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

<a NAME="E1:48"/>
( <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> <a href="tarski.html#R1">c=</a> <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>  &amp; <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>  is   <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#T29" target="_self">Th29</a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>1</sub></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="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="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E3:48"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="topgen_1.html#V2">dense-in-itself</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></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>,<a class="ref" href="topgen_1.html#D7">TOPGEN_1:def 7</a><br/>

<b>assume </b><a NAME="E1:48_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a>  ;<br/>
<a NAME="E3:48_1"/>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a> 
 
<b>by </b><i><a class="ref" href="topgen_5.html#T36" target="_self">Th36</a></i>;<br/>
<a NAME="E4:48_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="topgen_1.html#R1">is_an_accumulation_point_of</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="topgen_1.html#D2">TOPGEN_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:48_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topgen_1.html#K2">Der</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="topgen_1.html#D3">TOPGEN_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:48"/>
<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> is  <a href="topgen_1.html#V2">dense-in-itself</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a> 
 ;<br/>


</div>
