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

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a> ;<br/>

<b>assume </b><a NAME="E1:43"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  <a href="subset_1.html#K7">\</a> <a href="topgen_5.html#K1" target="_self">y=0-line</a> </span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><a href="numbers.html#K3">RAT</a> ,<a href="numbers.html#K3">RAT</a> </span><a href="finseq_1.html#K10">*&gt;</a></span></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E2:43"/>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E3:43"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a>  ;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">BB</font> being    <a href="topgen_2.html#M1">Neighborhood_System</a> of  <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a> <b> such that </b><br/><a NAME="E7:43"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  holds  <font color="Maroon">BB</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K1">Ball</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">q</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Olive">q</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,0</span><a href="euclid.html#K23">]|</a></span></span><a href="domain_1.html#K6">}</a></span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> 
 <b>and </b><br/><a NAME="E8:43"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>   st <font color="Olive">y</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/><font color="Maroon">BB</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K1">Ball</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Olive">q</font></span>)</span> <a href="subset_1.html#K9">/\</a> <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> </span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> 
 <b>by </b><i/>;<br/>
<a NAME="E9:43"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">a</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">b</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b</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="E10:43"/>
for <font color="Olive">U</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BB</font> <a href="topgen_2.html#K4">.</a> <font color="Maroon">a</font> holds <br/><font color="Olive">U</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">U</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:43_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BB</font> <a href="topgen_2.html#K4">.</a> <font color="Maroon">a</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_1_1"/>
( <font color="Maroon">b</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">b</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span><b>by </b><i><a class="ref" href="topgen_5.html#T1" target="_self">Th1</a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1"><b>suppose </b></a><a NAME="E1:43_1_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">b</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:43_1_1_1"/><b>then </b>
<font color="Maroon">BB</font> <a href="topgen_2.html#K4">.</a> <font color="Maroon">a</font> <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"><span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Olive">q</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Olive">q</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="euclid.html#K21">`1</a> </span>)</span>,0</span><a href="euclid.html#K23">]|</a></span></span><a href="domain_1.html#K6">}</a></span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> 
 <b>by </b><i>, <a class="ref" href="topgen_5.html#T1" target="_self">Th1</a></i>;<br/><b>then consider </b><font color="Maroon">q</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:43_1_1_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">U</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1">Ball</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">q</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">q</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">q</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i><a class="ref" href="topgen_5.html#T1" target="_self">Th1</a>, , </i>;<br/><b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as  <a href="xxreal_0.html#V2">positive</a> <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">w1</font> being  <a href="rat_1.html#V1">rational</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">v1</font> being  <a href="rat_1.html#V1">rational</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E7:43_1_1_1"/><i><font color="Green">E62</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">v1</font></span><a href="euclid.html#K23">]|</a></span> <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"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">q</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">q</font> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">v1</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">q</font></span><a href="euclid.html#K23">]|</a></span> )
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">q2</font> = <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">v1</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">q</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>;<br/><a NAME="E8:43_1_1_1"/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">v1</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">q</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:43"><i><font color="Green">E42</font></i></a>, <a class="ref" href="euclid.html#T47">EUCLID:47</a></i>;<br/><a NAME="E9:43_1_1_1"/><b>then </b>
( <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">v1</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">q</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">v1</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">q</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i><a class="ref" href="euclid_2.html#T60">EUCLID_2:60</a>, <a class="ref" href="euclid_2.html#T64">EUCLID_2:64</a></i>;<br/><b>then reconsider </b><font color="Maroon">q2</font> = <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">v1</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">q</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> as  <a href="xxreal_0.html#V2">positive</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/><b>consider </b><font color="Maroon">w2</font> being  <a href="rat_1.html#V1">rational</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">v2</font> being  <a href="rat_1.html#V1">rational</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E12:43_1_1_1"/><i><font color="Green">E63</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w2</font>,<font color="Maroon">v2</font></span><a href="euclid.html#K23">]|</a></span> <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"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">q</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">q2</font> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w2</font>,<font color="Maroon">v2</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">q</font></span><a href="euclid.html#K23">]|</a></span> )
 <b>by </b><i/>;<br/><a NAME="E13:43_1_1_1"/>
( <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w2</font>,<font color="Maroon">v2</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">q</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q2</font> &amp; <font color="Maroon">q2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:43"><i><font color="Green">E42</font></i></a>, <a class="txt" href="topgen_5.html#E7:43"><i><font color="Green">E43</font></i></a>, <a class="ref" href="topreal9.html#T7">TOPREAL9:7</a></i>;<br/><a NAME="E14:43_1_1_1"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w2</font>,<font color="Maroon">v2</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">q</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E15:43_1_1_1"/><b>then </b><i><font color="Green">E79</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w2</font>,<font color="Maroon">v2</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">v1</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">w2</font>,<font color="Maroon">v2</font></span><a href="euclid.html#K23">]|</a></span> <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"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">q</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">q</font> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E7:43"><i><font color="Green">E43</font></i></a>, <a class="ref" href="topreal9.html#T7">TOPREAL9:7</a></i>;<br/><a NAME="E16:43_1_1_1"/>
(  <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">q</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">q</font> <a href="tarski.html#R1">c=</a>  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  &amp;  <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">q</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">q</font> <a href="xboole_0.html#R1">misses</a>  <a href="topgen_5.html#K1" target="_self">y=0-line</a>  )
 <b>by </b><i>, </i>;<br/><a NAME="E17:43_1_1_1"/><b>then </b>
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">v1</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">v1</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w2</font>,<font color="Maroon">v2</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">w2</font>,<font color="Maroon">v2</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">w1</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K3">RAT</a>  &amp; <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K3">RAT</a>  &amp; <font color="Maroon">w2</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K3">RAT</a>  &amp; <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K3">RAT</a>  &amp;  <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">q</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">q</font> <a href="tarski.html#R1">c=</a> <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  <a href="subset_1.html#K7">\</a> <a href="topgen_5.html#K1" target="_self">y=0-line</a>  )
 <b>by </b><i><a class="ref" href="xboole_1.html#T86">XBOOLE_1:86</a>, <a class="ref" href="rat_1.html#D2">RAT_1:def 2</a></i>;<br/><a NAME="E18:43_1_1_1"/><b>then </b>
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">v1</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>  <a href="subset_1.html#K7">\</a> <a href="topgen_5.html#K1" target="_self">y=0-line</a>  &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">v1</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><a href="numbers.html#K3">RAT</a> ,<a href="numbers.html#K3">RAT</a> </span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w2</font>,<font color="Maroon">v2</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>  <a href="subset_1.html#K7">\</a> <a href="topgen_5.html#K1" target="_self">y=0-line</a>  &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w2</font>,<font color="Maroon">v2</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><a href="numbers.html#K3">RAT</a> ,<a href="numbers.html#K3">RAT</a> </span><a href="finseq_1.html#K10">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:43"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="funct_6.html#T2">FUNCT_6:2</a></i>;<br/><a NAME="E19:43_1_1_1"/><b>then </b><i><font color="Green">E80</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">v1</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w2</font>,<font color="Maroon">v2</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">v1</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w2</font>,<font color="Maroon">v2</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i>, , <a class="txt" href="topgen_5.html#E1:43"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E20:43_1_1_1"/>
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">v1</font></span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">v1</font></span><a href="euclid.html#K23">]|</a></span> )
 ;<br/><a NAME="E21:43_1_1_1"/><b>then </b>
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">v1</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span> or <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w2</font>,<font color="Maroon">v2</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/><b>hence </b><a NAME="E22:43_1_1_1"/>
<font color="Maroon">U</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_2"><b>suppose </b></a><a NAME="E1:43_1_1_2"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">b</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:43_1_1_2"/><b>then </b>
<font color="Maroon">BB</font> <a href="topgen_2.html#K4">.</a> <font color="Maroon">a</font> <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"><span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>,<font color="Olive">q</font></span>)</span> <a href="subset_1.html#K9">/\</a> <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> </span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> 
 <b>by </b><i>, <a class="ref" href="topgen_5.html#T1" target="_self">Th1</a></i>;<br/><b>then consider </b><font color="Maroon">q</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:43_1_1_2"/><i><font color="Green">E120</font></i>: 
( <font color="Maroon">U</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1">Ball</a> <font color="Maroon">b</font>,<font color="Maroon">q</font></span>)</span> <a href="subset_1.html#K9">/\</a> <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  &amp; <font color="Maroon">q</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i><a class="ref" href="topgen_5.html#T1" target="_self">Th1</a>, </i>;<br/><b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font>, <font color="Maroon">b2</font> = <font color="Maroon">b</font> <a href="euclid.html#K22">`2</a>  as  <a href="xxreal_0.html#V2">positive</a> <a href="real_1.html#NM1">Real</a> <b>by </b><i>, </i>;<br/><a NAME="E6:43_1_1_2"/>
( <font color="Maroon">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> or <font color="Maroon">q</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">b2</font> )
 ;<br/><b>then reconsider </b><font color="Maroon">q1</font> =  <a href="xxreal_0.html#K3">min</a> <font color="Maroon">q</font>,<font color="Maroon">b2</font> as  <a href="xxreal_0.html#V2">positive</a> <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D8">XXREAL_0:def 8</a></i>;<br/><b>consider </b><font color="Maroon">w1</font> being  <a href="rat_1.html#V1">rational</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">v1</font> being  <a href="rat_1.html#V1">rational</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E9:43_1_1_2"/><i><font color="Green">E121</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">v1</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">b</font>,<font color="Maroon">q1</font> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">v1</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="topgen_5.html#T1" target="_self">Th1</a>, </i>;<br/><b>set </b><font color="Maroon">q2</font> = <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">v1</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <font color="Maroon">b</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>;<br/><a NAME="E10:43_1_1_2"/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">v1</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:43"><i><font color="Green">E42</font></i></a>, <a class="ref" href="euclid.html#T47">EUCLID:47</a></i>;<br/><a NAME="E11:43_1_1_2"/><b>then </b>
( <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">v1</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <font color="Maroon">b</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">v1</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <font color="Maroon">b</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i><a class="ref" href="euclid_2.html#T60">EUCLID_2:60</a>, <a class="ref" href="euclid_2.html#T64">EUCLID_2:64</a></i>;<br/><b>then reconsider </b><font color="Maroon">q2</font> = <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">v1</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <font color="Maroon">b</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> as  <a href="xxreal_0.html#V2">positive</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/><b>consider </b><font color="Maroon">w2</font> being  <a href="rat_1.html#V1">rational</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">v2</font> being  <a href="rat_1.html#V1">rational</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E14:43_1_1_2"/><i><font color="Green">E123</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w2</font>,<font color="Maroon">v2</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">b</font>,<font color="Maroon">q2</font> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w2</font>,<font color="Maroon">v2</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="topgen_5.html#T1" target="_self">Th1</a>, </i>;<br/><a NAME="E15:43_1_1_2"/><i><font color="Green">E124</font></i>: 
( <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w2</font>,<font color="Maroon">v2</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <font color="Maroon">b</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q2</font> &amp; <font color="Maroon">q2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q1</font> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:43"><i><font color="Green">E42</font></i></a>, <a class="txt" href="topgen_5.html#E7:43"><i><font color="Green">E43</font></i></a>, <a class="ref" href="topreal9.html#T7">TOPREAL9:7</a></i>;<br/><a NAME="E16:43_1_1_2"/><b>then </b>
( <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w2</font>,<font color="Maroon">v2</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <font color="Maroon">b</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">q1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/><a NAME="E17:43_1_1_2"/><b>then </b><i><font color="Green">E125</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w2</font>,<font color="Maroon">v2</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">b</font>,<font color="Maroon">q1</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w2</font>,<font color="Maroon">v2</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <font color="Maroon">b</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q</font> &amp; <font color="Maroon">q2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q</font> )
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="topreal9.html#T7">TOPREAL9:7</a></i>;<br/><a NAME="E18:43_1_1_2"/><b>then </b><i><font color="Green">E126</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w2</font>,<font color="Maroon">v2</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">v1</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">w1</font>,<font color="Maroon">v1</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">b</font>,<font color="Maroon">q</font> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w2</font>,<font color="Maroon">v2</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">b</font>,<font color="Maroon">q</font> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E7:43"><i><font color="Green">E43</font></i></a>, <a class="ref" href="topreal9.html#T7">TOPREAL9:7</a></i>;<br/><a NAME="E19:43_1_1_2"/>
<font color="Maroon">q1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/><a NAME="E20:43_1_1_2"/><b>then </b><i><font color="Green">E127</font></i>: 
(  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">b</font>,<font color="Maroon">q1</font> <a href="tarski.html#R1">c=</a>  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  &amp;  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">b</font>,<font color="Maroon">q1</font> <a href="xboole_0.html#R1">misses</a>  <a href="topgen_5.html#K1" target="_self">y=0-line</a>  )
 <b>by </b><i><a class="ref" href="topgen_5.html#T1" target="_self">Th1</a>, , </i>;<br/><a NAME="E21:43_1_1_2"/><b>then </b>
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">v1</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">v1</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w2</font>,<font color="Maroon">v2</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">w2</font>,<font color="Maroon">v2</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">w1</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K3">RAT</a>  &amp; <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K3">RAT</a>  &amp; <font color="Maroon">w2</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K3">RAT</a>  &amp; <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K3">RAT</a>  &amp;  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">b</font>,<font color="Maroon">q1</font> <a href="tarski.html#R1">c=</a> <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  <a href="subset_1.html#K7">\</a> <a href="topgen_5.html#K1" target="_self">y=0-line</a>  )
 <b>by </b><i><a class="ref" href="xboole_1.html#T86">XBOOLE_1:86</a>, <a class="ref" href="rat_1.html#D2">RAT_1:def 2</a></i>;<br/><a NAME="E22:43_1_1_2"/><b>then </b>
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">v1</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>  <a href="subset_1.html#K7">\</a> <a href="topgen_5.html#K1" target="_self">y=0-line</a>  &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">v1</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><a href="numbers.html#K3">RAT</a> ,<a href="numbers.html#K3">RAT</a> </span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w2</font>,<font color="Maroon">v2</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>  <a href="subset_1.html#K7">\</a> <a href="topgen_5.html#K1" target="_self">y=0-line</a>  &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w2</font>,<font color="Maroon">v2</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><a href="numbers.html#K3">RAT</a> ,<a href="numbers.html#K3">RAT</a> </span><a href="finseq_1.html#K10">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:43"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="funct_6.html#T2">FUNCT_6:2</a></i>;<br/><a NAME="E23:43_1_1_2"/><b>then </b><i><font color="Green">E128</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">v1</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w2</font>,<font color="Maroon">v2</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">v1</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w2</font>,<font color="Maroon">v2</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i>, , <a class="txt" href="topgen_5.html#E1:43"><i><font color="Green">E42</font></i></a>, , , <a class="ref" href="topgen_5.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E24:43_1_1_2"/>
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">v1</font></span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">v1</font></span><a href="euclid.html#K23">]|</a></span> )
 ;<br/><a NAME="E25:43_1_1_2"/><b>then </b>
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">v1</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span> or <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">w2</font>,<font color="Maroon">v2</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/><b>hence </b><a NAME="E26:43_1_1_2"/>
<font color="Maroon">U</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E11:43"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="topgen_2.html#T11">TOPGEN_2:11</a></i>;<br/>


</div>
