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

<b>let </b><font color="Maroon" title="c1">x</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a> <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <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="c1">x</font>,<font color="Olive" title="b1">r</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#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">x</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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> 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> </span><br/><b>let </b><font color="Maroon" title="c2">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a> <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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="c1">x</font>,<font color="Maroon" title="c2">r</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Maroon" title="c2">r</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">x</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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> 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> </span><br/>



<a NAME="E1:32"/>
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="c3">a</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">x</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> 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>  <b>by </b><i><a class="ref" href="topgen_5.html#T22" target="_self" title="TOPGEN_5:th.22">Th22</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">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="E4:32"/><i><font color="Green" title="E26">A1</font></i>: 
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="c4">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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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">q</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Olive" title="b2">q</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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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">q</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">q</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> 
 <b>and </b><br/><a NAME="E5:32"/>
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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Maroon" title="c4">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">q</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">q</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">q</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </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="E6:32"/><i><font color="Green" title="E27">A2</font></i>: 
( <font color="Maroon" title="c1">x</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> &amp; <font color="Maroon" title="c2">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> )
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E7:32"/><b>then </b>
<font color="Maroon" title="c4">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="Maroon" title="c1">x</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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="Maroon" title="c1">x</font>,<font color="Olive" title="b1">q</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Olive" title="b1">q</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="Maroon" title="c1">x</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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="b1">q</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">q</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> 
 
<b>by </b><i><a class="txt" href="topgen_5.html#E4:32"><i><font color="Green" title="E26">A1</font></i></a></i>;<br/>
<a NAME="E8:32"/><b>then </b>
<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="c1">x</font>,<font color="Maroon" title="c2">r</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Maroon" title="c2">r</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">x</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">BB</font> <a href="topgen_2.html#K4" title="TOPGEN_2:func.4">.</a> <font color="Maroon" title="c3">a</font>
 
<b>by </b><i><a class="txt" href="topgen_5.html#E6:32"><i><font color="Green" title="E27">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:32"/>
<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="c1">x</font>,<font color="Maroon" title="c2">r</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Maroon" title="c2">r</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">x</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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> 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_8.html#T21" title="YELLOW_8:th.21">YELLOW_8:21</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
