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<div class="add">

<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></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="E2:91_1_1"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  holds <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="euclid.html#K23">]|</a></span>,<font color="Olive">b<sub>2</sub></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">b<sub>1</sub></font>,0</span><a href="euclid.html#K23">]|</a></span></span><a href="domain_1.html#K6">}</a></span></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> 
 <b>and </b><br/><a NAME="E3:91_1_1"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="euclid.html#K23">]|</a></span>,<font color="Olive">b<sub>3</sub></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 B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> 
 <b>by </b><i><a class="ref" href="topgen_5.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E4:91_1_1"/><i><font color="Green">E71</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></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="txt" href="topgen_5.html#E33"><i><font color="Green">Lemma27</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></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="E1:91_1_1_1"/><i><font color="Green">E72</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></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">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></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="E1:91_1_1_1_1"/>
<font color="Maroon">c<sub>7</sub></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><a class="txt" href="topgen_5.html#E33"><i><font color="Green">Lemma27</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></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="E3:91_1_1_1_1"/>
<font color="Maroon">c<sub>8</sub></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">c<sub>8</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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="E4:91_1_1_1_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="topgen_5.html#T22" target="_self">Th22</a>, <a class="txt" href="topgen_5.html#E33"><i><font color="Green">Lemma27</font></i></a></i>;<br/><b>assume </b><a NAME="E5:91_1_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>6</sub></font></span><a href="rcomp_2.html#K1">.[</a></span>
 ;<br/><a NAME="E6:91_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Maroon">c<sub>6</sub></font></span><a href="rcomp_2.html#K1">.[</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><a NAME="E7:91_1_1_1_1"/><b>then </b><i><font color="Green">E74</font></i>: 
( ( <span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0 or 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> ) &amp; <span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_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="E1:91_1_1_1_1_1"/>
( <span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0 or ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="topgen_5.html#E7:91_1_1_1_1"><i><font color="Green">E74</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:91_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:91_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E75</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="euclid.html#K23">]|</a></span> &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:91_1_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="topgen_5.html#E4:91_1_1_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="topgen_5.html#T64" target="_self">Th64</a></i>;<br/><a NAME="E3:91_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a>, <a class="txt" href="topgen_5.html#E1:91_1_1_1"><i><font color="Green">E72</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font> 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>  ;<br/><a NAME="E5:91_1_1_1_1_1_1"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</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">c<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="euclid.html#K23">]|</a></span></span><a href="domain_1.html#K6">}</a></span> &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"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="euclid.html#K23">]|</a></span></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#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>6</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E2:91_1_1_1_1_1_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="topgen_5.html#T71" target="_self">Th71</a>, <a class="ref" href="setwiseo.html#T6">SETWISEO:6</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></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"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="euclid.html#K23">]|</a></span></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="ref" href="topgen_5.html#T31" target="_self">Th31</a></i>;<br/><b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font>;<br/><a NAME="E7:91_1_1_1_1_1_1"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> is   <a href="real_1.html#NM1">Real</a> &amp; <font color="Maroon">c<sub>1</sub></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="E8:91_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></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"><font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="euclid.html#K23">]|</a></span>,<font color="Olive">b<sub>1</sub></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="Maroon">c<sub>1</sub></font>,0</span><a href="euclid.html#K23">]|</a></span></span><a href="domain_1.html#K6">}</a></span></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> 
 ;<br/><a NAME="E9:91_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</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><a class="txt" href="topgen_5.html#E2:91_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="topgen_5.html#E7:91_1_1_1_1_1_1"><i><font color="Green">E77</font></i></a></i>;<br/><b>hence </b><a NAME="E10:91_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="topgen_2.html#K3">Union</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>6</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E5:91_1_1_1_1_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="topgen_5.html#E4:91_1_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:91_1_1_1_1_1_2"/>
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div class="add"><a NAME="E2:91_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E75</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp;  <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<font color="Maroon">c<sub>6</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:91_1_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="topgen_5.html#E4:91_1_1_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="topgen_5.html#T72" target="_self">Th72</a>, <a class="ref" href="topgen_5.html#T70" target="_self">Th70</a></i>;<br/><a NAME="E3:91_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font> 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>  ;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> =  <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">c<sub>9</sub></font> 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="ref" href="topgen_5.html#T33" target="_self">Th33</a></i>;<br/><b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font>;<br/><a NAME="E6:91_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</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#T24" target="_self">Th24</a></i>;<br/><a NAME="E7:91_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E76</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="subset_1.html#K9">/\</a> <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> is   <a href="real_1.html#NM1">Real</a> &amp; <font color="Maroon">c<sub>1</sub></font> is   <a href="real_1.html#NM1">Real</a> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a>, <a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E8:91_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></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"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="euclid.html#K23">]|</a></span>,<font color="Olive">b<sub>1</sub></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 B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> 
 ;<br/><a NAME="E9:91_1_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font></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><a class="txt" href="topgen_5.html#E3:91_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="topgen_5.html#E7:91_1_1_1_1_1_2"><i><font color="Green">E76</font></i></a></i>;<br/><b>hence </b><a NAME="E10:91_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="topgen_2.html#K3">Union</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="topgen_5.html#E4:91_1_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/><a NAME="E11:91_1_1_1_1_1_2"/>
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<font color="Maroon">c<sub>6</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Maroon">c<sub>6</sub></font></span><a href="rcomp_2.html#K1">.[</a></span>
 <b>by </b><i><a class="ref" href="topreala.html#T20">TOPREALA:20</a></i>;<br/><a NAME="E12:91_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<font color="Maroon">c<sub>6</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>6</sub></font></span><a href="rcomp_2.html#K1">.[</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#T178">RELAT_1:178</a></i>;<br/><b>hence </b><a NAME="E13:91_1_1_1_1_1_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>6</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E2:91_1_1_1_1_1_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:91_1_1_1"/>
<span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>6</sub></font></span><a href="rcomp_2.html#K1">.[</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><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></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="E1:91_1_1_1_2"/>
<font color="Maroon">c<sub>7</sub></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><a class="txt" href="topgen_5.html#E33"><i><font color="Green">Lemma27</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></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="E3:91_1_1_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>8</sub></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">c<sub>8</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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="E4:91_1_1_1_2"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="topgen_5.html#T22" target="_self">Th22</a>, <a class="txt" href="topgen_5.html#E33"><i><font color="Green">Lemma27</font></i></a></i>;<br/><b>assume </b><a NAME="E5:91_1_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>5</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span>
 ;<br/><a NAME="E6:91_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">c<sub>5</sub></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="E7:91_1_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></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="E8:91_1_1_1_2"/><b>then </b><i><font color="Green">E75</font></i>: 
( ( <span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></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#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> ) &amp; <span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>5</sub></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="91_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:91_1_1_1_2_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or ( <span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="topgen_5.html#E4:91_1_1_1_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="topgen_5.html#E8:91_1_1_1_2"><i><font color="Green">E75</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:91_1_1_1_2_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>9</sub></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="E3:91_1_1_1_2_1_1"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>5</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:91_1_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="topgen_5.html#E8:91_1_1_1_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="topgen_5.html#T73" target="_self">Th73</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> =  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> 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#E3:91_1_1_1_2_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="topgen_5.html#E3:91_1_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="topgen_5.html#T33" target="_self">Th33</a></i>;<br/><b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font>;<br/><a NAME="E5:91_1_1_1_2_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a>  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> 
 <b>by </b><i><a class="txt" href="topgen_5.html#E3:91_1_1_1_2_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="topgen_5.html#E3:91_1_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="topgen_5.html#T24" target="_self">Th24</a></i>;<br/><a NAME="E6:91_1_1_1_2_1_1"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> <a href="subset_1.html#K9">/\</a> <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> is   <a href="real_1.html#NM1">Real</a> &amp; <font color="Maroon">c<sub>1</sub></font> is   <a href="real_1.html#NM1">Real</a> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a>, <a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E7:91_1_1_1_2_1_1"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></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">c<sub>8</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>,<font color="Olive">b<sub>1</sub></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 B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="topgen_5.html#E3:91_1_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="topgen_5.html#E3:91_1_1_1_2_1_1"><i><font color="Green">E76</font></i></a></i>;<br/><a NAME="E8:91_1_1_1_2_1_1"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></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">c<sub>8</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>8</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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><a class="txt" href="topgen_5.html#E3:91_1_1"><i><font color="Green">E70</font></i></a></i>;<br/><b>hence </b><a NAME="E9:91_1_1_1_2_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="topgen_2.html#K3">Union</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="topgen_5.html#E4:91_1_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/><b>thus </b><a NAME="E10:91_1_1_1_2_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>5</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E3:91_1_1_1_2_1_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="topgen_5.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:91_1_1_1_2_1_2"/><i><font color="Green">E76</font></i>: 
( <span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>8</sub></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">c<sub>9</sub></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="E3:91_1_1_1_2_1_2"/><i><font color="Green">E77</font></i>: 
<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">c<sub>8</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">c<sub>9</sub></font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">c<sub>9</sub></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">c<sub>8</sub></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#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="topgen_5.html#T74" target="_self">Th74</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></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">c<sub>8</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">c<sub>9</sub></font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">c<sub>9</sub></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">c<sub>8</sub></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#E3:91_1_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="topgen_5.html#E1:91_1_1_1_2_1_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="topgen_5.html#T31" target="_self">Th31</a></i>;<br/><b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font>;<br/><a NAME="E5:91_1_1_1_2_1_2"/>
<font color="Maroon">c<sub>9</sub></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="E6:91_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></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">c<sub>8</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Olive">b<sub>1</sub></font></span><a href="euclid.html#K23">]|</a></span>,<font color="Olive">b<sub>1</sub></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">c<sub>8</sub></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 B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Olive">b<sub>1</sub></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#E3:91_1_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="topgen_5.html#E1:91_1_1_1_2_1_2"><i><font color="Green">E76</font></i></a></i>;<br/><a NAME="E7:91_1_1_1_2_1_2"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></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">c<sub>8</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>8</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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><a class="txt" href="topgen_5.html#E1:91_1_1_1_2_1_2"><i><font color="Green">E76</font></i></a>, <a class="txt" href="topgen_5.html#E2:91_1_1"><i><font color="Green">E69</font></i></a></i>;<br/><b>hence </b><a NAME="E8:91_1_1_1_2_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="topgen_2.html#K3">Union</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="topgen_5.html#E4:91_1_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/><b>thus </b><a NAME="E9:91_1_1_1_2_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="setwiseo.html#T6">SETWISEO:6</a></i>;<br/><a NAME="E10:91_1_1_1_2_1_2"/>
1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span>
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:91_1_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/><a NAME="E11:91_1_1_1_2_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E12:91_1_1_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</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#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>5</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#T178">RELAT_1:178</a></i>;<br/><b>hence </b><a NAME="E13:91_1_1_1_2_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>5</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span>
 <b>by </b><i><a class="txt" href="topgen_5.html#E3:91_1_1_1_2_1_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_2_1_3"><b>suppose </b></a><a NAME="E1:91_1_1_1_2_1_3"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 ;<br/><div class="add"><a NAME="E2:91_1_1_1_2_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,1</span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></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="E4:91_1_1_1_2_1_3"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,1</span><a href="rcomp_1.html#K2">.[</a></span> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:91_1_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="topgen_5.html#E4:91_1_1_1_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="topgen_5.html#T69" target="_self">Th69</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> =  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> 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#E4:91_1_1_1_2_1_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="topgen_5.html#E3:91_1_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="topgen_5.html#T33" target="_self">Th33</a></i>;<br/><b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font>;<br/><a NAME="E6:91_1_1_1_2_1_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a>  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> 
 <b>by </b><i><a class="txt" href="topgen_5.html#E4:91_1_1_1_2_1_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="topgen_5.html#E3:91_1_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="topgen_5.html#T24" target="_self">Th24</a></i>;<br/><a NAME="E7:91_1_1_1_2_1_3"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> <a href="subset_1.html#K9">/\</a> <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> is   <a href="real_1.html#NM1">Real</a> &amp; <font color="Maroon">c<sub>1</sub></font> is   <a href="real_1.html#NM1">Real</a> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a>, <a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E8:91_1_1_1_2_1_3"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></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">c<sub>8</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>,<font color="Olive">b<sub>1</sub></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 B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="topgen_5.html#E3:91_1_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="topgen_5.html#E4:91_1_1_1_2_1_3"><i><font color="Green">E76</font></i></a></i>;<br/><a NAME="E9:91_1_1_1_2_1_3"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></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">c<sub>8</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>8</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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><a class="txt" href="topgen_5.html#E3:91_1_1"><i><font color="Green">E70</font></i></a></i>;<br/><b>hence </b><a NAME="E10:91_1_1_1_2_1_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="topgen_2.html#K3">Union</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="topgen_5.html#E4:91_1_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/><a NAME="E11:91_1_1_1_2_1_3"/>
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,1</span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span>
 <b>by </b><i><a class="ref" href="topreala.html#T21">TOPREALA:21</a></i>;<br/><a NAME="E12:91_1_1_1_2_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,1</span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>5</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#T178">RELAT_1:178</a></i>;<br/><b>hence </b><a NAME="E13:91_1_1_1_2_1_3"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>5</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E4:91_1_1_1_2_1_3"><i><font color="Green">E76</font></i></a>, <a class="ref" href="topgen_5.html#T17" target="_self">Th17</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:91_1_1_1"/>
<span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>5</sub></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="E6:91_1_1"/>
 <a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="topgen_5.html#T79" target="_self">Th79</a></i>;<br/>


</div>
