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<b>consider </b><font color="Maroon" title="c1">BB</font> being    <a href="topgen_2.html#M1" title="TOPGEN_2:mode.1">Neighborhood_System</a> of  <a href="topgen_5.html#K3" title="TOPGEN_5:func.3">Niemytzki-plane</a> <b> such that </b><br/><a NAME="E2:32"/>
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds  <font color="Maroon" title="c1">BB</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">r</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Olive" title="b2">r</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> where <font color="Olive" title="b2">r</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  : <font color="Olive" title="b2">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 </span>}</span> 
 <b>and </b><br/><a NAME="E3:32"/><i><font color="Green" title="E23">A1</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 holds <br/><font color="Maroon" title="c1">BB</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Olive" title="b3">r</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <a href="topgen_5.html#K2" title="TOPGEN_5:func.2">y&gt;=0-plane</a> </span>)</span> where <font color="Olive" title="b3">r</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  : <font color="Olive" title="b3">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 </span>}</span> 
 <b>by </b><i><a class="ref" href="topgen_5.html#D3" target="_self" title="TOPGEN_5:def.3">Def3</a></i>;<br/>
<a NAME="E4:32"/><i><font color="Green" title="E24">A2</font></i>: 
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>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topgen_5.html#K2" title="TOPGEN_5:func.2">y&gt;=0-plane</a> 
 
<b>by </b><i><a class="ref" href="topgen_5.html#D3" target="_self" title="TOPGEN_5:def.3">Def3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c2">A</font> = <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>  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#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">a</font> be   <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/><b>assume </b><a NAME="E1:32_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 ;<br/><a NAME="E2:32_1"/><b>then </b><i><font color="Green" title="E25">A3</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topgen_5.html#K2" title="TOPGEN_5:func.2">y&gt;=0-plane</a>  &amp; not <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topgen_5.html#K1" title="TOPGEN_5:func.1">y=0-line</a>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>then consider </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E4:32_1"/><i><font color="Green" title="E26">A4</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c5">y</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 )
 ;<br/><b>set </b><font color="Maroon" title="c6">B</font> = <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Maroon" title="c5">y</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <a href="topgen_5.html#K2" title="TOPGEN_5:func.2">y&gt;=0-plane</a> ;<br/><b>reconsider </b><font color="Maroon" title="c7">B</font> = <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Maroon" title="c5">y</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <a href="topgen_5.html#K2" title="TOPGEN_5:func.2">y&gt;=0-plane</a>  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="txt" href="topgen_5.html#E4:32"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><b>take </b><font color="Maroon" title="c8">B</font> = <font color="Maroon" title="c7">B</font>;<br/><a NAME="E6:32_1"/><i><font color="Green" title="E27">A5</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="topgen_5.html#E2:32_1"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="topgen_5.html#E4:32_1"><i><font color="Green" title="E26">A4</font></i></a></i>;<br/><a NAME="E7:32_1"/><b>then </b>
<font color="Maroon" title="c8">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Olive" title="b1">r</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <a href="topgen_5.html#K2" title="TOPGEN_5:func.2">y&gt;=0-plane</a> </span>)</span> where <font color="Olive" title="b1">r</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  : <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 </span>}</span> 
 <b>by </b><i><a class="txt" href="topgen_5.html#E4:32_1"><i><font color="Green" title="E26">A4</font></i></a></i>;<br/><a NAME="E8:32_1"/><b>then </b><i><font color="Green" title="E28">A6</font></i>: 
( <font color="Maroon" title="c8">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">BB</font> <a href="topgen_2.html#K4" title="TOPGEN_2:func.4">.</a> <font color="Maroon" title="c3">a</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">BB</font> <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="txt" href="topgen_5.html#E3:32"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="topgen_5.html#E4:32_1"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="topgen_5.html#E6:32_1"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><b>hence </b><a NAME="E9:32_1"/>
<font color="Maroon" title="c8">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topgen_2.html#K3" title="TOPGEN_2:func.3">Union</a> <font color="Maroon" title="c1">BB</font>
 <b>by </b><i><a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</a></i>;<br/><b>thus </b><a NAME="E10:32_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font>
 <b>by </b><i><a class="txt" href="topgen_5.html#E8:32_1"><i><font color="Green" title="E28">A6</font></i></a>, <a class="ref" href="yellow_8.html#T21" title="YELLOW_8:th.21">YELLOW_8:21</a></i>;<br/><a NAME="E11:32_1"/>
 <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Maroon" title="c5">y</font> <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> 
 <b>by </b><i><a class="txt" href="topgen_5.html#E4:32_1"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="topgen_5.html#E6:32_1"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="topgen_5.html#T24" target="_self" title="TOPGEN_5:th.24">Th24</a></i>;<br/><a NAME="E12:32_1"/><b>then </b>
<font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Maroon" title="c5">y</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/><a NAME="E13:32_1"/><b>then </b>
<font color="Maroon" title="c8">B</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <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#E4:32_1"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="topgen_5.html#E6:32_1"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="topgen_5.html#T25" target="_self" title="TOPGEN_5:th.25">Th25</a></i>;<br/><b>hence </b><a NAME="E14:32_1"/>
<font color="Maroon" title="c8">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="topgen_5.html#E4:32"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="xboole_1.html#T86" title="XBOOLE_1:th.86">XBOOLE_1:86</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:32"/>
<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="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>by </b><i><a class="ref" href="yellow_9.html#T31" title="YELLOW_9:th.31">YELLOW_9:31</a></i>;<br/>


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