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

<b>set </b><font color="Maroon" title="c1">T</font> =  <a href="topgen_5.html#K3" title="TOPGEN_5:func.3">Niemytzki-plane</a> ;<br/>
<b>assume </b><a NAME="E1:54"/><i><font color="Green" title="E38">A1</font></i>: 
for <font color="Olive" title="b1">W</font>, <font color="Olive" title="b2">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topgen_5.html#K3" title="TOPGEN_5:func.3">Niemytzki-plane</a>   st <font color="Olive" title="b1">W</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b2">V</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">W</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Olive" title="b2">V</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Olive" title="b1">W</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">V</font> holds <br/> ex <font color="Olive" title="b3">P</font>, <font color="Olive" title="b4">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topgen_5.html#K3" title="TOPGEN_5:func.3">Niemytzki-plane</a>  st <br/>( <font color="Olive" title="b3">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b4">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">P</font> &amp; <font color="Olive" title="b2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">Q</font> &amp; <font color="Olive" title="b3">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b4">Q</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D6" title="COMPTS_1:def.6">COMPTS_1:def 6</a><br/>

<b>reconsider </b><font color="Maroon" title="c2">C</font> = <span class="p1">(<span class="default"><a href="topgen_5.html#K2" title="TOPGEN_5:func.2">y&gt;=0-plane</a>  <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <a href="topgen_5.html#K1" title="TOPGEN_5:func.1">y=0-line</a> </span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> ,<a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span> as  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topgen_5.html#K3" title="TOPGEN_5:func.3">Niemytzki-plane</a>  <b>by </b><i><a class="ref" href="topgen_5.html#T40" target="_self" title="TOPGEN_5:th.40">Th40</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">U</font>, <font color="Olive" title="b2">V</font> being <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topgen_5.html#K3" title="TOPGEN_5:func.3">Niemytzki-plane</a>  st <br/>( $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">U</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">C</font> &amp; $1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">U</font> &amp; <a href="topgen_5.html#K1" title="TOPGEN_5:func.1">y=0-line</a>  <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> $1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> &amp; <font color="Olive" title="b1">U</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">V</font> );<br/>
<a NAME="E3:54"/><i><font color="Green" title="E39">A2</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="topgen_5.html#K1" title="TOPGEN_5:func.1">y=0-line</a>  holds <br/> ex <font color="Olive" title="b2">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:54_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="topgen_5.html#K1" title="TOPGEN_5:func.1">y=0-line</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c4">aa</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c5">a'</font> = <a href="topgen_5.html#K1" title="TOPGEN_5:func.1">y=0-line</a>  <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">a</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="topgen_5.html#K1" title="TOPGEN_5:func.1">y=0-line</a>  <b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">A</font> = <font color="Maroon" title="c4">aa</font>, <font color="Maroon" title="c7">B</font> = <font color="Maroon" title="c5">a'</font> as  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topgen_5.html#K3" title="TOPGEN_5:func.3">Niemytzki-plane</a>  <b>by </b><i><a class="ref" href="topgen_5.html#T46" target="_self" title="TOPGEN_5:th.46">Th46</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_1_1"/>
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topgen_5.html#K1" title="TOPGEN_5:func.1">y=0-line</a>  or ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="topgen_5.html#K1" title="TOPGEN_5:func.1">y=0-line</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_1"><b>suppose </b></a><a NAME="E1:54_1_1_1"/><i><font color="Green" title="E40">A3</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c8">b</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/><b>take </b><font color="Maroon" title="c9">U</font> =  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <a href="topgen_5.html#K3" title="TOPGEN_5:func.3">Niemytzki-plane</a> ;<br/><b>take </b><font color="Maroon" title="c10">V</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topgen_5.html#K3" title="TOPGEN_5:func.3">Niemytzki-plane</a> ;<br/><b>thus </b><a NAME="E2:54_1_1_1"/>
( <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">U</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">C</font> &amp; <font color="Maroon" title="c3">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">U</font> &amp; <a href="topgen_5.html#K1" title="TOPGEN_5:func.1">y=0-line</a>  <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">V</font> &amp; <font color="Maroon" title="c9">U</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c10">V</font> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:54_1_1_1"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="topgen_5.html#D3" target="_self" title="TOPGEN_5:def.3">Def3</a>, <a class="ref" href="topgen_5.html#T23" target="_self" title="TOPGEN_5:th.23">Th23</a>, <a class="ref" href="xboole_1.html#T65" title="XBOOLE_1:th.65">XBOOLE_1:65</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_2"><b>suppose </b></a><a NAME="E1:54_1_1_2"/><i><font color="Green" title="E40">A4</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topgen_5.html#K1" title="TOPGEN_5:func.1">y=0-line</a> 
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c8">b</font> = <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topgen_5.html#K3" title="TOPGEN_5:func.3">Niemytzki-plane</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">C</font>;<br/><b>take </b><font color="Maroon" title="c9">U</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topgen_5.html#K3" title="TOPGEN_5:func.3">Niemytzki-plane</a> ;<br/><b>take </b><font color="Maroon" title="c10">V</font> =  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <a href="topgen_5.html#K3" title="TOPGEN_5:func.3">Niemytzki-plane</a> ;<br/><b>thus </b><a NAME="E2:54_1_1_2"/>
( <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">U</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">C</font> &amp; <font color="Maroon" title="c3">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">U</font> &amp; <a href="topgen_5.html#K1" title="TOPGEN_5:func.1">y=0-line</a>  <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">V</font> &amp; <font color="Maroon" title="c9">U</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c10">V</font> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:54_1_1_2"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="topgen_5.html#D3" target="_self" title="TOPGEN_5:def.3">Def3</a>, <a class="ref" href="topgen_5.html#T23" target="_self" title="TOPGEN_5:th.23">Th23</a>, <a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a>, <a class="ref" href="xboole_1.html#T65" title="XBOOLE_1:th.65">XBOOLE_1:65</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_3"><b>suppose </b></a><a NAME="E1:54_1_1_3"/><i><font color="Green" title="E40">A5</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="topgen_5.html#K1" title="TOPGEN_5:func.1">y=0-line</a>  )
 ;<br/><div class="add"><a NAME="E2:54_1_1_3"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">aa</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a'</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">aa</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <span class="p1">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="topgen_5.html#K1" title="TOPGEN_5:func.1">y=0-line</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <a href="topgen_5.html#K1" title="TOPGEN_5:func.1">y=0-line</a>  )
 ;<br/><a NAME="E3:54_1_1_3"/><b>then </b>
( <font color="Maroon" title="c7">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="topgen_5.html#K1" title="TOPGEN_5:func.1">y=0-line</a>  &amp; <font color="Maroon" title="c6">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">B</font> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:54_1_1_3"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">P</font>, <font color="Maroon" title="c9">Q</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topgen_5.html#K3" title="TOPGEN_5:func.3">Niemytzki-plane</a> <b> such that </b><br/><a NAME="E5:54_1_1_3"/><i><font color="Green" title="E41">A6</font></i>: 
( <font color="Maroon" title="c8">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c9">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">P</font> &amp; <font color="Maroon" title="c7">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">Q</font> &amp; <font color="Maroon" title="c8">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c9">Q</font> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:54"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="topgen_5.html#E1:54_1_1_3"><i><font color="Green" title="E40">A5</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c10">b</font> = <font color="Maroon" title="c8">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">C</font>;<br/><b>thus </b><a NAME="E6:54_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c10">b</font>]
 <b>by </b><i><a class="txt" href="topgen_5.html#E5:54_1_1_3"><i><font color="Green" title="E41">A6</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c3">G</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:54"/><i><font color="Green" title="E40">A7</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="topgen_5.html#K1" title="TOPGEN_5:func.1">y=0-line</a> 
 <b>and </b><br/><a NAME="E6:54"/><i><font color="Green" title="E41">A8</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="topgen_5.html#K1" title="TOPGEN_5:func.1">y=0-line</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font>]
 <b>from </b><i><a class="ref" href="classes1.html#S1" title="CLASSES1:sch.1">CLASSES1:sch 1</a>(<a class="txt" href="topgen_5.html#E3:54"><i><font color="Green" title="E39">A2</font></i></a>);<br/></i>
<a NAME="E7:54"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c2">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:54_2"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">G</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:54_2"/><i><font color="Green" title="E42">A9</font></i>: 
( <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">G</font> &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:54_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font>]
 
<b>by </b><i><a class="txt" href="topgen_5.html#E5:54"><i><font color="Green" title="E40">A7</font></i></a>, <a class="txt" href="topgen_5.html#E6:54"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="topgen_5.html#E3:54_2"><i><font color="Green" title="E42">A9</font></i></a></i>;<br/>
<a NAME="E5:54_2"/><b>then </b>
<font color="Maroon" title="c4">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">C</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E6:54_2"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c2">C</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:54"/><b>then </b><i><font color="Green" title="E42">A10</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">G</font></span>)</span> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c2">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></i>;<br/>
<a NAME="E9:54"/>
<font color="Maroon" title="c2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> ,<a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E10:54"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">C</font> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> ,<a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></i>;<br/>
<a NAME="E11:54"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">C</font> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="topgen_5.html#T12" target="_self" title="TOPGEN_5:th.12">Th12</a>, <a class="ref" href="card_4.html#T54" title="CARD_4:th.54">CARD_4:54</a>, <a class="ref" href="topgen_3.html#T17" title="TOPGEN_3:th.17">TOPGEN_3:17</a></i>;<br/>
<a NAME="E12:54"/><b>then </b><i><font color="Green" title="E43">A11</font></i>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> 2,<span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">C</font></span>)</span> &amp;  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> 2,<span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> 2,<span class="p1">(<span class="default"><a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="card_2.html#T44" title="CARD_2:th.44">CARD_2:44</a>, <a class="ref" href="card_4.html#T71" title="CARD_4:th.71">CARD_4:71</a></i>;<br/>
<a NAME="E13:54"/>
<font color="Maroon" title="c3">G</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="54_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E1:54_3"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">G</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">G</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">A</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c7">B</font> = <font color="Maroon" title="c5">y</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="topgen_5.html#K1" title="TOPGEN_5:func.1">y=0-line</a>  <b>by </b><i><a class="txt" href="topgen_5.html#E5:54"><i><font color="Green" title="E40">A7</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">UA</font>, <font color="Maroon" title="c9">VA</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topgen_5.html#K3" title="TOPGEN_5:func.3">Niemytzki-plane</a> <b> such that </b><br/><a NAME="E4:54_3"/><i><font color="Green" title="E44">A12</font></i>: 
( <font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">UA</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">C</font> &amp; <font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">UA</font> &amp; <a href="topgen_5.html#K1" title="TOPGEN_5:func.1">y=0-line</a>  <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">VA</font> &amp; <font color="Maroon" title="c8">UA</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c9">VA</font> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E6:54"><i><font color="Green" title="E41">A8</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">UB</font>, <font color="Maroon" title="c11">VB</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topgen_5.html#K3" title="TOPGEN_5:func.3">Niemytzki-plane</a> <b> such that </b><br/><a NAME="E6:54_3"/><i><font color="Green" title="E45">A13</font></i>: 
( <font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">UB</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">C</font> &amp; <font color="Maroon" title="c7">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">UB</font> &amp; <a href="topgen_5.html#K1" title="TOPGEN_5:func.1">y=0-line</a>  <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">VB</font> &amp; <font color="Maroon" title="c10">UB</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c11">VB</font> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E6:54"><i><font color="Green" title="E41">A8</font></i></a></i>;<br/>
<b>assume </b><a NAME="E7:54_3"/><i><font color="Green" title="E46">A14</font></i>: 
( <font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">y</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:54_3"/><i><font color="Green" title="E47">A15</font></i>: 
( ( <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> &amp; not <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> ) or ( <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> &amp; not <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> ) )
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<a NAME="E10:54_3"/><i><font color="Green" title="E48">A16</font></i>: 
( <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">B</font> or <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A</font> )
 
<b>by </b><i><a class="txt" href="topgen_5.html#E9:54_3"><i><font color="Green" title="E47">A15</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E11:54_3"/>
( <font color="Maroon" title="c6">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="topgen_5.html#K1" title="TOPGEN_5:func.1">y=0-line</a>  <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Maroon" title="c7">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="topgen_5.html#K1" title="TOPGEN_5:func.1">y=0-line</a>  <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">B</font> &amp; <font color="Maroon" title="c6">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Maroon" title="c7">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">B</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="subset_1.html#T32" title="SUBSET_1:th.32">SUBSET_1:32</a></i>;<br/>
<a NAME="E12:54_3"/><b>then </b>
( <font color="Maroon" title="c6">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">UA</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c11">VB</font> &amp; <font color="Maroon" title="c7">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">UB</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">VA</font> )
 
<b>by </b><i><a class="txt" href="topgen_5.html#E4:54_3"><i><font color="Green" title="E44">A12</font></i></a>, <a class="txt" href="topgen_5.html#E6:54_3"><i><font color="Green" title="E45">A13</font></i></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></i>;<br/>
<a NAME="E13:54_3"/><b>then </b>
( ( <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">UA</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c11">VB</font> &amp; <font color="Maroon" title="c8">UA</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c11">VB</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) or ( <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">UB</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">VA</font> &amp; <font color="Maroon" title="c10">UB</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">VA</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) )
 
<b>by </b><i><a class="txt" href="topgen_5.html#E10:54_3"><i><font color="Green" title="E48">A16</font></i></a>, <a class="ref" href="tops_1.html#T38" title="TOPS_1:th.38">TOPS_1:38</a></i>;<br/>
<a NAME="E14:54_3"/><b>then </b>
( <font color="Maroon" title="c2">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c8">UA</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c11">VB</font> or <font color="Maroon" title="c2">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c10">UB</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">VA</font> )
 
<b>by </b><i><a class="ref" href="tops_1.html#T80" title="TOPS_1:th.80">TOPS_1:80</a></i>;<br/>
<a NAME="E15:54_3"/><b>then </b>
(  ex <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> &amp; <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">UA</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c11">VB</font> ) or  ex <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> &amp; <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">UB</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">VA</font> ) )
 
<b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E17:54_3"/><i><font color="Green" title="E49">A17</font></i>: 
( <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> &amp; ( <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">UA</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c11">VB</font> or <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">UB</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">VA</font> ) )
 ;<br/>
<a NAME="E18:54_3"/>
( ( <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">UA</font> &amp; <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">VB</font> ) or ( <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">UB</font> &amp; <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">VA</font> ) )
 
<b>by </b><i><a class="txt" href="topgen_5.html#E17:54_3"><i><font color="Green" title="E49">A17</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E19:54_3"/><b>then </b>
( ( <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">UA</font> &amp; not <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">UB</font> ) or ( <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">UB</font> &amp; not <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">UA</font> ) )
 
<b>by </b><i><a class="txt" href="topgen_5.html#E4:54_3"><i><font color="Green" title="E44">A12</font></i></a>, <a class="txt" href="topgen_5.html#E6:54_3"><i><font color="Green" title="E45">A13</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E20:54_3"/><b>then </b>
( ( <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">A</font> &amp; not <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">B</font> ) or ( <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">B</font> &amp; not <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">A</font> ) )
 
<b>by </b><i><a class="txt" href="topgen_5.html#E4:54_3"><i><font color="Green" title="E44">A12</font></i></a>, <a class="txt" href="topgen_5.html#E6:54_3"><i><font color="Green" title="E45">A13</font></i></a>, <a class="txt" href="topgen_5.html#E17:54_3"><i><font color="Green" title="E49">A17</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E21:54_3"/>
contradiction
 <b>by </b><i><a class="txt" href="topgen_5.html#E7:54_3"><i><font color="Green" title="E46">A14</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:54"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">G</font></span>)</span> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T26" title="CARD_1:th.26">CARD_1:26</a></i>;<br/>
<a NAME="E15:54"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="topgen_5.html#K1" title="TOPGEN_5:func.1">y=0-line</a> </span>)</span> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c2">C</font></span>)</span>
 
<b>by </b><i><a class="txt" href="topgen_5.html#E5:54"><i><font color="Green" title="E40">A7</font></i></a>, <a class="txt" href="topgen_5.html#E8:54"><i><font color="Green" title="E42">A10</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E16:54"/><b>then </b>
 <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> 2,<a href="topgen_3.html#K3" title="TOPGEN_3:func.3">continuum</a>  <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c2">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topgen_5.html#T20" target="_self" title="TOPGEN_5:th.20">Th20</a>, <a class="ref" href="card_2.html#T44" title="CARD_2:th.44">CARD_2:44</a></i>;<br/>
<a NAME="E17:54"/><b>then </b>
(  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> 2,<a href="topgen_3.html#K3" title="TOPGEN_3:func.3">continuum</a>  <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> 2,<span class="p1">(<span class="default"><a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp;  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> 2,<span class="p1">(<span class="default"><a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> 2,<span class="p1">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> 2,<span class="p2">(<span class="default"><a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="topgen_5.html#E12:54"><i><font color="Green" title="E43">A11</font></i></a>, <a class="ref" href="card_5.html#T23" title="CARD_5:th.23">CARD_5:23</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E18:54"/><b>then </b>
 <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> 2,<span class="p1">(<span class="default"><a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> 2,<span class="p1">(<span class="default"><a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="topgen_3.html#T29" title="TOPGEN_3:th.29">TOPGEN_3:29</a></i>;<br/>
<b>hence </b><a NAME="E19:54"/>
contradiction
 ;<br/>


</div>
