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

<span class="kw">set </span><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/>
<span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E2:90_1_1"/><span class="lab"><font color="Green" title="E71">A1</font></span>: 
 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#K19" title="EUCLID:func.19">|[</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#K19" title="EUCLID:func.19">]|</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#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">q</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</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#K19" title="EUCLID:func.19">|[</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#K19" title="EUCLID:func.19">]|</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> 
 <span class="kw">and </span><br/><a NAME="E3:90_1_1"/><span class="lab"><font color="Green" title="E72">A2</font></span>: 
 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#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</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#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topgen_5.html#D3" target="_self" title="TOPGEN_5:def.3">Def3</a></span>;<br/>
<a NAME="E4:90_1_1"/><span class="lab"><font color="Green" title="E73">A3</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="topgen_5.html#E29"><span class="lab"><font color="Green" title="E23">Lm1</font></span></a>, <a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <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 implies ( <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#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</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_1.html#K3" title="RCOMP_1:func.3">.[</a></span> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</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#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c6">a</font>,1</span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:90_1_1_1"/><span class="lab"><font color="Green" title="E74">A4</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E2:90_1_1_1"/>
<font color="Maroon" title="c6">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <span class="kw">and </span><br/><a NAME="E3:90_1_1_1"/><span class="lab"><font color="Green" title="E75">A5</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E4:90_1_1_1"/><span class="lab"><font color="Green" title="E76">A6</font></span>: 
<font color="Maroon" title="c7">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</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_1.html#K3" title="RCOMP_1:func.3">.[</a></span> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</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#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c6">a</font>,1</span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/><a NAME="E5:90_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#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</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_1.html#K3" title="RCOMP_1:func.3">.[</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#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_1_1_1"><span class="lab"><font color="Green" title="E75">A5</font></span></a>, <a class="txt" href="#E4:90_1_1_1"><span class="lab"><font color="Green" title="E76">A6</font></span></a>, <a class="ref" href="topgen_5.html#T78" target="_self" title="TOPGEN_5:th.78">Th82</a></span>;<br/><span class="kw">hence </span><a NAME="E6:90_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#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</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_1.html#K3" title="RCOMP_1:func.3">.[</a></span> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_1_1_1"><span class="lab"><font color="Green" title="E75">A5</font></span></a>, <a class="ref" href="topgen_5.html#T28" target="_self" title="TOPGEN_5:th.28">Th32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c6">a</font>,1</span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c6">a</font>,1</span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> implies  ex <font color="Olive" title="b1">U</font> being   <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> st <br/>( <font color="Olive">b<sub>2</sub></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> &amp; <font color="Olive" title="b1">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></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#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c6">a</font>,1</span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> ) )</span><br/><a NAME="E1:90_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="topgen_5.html#E29"><span class="lab"><font color="Green" title="E23">Lm1</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><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="E3:90_1_1_1_1"/><span class="lab"><font color="Green" title="E77">A7</font></span>: 
<font color="Maroon" title="c9">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" title="EUCLID:th.53">EUCLID:53</a></span>;<br/><span class="kw">assume </span><a NAME="E4:90_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#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c6">a</font>,1</span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">U</font> being   <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> st <br/>( <font color="Olive">b<sub>2</sub></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> &amp; <font color="Olive" title="b1">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></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#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c6">a</font>,1</span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> )</span><br/><a NAME="E5:90_1_1_1_1"/><span class="kw">then </span>
<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="funct_2.html#K3" title="FUNCT_2:func.3">.</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_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c6">a</font>,1</span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D7" title="FUNCT_1:def.7">FUNCT_1:def 7</a></span>;<br/><a NAME="E6:90_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A8</font></span>: 
<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="funct_2.html#K3" title="FUNCT_2:func.3">.</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T2" title="XXREAL_1:th.2">XXREAL_1:2</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:90_1_1_1_1_1"/>
( <font color="Maroon" title="c9">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`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#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_1_1_1_1"><span class="lab"><font color="Green" title="E77">A7</font></span></a>, <a class="txt" href="topgen_5.html#E29"><span class="lab"><font color="Green" title="E23">Lm1</font></span></a>, <a class="ref" href="topgen_5.html#T18" target="_self" title="TOPGEN_5:th.18">Th22</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:90_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E79">A9</font></span>: 
<font color="Maroon" title="c9">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">U</font> being   <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> st <br/>( <font color="Olive">b<sub>2</sub></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> &amp; <font color="Olive" title="b1">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></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#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c6">a</font>,1</span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><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#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_1_1_1"><span class="lab"><font color="Green" title="E74">A4</font></span></a>, <a class="txt" href="#E6:90_1_1_1_1"><span class="lab"><font color="Green" title="E78">A8</font></span></a>, <a class="ref" href="topgen_5.html#T79" target="_self" title="TOPGEN_5:th.79">Th83</a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_1_1_1_1"><span class="lab"><font color="Green" title="E77">A7</font></span></a>, <a class="txt" href="#E1:90_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E79">A9</font></span></a>, <a class="ref" href="topgen_5.html#T28" target="_self" title="TOPGEN_5:th.28">Th32</a></span>;<br/><a NAME="E4:90_1_1_1_1_1_1"/>
<font color="Maroon" title="c11">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#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_1_1_1_1"><span class="lab"><font color="Green" title="E77">A7</font></span></a></span>;<br/><a NAME="E5:90_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A10</font></span>: 
<font color="Maroon" title="c11">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#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_1_1"><span class="lab"><font color="Green" title="E72">A2</font></span></a>, <a class="txt" href="#E1:90_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E79">A9</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c12">U</font> = <font color="Maroon" title="c11">U</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; <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#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c6">a</font>,1</span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> )</span><br/><a NAME="E6:90_1_1_1_1_1_1"/>
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E79">A9</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:90_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:90_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E80">A10</font></span></a>, <a class="txt" href="#E4:90_1_1"><span class="lab"><font color="Green" title="E73">A3</font></span></a>, <a class="ref" href="card_5.html#T2" title="CARD_5:th.2">CARD_5:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c6">a</font>,1</span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> )</span><br/><a NAME="E8:90_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>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topgen_5.html#T13" target="_self" title="TOPGEN_5:th.13">Th17</a></span>;<br/><span class="kw">hence </span><a NAME="E9:90_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#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c6">a</font>,1</span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_1_1_1"><span class="lab"><font color="Green" title="E74">A4</font></span></a>, <a class="txt" href="#E6:90_1_1_1_1"><span class="lab"><font color="Green" title="E78">A8</font></span></a>, <a class="txt" href="#E1:90_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E79">A9</font></span></a>, <a class="txt" href="topgen_5.html#E29"><span class="lab"><font color="Green" title="E23">Lm1</font></span></a>, <a class="ref" href="topgen_5.html#T79" target="_self" title="TOPGEN_5:th.79">Th83</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:90_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E79">A11</font></span>: 
<font color="Maroon" title="c9">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">U</font> being   <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> st <br/>( <font color="Olive">b<sub>2</sub></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> &amp; <font color="Olive" title="b1">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></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#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c6">a</font>,1</span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:90_1_1_1_1_1_2"/>
<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#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</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
 <span class="kw">and </span><br/><a NAME="E4:90_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E80">A12</font></span>: 
<span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<font color="Maroon" title="c10">r1</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</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#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c6">a</font>,1</span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_1_1_1"><span class="lab"><font color="Green" title="E74">A4</font></span></a>, <a class="txt" href="#E6:90_1_1_1_1"><span class="lab"><font color="Green" title="E78">A8</font></span></a>, <a class="ref" href="topgen_5.html#T80" target="_self" title="TOPGEN_5:th.80">Th84</a></span>;<br/><span class="kw">reconsider </span><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#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<font color="Maroon" title="c10">r1</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_1_1_1_1"><span class="lab"><font color="Green" title="E77">A7</font></span></a>, <a class="txt" href="#E1:90_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E79">A11</font></span></a>, <a class="ref" href="topgen_5.html#T27" target="_self" title="TOPGEN_5:th.27">Th31</a></span>;<br/><a NAME="E6:90_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><a NAME="E7:90_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">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#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<font color="Olive" title="b1">q</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</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#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_1_1_1_1"><span class="lab"><font color="Green" title="E77">A7</font></span></a>, <a class="txt" href="#E1:90_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E79">A11</font></span></a></span>;<br/><a NAME="E8:90_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A13</font></span>: 
<font color="Maroon" title="c11">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#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:90_1_1"><span class="lab"><font color="Green" title="E71">A1</font></span></a>, <a class="txt" href="#E1:90_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E79">A11</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c12">U</font> = <font color="Maroon" title="c11">U</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; <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#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c6">a</font>,1</span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> )</span><br/><a NAME="E9:90_1_1_1_1_1_2"/>
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E79">A11</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E10:90_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:90_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E81">A13</font></span></a>, <a class="txt" href="#E4:90_1_1"><span class="lab"><font color="Green" title="E73">A3</font></span></a>, <a class="ref" href="card_5.html#T2" title="CARD_5:th.2">CARD_5:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c6">a</font>,1</span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> )</span><br/><span class="kw">thus </span><a NAME="E11:90_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setwiseo.html#T4" title="SETWISEO:th.4">SETWISEO:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c6">a</font>,1</span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span></span><br/><span class="kw">thus </span><a NAME="E12:90_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#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c6">a</font>,1</span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:90_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E80">A12</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E8:90_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#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c6">a</font>,1</span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#T31" title="YELLOW_9:th.31">YELLOW_9:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:90_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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topgen_5.html#T75" target="_self" title="TOPGEN_5:th.75">Th79</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
