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

<b>set </b><font color="Maroon">T</font> =  <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a> ;<br/>
<b>assume </b><a NAME="E1:58"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">W</font>, <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a>   st <font color="Olive">W</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">V</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">W</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Olive">V</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Olive">W</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">V</font> holds <br/> ex <font color="Olive">P</font>, <font color="Olive">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a>  st <br/>( <font color="Olive">P</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Q</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Olive">P</font> &amp; <font color="Olive">V</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Q</font> &amp; <font color="Olive">P</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Q</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D6">COMPTS_1:def 6</a><br/>

<b>reconsider </b><font color="Maroon">C</font> = <span class="p1">(<span class="default"><a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  <a href="subset_1.html#K7">\</a> <a href="topgen_5.html#K1" target="_self">y=0-line</a> </span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><a href="numbers.html#K3">RAT</a> ,<a href="numbers.html#K3">RAT</a> </span><a href="finseq_1.html#K10">*&gt;</a></span></span>)</span> as  <a href="tops_1.html#V1">dense</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a>  <b>by </b><i><a class="ref" href="topgen_5.html#T1" target="_self">Th1</a></i>;<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">U</font>, <font color="Olive">V</font> being <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a>  st <br/>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">U</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">U</font> &amp; <a href="topgen_5.html#K1" target="_self">y=0-line</a>  <a href="subset_1.html#K7">\</a> <font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">V</font> &amp; <font color="Olive">U</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">V</font> );<br/>
<a NAME="E3:58"/><i><font color="Green">E43</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="zfmisc_1.html#K1">bool</a> <a href="topgen_5.html#K1" target="_self">y=0-line</a>  holds <br/> ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <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="58_1"><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="E1:58_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <a href="topgen_5.html#K1" target="_self">y=0-line</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">aa</font> = <font color="Maroon">a</font>, <font color="Maroon">a'</font> = <a href="topgen_5.html#K1" target="_self">y=0-line</a>  <a href="subset_1.html#K7">\</a> <font color="Maroon">a</font> as   <a href="subset_1.html#NM2">Subset</a> of <a href="topgen_5.html#K1" target="_self">y=0-line</a>  <b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">aa</font>, <font color="Maroon">B</font> = <font color="Maroon">a'</font> as  <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a>  <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:58_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="topgen_5.html#K1" target="_self">y=0-line</a>  or ( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="topgen_5.html#K1" target="_self">y=0-line</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1"><b>suppose </b></a><a NAME="E1:58_1_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>take </b><font color="Maroon">b</font> =  <a href="xboole_0.html#K1">{}</a> ;<br/><b>take </b><font color="Maroon">U</font> =  <a href="pre_topc.html#K1">{}</a> <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a> ;<br/><b>take </b><font color="Maroon">V</font> =  <a href="pre_topc.html#K2">[#]</a> <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a> ;<br/><b>thus </b><a NAME="E2:58_1_1_1"/>
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">U</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font> &amp; <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font> &amp; <a href="topgen_5.html#K1" target="_self">y=0-line</a>  <a href="subset_1.html#K7">\</a> <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> &amp; <font color="Maroon">U</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">V</font> )
 <b>by </b><i>, , , <a class="ref" href="xboole_1.html#T65">XBOOLE_1:65</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_2"><b>suppose </b></a><a NAME="E1:58_1_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="topgen_5.html#K1" target="_self">y=0-line</a> 
 ;<br/><div class="add"><b>take </b><font color="Maroon">b</font> = <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font>;<br/><b>take </b><font color="Maroon">U</font> =  <a href="pre_topc.html#K2">[#]</a> <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a> ;<br/><b>take </b><font color="Maroon">V</font> =  <a href="pre_topc.html#K1">{}</a> <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a> ;<br/><b>thus </b><a NAME="E2:58_1_1_2"/>
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">U</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font> &amp; <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font> &amp; <a href="topgen_5.html#K1" target="_self">y=0-line</a>  <a href="subset_1.html#K7">\</a> <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> &amp; <font color="Maroon">U</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">V</font> )
 <b>by </b><i>, , , <a class="ref" href="xboole_1.html#T65">XBOOLE_1:65</a>, <a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_3"><b>suppose </b></a><a NAME="E1:58_1_1_3"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="topgen_5.html#K1" target="_self">y=0-line</a>  )
 ;<br/><div class="add"><a NAME="E2:58_1_1_3"/>
( <span class="p1">(<span class="default"><font color="Maroon">aa</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> <a href="subset_1.html#K3">`</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">aa</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <span class="p1">(<span class="default"><a href="subset_1.html#K1">{}</a> <a href="topgen_5.html#K1" target="_self">y=0-line</a> </span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K2">[#]</a> <a href="topgen_5.html#K1" target="_self">y=0-line</a>  )
 ;<br/><a NAME="E3:58_1_1_3"/><b>then </b>
( <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="subset_1.html#K1">{}</a> <a href="topgen_5.html#K1" target="_self">y=0-line</a>  &amp; <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> )
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/><b>then consider </b><font color="Maroon">P</font> being   <a href="struct_0.html#NM2">Subset</a> of <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a> , <font color="Maroon">Q</font> being   <a href="struct_0.html#NM2">Subset</a> of <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a> <b> such that </b><br/><a NAME="E5:58_1_1_3"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">Q</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">P</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Q</font> )
 <b>by </b><i>, <a class="txt" href="topgen_5.html#E1:58"><i><font color="Green">E42</font></i></a></i>;<br/><b>take </b><font color="Maroon">b</font> = <font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font>;<br/><b>thus </b><a NAME="E6:58_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a</font>,<font color="Maroon">b</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">G</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:58"/><i><font color="Green">E61</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="zfmisc_1.html#K1">bool</a> <a href="topgen_5.html#K1" target="_self">y=0-line</a> 
 <b>and </b><br/><a NAME="E6:58"/><i><font color="Green">E62</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="zfmisc_1.html#K1">bool</a> <a href="topgen_5.html#K1" target="_self">y=0-line</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font>]
 <b>from </b><i><a class="ref" href="zfrefle1.html#S1">ZFREFLE1:sch 1</a>(<a class="txt" href="topgen_5.html#E3:58"><i><font color="Green">E43</font></i></a>);<br/></i>
<a NAME="E7:58"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2"><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="E1:58_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font>
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:58_2"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:58_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">b</font>,<font color="Maroon">a</font>]
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E5:58_2"/><b>then </b>
<font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E6:58_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">C</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:58"/><b>then </b><i><font color="Green">E79</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font></span>)</span> <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="zfmisc_1.html#K1">bool</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T27">CARD_1:27</a></i>;<br/>
<a NAME="E9:58"/>
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><a href="numbers.html#K3">RAT</a> ,<a href="numbers.html#K3">RAT</a> </span><a href="finseq_1.html#K10">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E10:58"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">C</font> <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="card_3.html#K4">product</a> <span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><a href="numbers.html#K3">RAT</a> ,<a href="numbers.html#K3">RAT</a> </span><a href="finseq_1.html#K10">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T27">CARD_1:27</a></i>;<br/>
<a NAME="E11:58"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">C</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K3">alef</a> 0
 
<b>by </b><i><a class="ref" href="card_4.html#T54">CARD_4:54</a>, <a class="ref" href="topgen_3.html#T17">TOPGEN_3:17</a>, </i>;<br/>
<a NAME="E12:58"/><b>then </b><i><font color="Green">E80</font></i>: 
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_2.html#K3">exp</a> 2,<span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">C</font></span>)</span> &amp;  <a href="card_2.html#K3">exp</a> 2,<span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">C</font></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_2.html#T44">CARD_2:44</a>, <a class="ref" href="card_4.html#T71">CARD_4:71</a></i>;<br/>
<a NAME="E13:58"/>
<font color="Maroon">G</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</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>assume </b><a NAME="E1:58_3"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">x</font>, <font color="Maroon">B</font> = <font color="Maroon">y</font> as   <a href="subset_1.html#NM2">Subset</a> of <a href="topgen_5.html#K1" target="_self">y=0-line</a>  <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">UA</font> being  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a> , <font color="Maroon">VA</font> being  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a> <b> such that </b><br/><a NAME="E4:58_3"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">UA</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">UA</font> &amp; <a href="topgen_5.html#K1" target="_self">y=0-line</a>  <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">VA</font> &amp; <font color="Maroon">UA</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">VA</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">UB</font> being  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a> , <font color="Maroon">VB</font> being  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a> <b> such that </b><br/><a NAME="E6:58_3"/><i><font color="Green">E120</font></i>: 
( <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">UB</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">UB</font> &amp; <a href="topgen_5.html#K1" target="_self">y=0-line</a>  <a href="subset_1.html#K7">\</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">VB</font> &amp; <font color="Maroon">UB</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">VB</font> )
 <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E7:58_3"/><i><font color="Green">E121</font></i>: 
( <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:58_3"/><i><font color="Green">E123</font></i>: 
( ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) or ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) )
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<a NAME="E10:58_3"/><i><font color="Green">E124</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">B</font> or <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E11:58_3"/>
( <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <a href="topgen_5.html#K1" target="_self">y=0-line</a>  <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font> &amp; <font color="Maroon">B</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <a href="topgen_5.html#K1" target="_self">y=0-line</a>  <a href="subset_1.html#K7">\</a> <font color="Maroon">B</font> &amp; <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="subset_1.html#T32">SUBSET_1:32</a></i>;<br/>
<a NAME="E12:58_3"/><b>then </b>
( <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">UA</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">VB</font> &amp; <font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">UB</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">VA</font> )
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/>
<a NAME="E13:58_3"/><b>then </b>
( ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">UA</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">VB</font> &amp; <font color="Maroon">UA</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">VB</font> is <a href="pre_topc.html#V3">open</a> ) or ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">UB</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">VA</font> &amp; <font color="Maroon">UB</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">VA</font> is <a href="pre_topc.html#V3">open</a> ) )
 
<b>by </b><i>, <a class="ref" href="tops_1.html#T38">TOPS_1:38</a></i>;<br/>
<a NAME="E14:58_3"/><b>then </b>
( <font color="Maroon">C</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">UA</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">VB</font> or <font color="Maroon">C</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">UB</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">VA</font> )
 
<b>by </b><i><a class="ref" href="tops_1.html#T80">TOPS_1:80</a></i>;<br/>
<a NAME="E15:58_3"/><b>then </b>
(  ex <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">UA</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">VB</font> ) or  ex <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">UB</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">VA</font> ) )
 
<b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E17:58_3"/><i><font color="Green">E125</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">UA</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">VB</font> or <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">UB</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">VA</font> ) )
 ;<br/>
<a NAME="E18:58_3"/>
( ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">UA</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">VB</font> ) or ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">UB</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">VA</font> ) )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E19:58_3"/><b>then </b>
( ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">UA</font> &amp; not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">UB</font> ) or ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">UB</font> &amp; not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">UA</font> ) )
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E20:58_3"/><b>then </b>
( ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font> ) or ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font> &amp; not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> ) )
 
<b>by </b><i>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E21:58_3"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E14:58"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font></span>)</span> <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="relat_1.html#K2">rng</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T26">CARD_1:26</a></i>;<br/>
<a NAME="E15:58"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <a href="topgen_5.html#K1" target="_self">y=0-line</a> </span>)</span> <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="zfmisc_1.html#K1">bool</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E16:58"/><b>then </b>
 <a href="card_2.html#K3">exp</a> 2,<a href="topgen_3.html#K3">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="zfmisc_1.html#K1">bool</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="card_2.html#T44">CARD_2:44</a></i>;<br/>
<a NAME="E17:58"/><b>then </b>
(  <a href="card_2.html#K3">exp</a> 2,<a href="topgen_3.html#K3">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> &amp;  <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="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_2.html#K3">exp</a> 2,<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="topgen_5.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a>, <a class="ref" href="card_5.html#T23">CARD_5:23</a></i>;<br/>
<a NAME="E18:58"/><b>then </b>
 <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="card_1.html#NR2" 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#T29">TOPGEN_3:29</a></i>;<br/>
<b>hence </b><a NAME="E19:58"/>
contradiction
 ;<br/>


</div>
