class="ref" href="card_4.html#T76">CARD_4:76</a></i>;<br/>
<b>hence </b><a NAME="E34:43"/>
 <a href="card_2.html#K3">exp</a> 2,<span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topgen_3.html#K3" target="_self">continuum</a> 
 <b>by </b><i><a class="ref" href="card_1.html#T83">CARD_1:83</a>, <a class="ref" href="card_1.html#T84">CARD_1:84</a>, <a class="ref" href="card_2.html#T44">CARD_2:44</a></i>;<br/>


</div>
