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

<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Olive">x</font>,<font color="Olive">q</font></span><a href="rcomp_2.html#K1">.[</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">q</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : ( <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> is <a href="rat_1.html#V1">rational</a> ) </span>}</span> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<a href="numbers.html#K3">RAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K1">REAL</a>  = <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="rcomp_2.html#K1">.[</a></span>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E2:32"/><i><font color="Green">E37</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<a href="numbers.html#K3">RAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E3:32"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<a href="numbers.html#K3">RAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span> holds  <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">z</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S4">FUNCT_1:sch 4</a>();<br/></i>
<a NAME="E4:32"/><i><font color="Green">E39</font></i>: 
<span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Olive">x</font>,<font color="Olive">q</font></span><a href="rcomp_2.html#K1">.[</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">q</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : ( <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> is <a href="rat_1.html#V1">rational</a> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:32_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Olive">x</font>,<font color="Olive">q</font></span><a href="rcomp_2.html#K1">.[</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">q</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : ( <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> is <a href="rat_1.html#V1">rational</a> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">q</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:32_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">q</font></span><a href="rcomp_2.html#K1">.[</a></span> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q</font> &amp; <font color="Maroon">q</font> is <a href="rat_1.html#V1">rational</a> )
 ;<br/>
<a NAME="E4:32_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K3">RAT</a> 
 
<b>by </b><i>, <a class="ref" href="rat_1.html#D2">RAT_1:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">b</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">q</font></span><a href="domain_1.html#K1">]</a></span> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<a href="numbers.html#K3">RAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E6:32_1"/>
( <font color="Maroon">b</font> <a href="domain_1.html#K2">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">b</font> <a href="domain_1.html#K3">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E7:32_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">q</font></span><a href="rcomp_2.html#K1">.[</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:32_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:32"/><i><font color="Green">E46</font></i>: 
(  <a href="arytm_3.html#K13">{}</a>  <a href="hidden.html#R2">in</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#NR1" target="_self">&lt;=`</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_1.html#T27">CARD_1:27</a></i>;<br/>
 <a href="card_1.html#K1">Card</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<a href="numbers.html#K3">RAT</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"><span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <a href="numbers.html#K1">REAL</a> </span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <a href="numbers.html#K3">RAT</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="card_2.html#T14">CARD_2:14</a></i>
<br/>.= 
<a href="topgen_3.html#K3" target="_self">continuum</a>  <a href="card_2.html#K2">*`</a> <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_2.html#D2">CARD_2:def 2</a></i>
<br/>.= 
 <a href="topgen_3.html#K3" target="_self">continuum</a> 
<b>by </b><i>, <a class="ref" href="card_4.html#T78">CARD_4:78</a></i>
;<br/>
<b>hence </b><a NAME="E7:32"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Olive">x</font>,<font color="Olive">q</font></span><a href="rcomp_2.html#K1">.[</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">q</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : ( <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> is <a href="rat_1.html#V1">rational</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="ref" href="card_1.html#T28">CARD_1:28</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>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">x</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Olive">x</font>,<font color="Olive">q</font></span><a href="rcomp_2.html#K1">.[</a></span> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> is <a href="rat_1.html#V1">rational</a> );<br/>
<a NAME="E8:32"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  holds <br/> ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Olive">x</font>,<font color="Olive">q</font></span><a href="rcomp_2.html#K1">.[</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">q</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : ( <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> is <a href="rat_1.html#V1">rational</a> ) </span>}</span>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:32_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E3:32_3"/>
<font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon">q</font> being  <a href="rat_1.html#V1">rational</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E5:32_3"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q</font> &amp; <font color="Maroon">q</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> 1 )
 <b>by </b><i><a class="ref" href="rat_1.html#T22">RAT_1:22</a></i>;<br/>
<a NAME="E6:32_3"/>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K3">RAT</a>  &amp;  <a href="numbers.html#K3">RAT</a>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a>  )
 
<b>by </b><i><a class="ref" href="rat_1.html#D2">RAT_1:def 2</a>, <a class="ref" href="numbers.html#T12">NUMBERS:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E8:32_3"/>
<span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">q</font></span><a href="rcomp_2.html#K1">.[</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Olive">x</font>,<font color="Olive">q</font></span><a href="rcomp_2.html#K1">.[</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">q</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : ( <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> is <a href="rat_1.html#V1">rational</a> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:32_3"/>
 ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Olive">x</font>,<font color="Olive">q</font></span><a href="rcomp_2.html#K1">.[</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">q</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : ( <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> is <a href="rat_1.html#V1">rational</a> ) </span>}</span>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a</font>,<font color="Olive">b</font>] )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E10:32"/><i><font color="Green">E49</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a>  &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Olive">x</font>,<font color="Olive">q</font></span><a href="rcomp_2.html#K1">.[</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">q</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : ( <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> is <a href="rat_1.html#V1">rational</a> ) </span>}</span>  &amp; ( for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font>] ) )
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>();<br/></i>
<a NAME="E11:32"/>
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_4"><b>proof </b></a><div class="add">

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



<b>assume </b><a NAME="E1:32_4"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">q</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:32_4"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">q</font></span><a href="rcomp_2.html#K1">.[</a></span> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q</font> &amp; <font color="Maroon">q</font> is <a href="rat_1.html#V1">rational</a> )
 <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E4:32_4"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">r</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E6:32_4"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">r</font></span><a href="rcomp_2.html#K1">.[</a></span> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> is <a href="rat_1.html#V1">rational</a> )
 <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E7:32_4"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E8:32_4"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">r</font></span><a href="rcomp_2.html#K1">.[</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">q</font></span><a href="rcomp_2.html#K1">.[</a></span> )
 
<b>by </b><i>, <a class="ref" href="topgen_3.html#D1" target="_self">Def1</a>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/>
<a NAME="E9:32_4"/><b>then </b>
( <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> )
 
<b>by </b><i><a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/>
<b>hence </b><a NAME="E10:32_4"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="topgen_3.html#D1" target="_self">Def1</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:32"/>
 <a href="topgen_3.html#K3" target="_self">continuum</a>  <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Olive">x</font>,<font color="Olive">q</font></span><a href="rcomp_2.html#K1">.[</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">q</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : ( <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> is <a href="rat_1.html#V1">rational</a> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="card_1.html#T26">CARD_1:26</a></i>;<br/>


</div>
