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<b>set </b><font color="Maroon" title="c4">f</font> =  <a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">r</font>;<br/>
<b>consider </b><font color="Maroon" title="c5">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="E2:94_1_1"/><i><font color="Green" title="E71">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="c5">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="E3:94_1_1"/><i><font color="Green" title="E72">A2</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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Maroon" title="c5">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="E4:94_1_1"/><i><font color="Green" title="E73">A3</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">BB</font> <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="txt" href="topgen_5.html#E33"><i><font color="Green" title="E23">Lm1</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</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> ;<br/><b>assume </b><a NAME="E1:94_1_1_1"/><i><font color="Green" title="E74">A4</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c6">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c7">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/><a NAME="E2:94_1_1_1"/><b>then </b>
( <font color="Maroon" title="c3">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c7">b</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">b</font></span>)</span></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>  )
 <b>by </b><i><a class="ref" href="topgen_5.html#T82" target="_self" title="TOPGEN_5:th.82">Th82</a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><b>hence </b><a NAME="E3:94_1_1_1"/>
<span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c7">b</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="topgen_5.html#T32" target="_self" title="TOPGEN_5:th.32">Th32</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="topgen_5.html#K3" title="TOPGEN_5:func.3">Niemytzki-plane</a> ;<br/><a NAME="E1:94_1_1_1_1"/>
<font color="Maroon" title="c8">c</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> 
 <b>by </b><i><a class="txt" href="topgen_5.html#E33"><i><font color="Green" title="E23">Lm1</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c9">z</font> = <font color="Maroon" title="c8">c</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E3:94_1_1_1_1"/><i><font color="Green" title="E75">A5</font></i>: 
<font color="Maroon" title="c9">z</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"><span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/><b>assume </b><a NAME="E4:94_1_1_1_1"/>
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><font color="Maroon" title="c6">a</font>,1</span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span>
 ;<br/><a NAME="E5:94_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><font color="Maroon" title="c6">a</font>,1</span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/><a NAME="E6:94_1_1_1_1"/><b>then </b><i><font color="Green" title="E76">A6</font></i>: 
( <span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">a</font> &amp; <span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="ref" href="rcomp_2.html#T4" title="RCOMP_2:th.4">RCOMP_2:4</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_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="E1:94_1_1_1_1_1"/>
( <font color="Maroon" title="c9">z</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c9">z</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span><b>by </b><i><a class="txt" href="topgen_5.html#E3:94_1_1_1_1"><i><font color="Green" title="E75">A5</font></i></a>, <a class="txt" href="topgen_5.html#E33"><i><font color="Green" title="E23">Lm1</font></i></a>, <a class="ref" href="topgen_5.html#T22" target="_self" title="TOPGEN_5:th.22">Th22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:94_1_1_1_1_1_1"/><i><font color="Green" title="E77">A7</font></i>: 
<font color="Maroon" title="c9">z</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:94_1_1_1_1_1_1"/><b>then </b>
( <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><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">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c9">z</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">a</font> &amp; <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c9">z</font>,<span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5"><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">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c9">z</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span></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>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><font color="Maroon" title="c6">a</font>,1</span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:94_1_1_1"><i><font color="Green" title="E74">A4</font></i></a>, <a class="txt" href="topgen_5.html#E6:94_1_1_1_1"><i><font color="Green" title="E76">A6</font></i></a>, <a class="ref" href="topgen_5.html#T83" target="_self" title="TOPGEN_5:th.83">Th83</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">r1</font> = <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><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">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c9">z</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">a</font></span>)</span> as  <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>  <b>by </b><i><a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">U</font> = <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c9">z</font>,<font color="Maroon" title="c10">r1</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#E3:94_1_1_1_1"><i><font color="Green" title="E75">A5</font></i></a>, <a class="txt" href="topgen_5.html#E1:94_1_1_1_1_1_1"><i><font color="Green" title="E77">A7</font></i></a>, <a class="ref" href="topgen_5.html#T32" target="_self" title="TOPGEN_5:th.32">Th32</a></i>;<br/><b>take </b><font color="Maroon" title="c12">U</font> = <font color="Maroon" title="c11">U</font>;<br/><a NAME="E5:94_1_1_1_1_1_1"/>
( <font color="Maroon" title="c10">r1</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> &amp; <font color="Maroon" title="c1">x</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="E6:94_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">U</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"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></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#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">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#E3:94_1_1_1_1"><i><font color="Green" title="E75">A5</font></i></a></i>;<br/><a NAME="E7:94_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c12">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">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"><span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <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>  )
 <b>by </b><i><a class="txt" href="topgen_5.html#E3:94_1_1"><i><font color="Green" title="E72">A2</font></i></a>, <a class="txt" href="topgen_5.html#E1:94_1_1_1_1_1_1"><i><font color="Green" title="E77">A7</font></i></a></i>;<br/><b>hence </b><a NAME="E8:94_1_1_1_1_1_1"/>
<font color="Maroon" title="c12">U</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="c5">BB</font>
 <b>by </b><i><a class="txt" href="topgen_5.html#E4:94_1_1"><i><font color="Green" title="E73">A3</font></i></a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</a></i>;<br/><a NAME="E9:94_1_1_1_1_1_1"/>
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c9">z</font>,<font color="Maroon" title="c10">r1</font>
 <b>by </b><i><a class="ref" href="topgen_5.html#T17" target="_self" title="TOPGEN_5:th.17">Th17</a></i>;<br/><b>hence </b><a NAME="E10:94_1_1_1_1_1_1"/>
( <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">U</font> &amp; <font color="Maroon" title="c12">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><font color="Maroon" title="c6">a</font>,1</span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:94_1_1_1"><i><font color="Green" title="E74">A4</font></i></a>, <a class="txt" href="topgen_5.html#E6:94_1_1_1_1"><i><font color="Green" title="E76">A6</font></i></a>, <a class="txt" href="topgen_5.html#E1:94_1_1_1_1_1_1"><i><font color="Green" title="E77">A7</font></i></a>, <a class="txt" href="topgen_5.html#E33"><i><font color="Green" title="E23">Lm1</font></i></a>, <a class="ref" href="topgen_5.html#T83" target="_self" title="TOPGEN_5:th.83">Th83</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:94_1_1_1_1_1_2"/><i><font color="Green" title="E77">A8</font></i>: 
<font color="Maroon" title="c9">z</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c10">r1</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> <b> such that </b><br/><a NAME="E3:94_1_1_1_1_1_2"/><i><font color="Green" title="E78">A9</font></i>: 
( <font color="Maroon" title="c10">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p3">(<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>,<font color="Maroon" title="c2">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c9">z</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp; <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"><span class="p3">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<font color="Maroon" title="c10">r1</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Maroon" title="c10">r1</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"><font color="Maroon" title="c9">z</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><font color="Maroon" title="c6">a</font>,1</span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:94_1_1_1"><i><font color="Green" title="E74">A4</font></i></a>, <a class="txt" href="topgen_5.html#E6:94_1_1_1_1"><i><font color="Green" title="E76">A6</font></i></a>, <a class="ref" href="topgen_5.html#T84" target="_self" title="TOPGEN_5:th.84">Th84</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">U</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"><span class="p3">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<font color="Maroon" title="c10">r1</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Maroon" title="c10">r1</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"><font color="Maroon" title="c9">z</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></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="txt" href="topgen_5.html#E3:94_1_1_1_1"><i><font color="Green" title="E75">A5</font></i></a>, <a class="txt" href="topgen_5.html#E1:94_1_1_1_1_1_2"><i><font color="Green" title="E77">A8</font></i></a>, <a class="ref" href="topgen_5.html#T31" target="_self" title="TOPGEN_5:th.31">Th31</a></i>;<br/><b>take </b><font color="Maroon" title="c12">U</font> = <font color="Maroon" title="c11">U</font>;<br/><a NAME="E5:94_1_1_1_1_1_2"/>
<font color="Maroon" title="c10">r1</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="E6:94_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c12">U</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"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<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"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<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#E3:94_1_1_1_1"><i><font color="Green" title="E75">A5</font></i></a>, <a class="txt" href="topgen_5.html#E1:94_1_1_1_1_1_2"><i><font color="Green" title="E77">A8</font></i></a></i>;<br/><a NAME="E7:94_1_1_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c12">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">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"><span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <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>  )
 <b>by </b><i><a class="txt" href="topgen_5.html#E2:94_1_1"><i><font color="Green" title="E71">A1</font></i></a>, <a class="txt" href="topgen_5.html#E1:94_1_1_1_1_1_2"><i><font color="Green" title="E77">A8</font></i></a></i>;<br/><b>hence </b><a NAME="E8:94_1_1_1_1_1_2"/>
<font color="Maroon" title="c12">U</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="c5">BB</font>
 <b>by </b><i><a class="txt" href="topgen_5.html#E4:94_1_1"><i><font color="Green" title="E73">A3</font></i></a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</a></i>;<br/><b>thus </b><a NAME="E9:94_1_1_1_1_1_2"/>
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">U</font>
 <b>by </b><i><a class="ref" href="setwiseo.html#T6" title="SETWISEO:th.6">SETWISEO:6</a></i>;<br/><b>thus </b><a NAME="E10:94_1_1_1_1_1_2"/>
<font color="Maroon" title="c12">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><font color="Maroon" title="c6">a</font>,1</span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span>
 <b>by </b><i><a class="txt" href="topgen_5.html#E3:94_1_1_1_1_1_2"><i><font color="Green" title="E78">A9</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:94_1_1_1"/>
<span class="p1">(<span class="default"><a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><font color="Maroon" title="c6">a</font>,1</span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="yellow_9.html#T31" title="YELLOW_9:th.31">YELLOW_9:31</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:94_1_1"/>
 <a href="topgen_5.html#K5" title="TOPGEN_5:func.5">+</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">r</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="topgen_5.html#T79" target="_self" title="TOPGEN_5:th.79">Th79</a></i>;<br/>


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