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<a NAME="E1:44"/>
( <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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <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>  &amp; <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>  is   <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#T29" target="_self" title="TOPGEN_5:th.29">Th29</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c1">A</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="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="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E3:44"/>
<font color="Maroon" title="c1">A</font> is <a href="topgen_1.html#V2" title="TOPGEN_1:attr.2">dense-in-itself</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">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>,<a class="ref" href="topgen_1.html#D7" title="TOPGEN_1:def.7">TOPGEN_1:def 7</a><br/>

<b>assume </b><a NAME="E1:44_1"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c3">x</font> = <font color="Maroon" title="c2">a</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topgen_5.html#K3" title="TOPGEN_5:func.3">Niemytzki-plane</a>  ;<br/>
<a NAME="E3:44_1"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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#T36" target="_self" title="TOPGEN_5:th.36">Th36</a></i>;<br/>
<a NAME="E4:44_1"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="topgen_1.html#R1" title="TOPGEN_1:pred.1">is_an_accumulation_point_of</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="topgen_1.html#D2" title="TOPGEN_1:def.2">TOPGEN_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:44_1"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topgen_1.html#K2" title="TOPGEN_1:func.2">Der</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="ref" href="topgen_1.html#D3" title="TOPGEN_1:def.3">TOPGEN_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:44"/>
<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> is  <a href="topgen_1.html#V2" title="TOPGEN_1:attr.2">dense-in-itself</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> 
 ;<br/>


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