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

<b>set </b><font color="Maroon">f</font> =  <a href="topgen_5.html#K5" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">r</font>;<br/>
<b>consider </b><font color="Maroon">BB</font> being    <a href="topgen_2.html#M1">Neighborhood_System</a> of  <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a> <b> such that </b><br/><a NAME="E2:98_1_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  holds  <font color="Maroon">BB</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K1">Ball</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">q</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Olive">q</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,0</span><a href="euclid.html#K23">]|</a></span></span><a href="domain_1.html#K6">}</a></span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> 
 <b>and </b><br/><a NAME="E3:98_1_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>   st <font color="Olive">y</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/><font color="Maroon">BB</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K1">Ball</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Olive">q</font></span>)</span> <a href="subset_1.html#K9">/\</a> <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> </span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> 
 <b>by </b><i/>;<br/>
<a NAME="E4:98_1_1"/><i><font color="Green">E49</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">BB</font> <a href="hidden.html#R1">=</a>  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> 
 
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">b</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:98_1_1_1"/><i><font color="Green">E53</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><a NAME="E2:98_1_1_1"/><b>then </b>
( <font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> 0 &amp; <span class="p1">(<span class="default"><a href="topgen_5.html#K5" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">r</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Maroon">b</font></span><a href="rcomp_2.html#K1">.[</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1">Ball</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  )
 <b>by </b><i><a class="ref" href="topgen_5.html#T12" target="_self">Th12</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><b>hence </b><a NAME="E3:98_1_1_1"/>
<span class="p1">(<span class="default"><a href="topgen_5.html#K5" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">r</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Maroon">b</font></span><a href="rcomp_2.html#K1">.[</a></span> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a> ;<br/><a NAME="E1:98_1_1_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> 
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">c</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E3:98_1_1_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/><a NAME="E4:98_1_1_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">z</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, </i>;<br/><b>assume </b><a NAME="E5:98_1_1_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K5" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">r</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">a</font>,1</span><a href="rcomp_2.html#K2">.]</a></span>
 ;<br/><a NAME="E6:98_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topgen_5.html#K5" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">a</font>,1</span><a href="rcomp_2.html#K2">.]</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><a NAME="E7:98_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="topgen_5.html#K5" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> &amp; <span class="p1">(<span class="default"><a href="topgen_5.html#K5" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/><a NAME="E8:98_1_1_1_1"/><b>then </b><i><font color="Green">E61</font></i>: 
( ( <span class="p1">(<span class="default"><a href="topgen_5.html#K5" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 1 or 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K5" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> ) &amp; <span class="p1">(<span class="default"><a href="topgen_5.html#K5" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:98_1_1_1_1_1"/>
( <font color="Maroon">z</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">z</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:98_1_1_1_1_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">z</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:98_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E63</font></i>: 
( <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <font color="Maroon">z</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font> &amp; <span class="p1">(<span class="default"><a href="topreal9.html#K1">Ball</a> <font color="Maroon">z</font>,<span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <font color="Maroon">z</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K5" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">r</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">a</font>,1</span><a href="rcomp_2.html#K2">.]</a></span> )
 <b>by </b><i>, , </i>;<br/><b>then reconsider </b><font color="Maroon">r1</font> = <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <font color="Maroon">z</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> as  <a href="xxreal_0.html#V2">positive</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i><a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><b>reconsider </b><font color="Maroon">U</font> = <span class="p1">(<span class="default"><a href="topreal9.html#K1">Ball</a> <font color="Maroon">z</font>,<font color="Maroon">r1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  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>, , </i>;<br/><b>take </b><font color="Maroon">U</font> = <font color="Maroon">U</font>;<br/><a NAME="E5:98_1_1_1_1_1_1"/>
( <font color="Maroon">r1</font> is   <a href="real_1.html#NM1">Real</a> &amp; <font color="Maroon">x</font> is   <a href="real_1.html#NM1">Real</a> )
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E6:98_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">U</font> <a href="hidden.html#R2">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">Ball</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>,<font color="Olive">q</font></span>)</span> <a href="subset_1.html#K9">/\</a> <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> </span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E7:98_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BB</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E8:98_1_1_1_1_1_1"/>
<font color="Maroon">U</font> <a href="hidden.html#R2">in</a>  <a href="topgen_2.html#K3">Union</a> <font color="Maroon">BB</font>
 <b>by </b><i>, <a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/><a NAME="E9:98_1_1_1_1_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">z</font>,<font color="Maroon">r1</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E10:98_1_1_1_1_1_1"/>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font> &amp; <font color="Maroon">U</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K5" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">r</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">a</font>,1</span><a href="rcomp_2.html#K2">.]</a></span> )
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:98_1_1_1_1_1_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">z</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">r1</font> being  <a href="xxreal_0.html#V2">positive</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:98_1_1_1_1_1_2"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <font color="Maroon">z</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 &amp; <span class="p1">(<span class="default"><a href="topreal9.html#K1">Ball</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">r1</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">r1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">z</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K5" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">r</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">a</font>,1</span><a href="rcomp_2.html#K2">.]</a></span> )
 <b>by </b><i>, , </i>;<br/><b>reconsider </b><font color="Maroon">U</font> = <span class="p1">(<span class="default"><a href="topreal9.html#K1">Ball</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">r1</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">r1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">z</font></span><a href="domain_1.html#K6">}</a></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="txt" href="topgen_5.html#E1:98_1_1_1_1_1_2"><i><font color="Green">E79</font></i></a>, , </i>;<br/><b>take </b><font color="Maroon">U</font> = <font color="Maroon">U</font>;<br/><a NAME="E5:98_1_1_1_1_1_2"/>
<font color="Maroon">r1</font> is   <a href="real_1.html#NM1">Real</a>
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E6:98_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">U</font> <a href="hidden.html#R2">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">Ball</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Olive">q</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Olive">q</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span>,0</span><a href="euclid.html#K23">]|</a></span></span><a href="domain_1.html#K6">}</a></span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> 
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:98_1_1_1_1_1_2"><i><font color="Green">E79</font></i></a>, </i>;<br/><a NAME="E7:98_1_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BB</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E8:98_1_1_1_1_1_2"/>
<font color="Maroon">U</font> <a href="hidden.html#R2">in</a>  <a href="topgen_2.html#K3">Union</a> <font color="Maroon">BB</font>
 <b>by </b><i>, <a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/><b>thus </b><a NAME="E9:98_1_1_1_1_1_2"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font>
 <b>by </b><i><a class="ref" href="setwiseo.html#T6">SETWISEO:6</a></i>;<br/><b>thus </b><a NAME="E10:98_1_1_1_1_1_2"/>
<font color="Maroon">U</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K5" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">r</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">a</font>,1</span><a href="rcomp_2.html#K2">.]</a></span>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:98_1_1_1"/>
<span class="p1">(<span class="default"><a href="topgen_5.html#K5" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">r</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">a</font>,1</span><a href="rcomp_2.html#K2">.]</a></span> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="yellow_9.html#T31">YELLOW_9:31</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:98_1_1"/>
 <a href="topgen_5.html#K5" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">r</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i/>;<br/>


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