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<b>let </b><font color="Maroon" title="c1">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies for <font color="Olive" title="b1">x</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <br/> for <font color="Olive" title="b2">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a> <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <span class="p1">(<span class="default"><a href="topgen_5.html#K4" title="TOPGEN_5:func.4">+</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/> ex <font color="Olive" title="b3">r1</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a> <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<font color="Olive" title="b3">r1</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Olive" title="b3">r1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K4" title="TOPGEN_5:func.4">+</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">r</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><br/>

<b>assume </b><a NAME="E1:79"/><i><font color="Green" title="E60">A1</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <br/> for <font color="Olive" title="b2">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a> <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <span class="p1">(<span class="default"><a href="topgen_5.html#K4" title="TOPGEN_5:func.4">+</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/> ex <font color="Olive" title="b3">r1</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a> <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<font color="Olive" title="b3">r1</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Olive" title="b3">r1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K4" title="TOPGEN_5:func.4">+</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">r</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/>

<b>let </b><font color="Maroon" title="c2">x</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a> <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <span class="p1">(<span class="default"><a href="topgen_5.html#K4" title="TOPGEN_5:func.4">+</a> <font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/> ex <font color="Olive" title="b2">r1</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a> <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<font color="Olive" title="b2">r1</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Olive" title="b2">r1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K4" title="TOPGEN_5:func.4">+</a> <font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">r</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/><b>let </b><font color="Maroon" title="c3">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a> <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="topgen_5.html#K4" title="TOPGEN_5:func.4">+</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies  ex <font color="Olive" title="b1">r1</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a> <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<font color="Olive" title="b1">r1</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Olive" title="b1">r1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K4" title="TOPGEN_5:func.4">+</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><br/>



<b>set </b><font color="Maroon" title="c4">r1</font> = <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2;<br/>
<b>set </b><font color="Maroon" title="c5">f</font> =  <a href="topgen_5.html#K4" title="TOPGEN_5:func.4">+</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font>;<br/>
<b>assume </b><a NAME="E2:79"/><i><font color="Green" title="E61">A2</font></i>: 
<span class="p1">(<span class="default"><a href="topgen_5.html#K4" title="TOPGEN_5:func.4">+</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">r1</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a> <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<font color="Olive" title="b1">r1</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Olive" title="b1">r1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K4" title="TOPGEN_5:func.4">+</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/>

<a NAME="E3:79"/><i><font color="Green" title="E62">A3</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/>
<a NAME="E4:79"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="topgen_5.html#E1:79"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="topgen_5.html#E2:79"><i><font color="Green" title="E61">A2</font></i></a>, <a class="ref" href="topgen_5.html#D5" target="_self" title="TOPGEN_5:def.5">Def5</a></i>;<br/>
<a NAME="E5:79"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="square_1.html#T74" title="SQUARE_1:th.74">SQUARE_1:74</a></i>;<br/>
<a NAME="E6:79"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> &amp; <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E7:79"/><b>then </b>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">x</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">r</font></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> 
 
<b>by </b><i><a class="txt" href="topgen_5.html#E1:79"><i><font color="Green" title="E60">A1</font></i></a>, <a class="ref" href="topgen_5.html#T13" target="_self" title="TOPGEN_5:th.13">Th13</a></i>;<br/>
<a NAME="E8:79"/><b>then </b>
( <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a>  )
 
<b>by </b><i><a class="txt" href="topgen_5.html#E3:79"><i><font color="Green" title="E62">A3</font></i></a>, <a class="ref" href="euclid.html#T66" title="EUCLID:th.66">EUCLID:66</a></i>;<br/>
<a NAME="E9:79"/><b>then </b>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="ref" href="square_1.html#T77" title="SQUARE_1:th.77">SQUARE_1:77</a></i>;<br/>
<a NAME="E10:79"/><b>then </b>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/>
<a NAME="E11:79"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T76" title="XREAL_1:th.76">XREAL_1:76</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">r1</font> = <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 as  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a> <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  ;<br/>
<b>take </b>
<font color="Maroon" title="c6">r1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<font color="Maroon" title="c6">r1</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Maroon" title="c6">r1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K4" title="TOPGEN_5:func.4">+</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/>

<b>let </b><font color="Maroon" title="c7">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<font color="Maroon" title="c6">r1</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Maroon" title="c6">r1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K4" title="TOPGEN_5:func.4">+</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><br/>

<b>assume </b><a NAME="E13:79"/><i><font color="Green" title="E63">A4</font></i>: 
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<font color="Maroon" title="c6">r1</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Maroon" title="c6">r1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K4" title="TOPGEN_5:func.4">+</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/>

<a NAME="E14:79"/><b>then </b><i><font color="Green" title="E64">A5</font></i>: 
( <font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<font color="Maroon" title="c6">r1</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Maroon" title="c6">r1</font> or <font color="Maroon" title="c7">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> )
 
<b>by </b><i><a class="ref" href="setwiseo.html#T6" title="SETWISEO:th.6">SETWISEO:6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">q</font> = <font color="Maroon" title="c7">u</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="topgen_5.html#E13:79"><i><font color="Green" title="E63">A4</font></i></a></i>;<br/>
<a NAME="E16:79"/>
 <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<font color="Maroon" title="c6">r1</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Maroon" title="c6">r1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topgen_5.html#K2" title="TOPGEN_5:func.2">y&gt;=0-plane</a> 
 
<b>by </b><i><a class="ref" href="topgen_5.html#T24" target="_self" title="TOPGEN_5:th.24">Th24</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">z</font> = <font color="Maroon" title="c8">q</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topgen_5.html#K3" title="TOPGEN_5:func.3">Niemytzki-plane</a>  <b>by </b><i><a class="txt" href="topgen_5.html#E1:79"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="topgen_5.html#E3:79"><i><font color="Green" title="E62">A3</font></i></a>, <a class="txt" href="topgen_5.html#E14:79"><i><font color="Green" title="E64">A5</font></i></a>, <a class="txt" href="topgen_5.html#E33"><i><font color="Green" title="E23">Lm1</font></i></a>, <a class="ref" href="topgen_5.html#T22" target="_self" title="TOPGEN_5:th.22">Th22</a></i>;<br/>
<a NAME="E18:79"/><i><font color="Green" title="E65">A6</font></i>: 
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/>
<a NAME="E19:79"/>
<font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topgen_5.html#K2" title="TOPGEN_5:func.2">y&gt;=0-plane</a> 
 
<b>by </b><i><a class="txt" href="topgen_5.html#E33"><i><font color="Green" title="E23">Lm1</font></i></a></i>;<br/>
<a NAME="E20:79"/><b>then </b><i><font color="Green" title="E66">A7</font></i>: 
<font color="Maroon" title="c8">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="topgen_5.html#E18:79"><i><font color="Green" title="E65">A6</font></i></a>, <a class="ref" href="topgen_5.html#T22" target="_self" title="TOPGEN_5:th.22">Th22</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:79_1"/><i><font color="Green" title="E67">A8</font></i>: 
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<font color="Maroon" title="c6">r1</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Maroon" title="c6">r1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Maroon" title="c3">r</font> &amp; <font color="Maroon" title="c8">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><a NAME="E2:79_1"/><b>then </b>
( <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<font color="Maroon" title="c6">r1</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r1</font> &amp; <font color="Maroon" title="c6">r1</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> )
 <b>by </b><i><a class="ref" href="topreal9.html#T7" title="TOPREAL9:th.7">TOPREAL9:7</a></i>;<br/><a NAME="E3:79_1"/><b>then </b>
( <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<font color="Maroon" title="c6">r1</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<font color="Maroon" title="c6">r1</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<font color="Maroon" title="c6">r1</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<font color="Maroon" title="c6">r1</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<font color="Maroon" title="c6">r1</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> &amp; <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">r1</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r1</font> &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<font color="Maroon" title="c6">r1</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">r1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c6">r1</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">r1</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:79"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="topgen_5.html#E3:79"><i><font color="Green" title="E62">A3</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a>, <a class="ref" href="euclid.html#T66" title="EUCLID:th.66">EUCLID:66</a>, <a class="ref" href="topreal6.html#T31" title="TOPREAL6:th.31">TOPREAL6:31</a>, <a class="ref" href="toprns_1.html#T35" title="TOPRNS_1:th.35">TOPRNS_1:35</a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E4:79_1"/><b>then </b>
<font color="Maroon" title="c6">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">r1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E5:79_1"/><b>then </b>
( <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c8">q</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c8">q</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c8">q</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> )
 <b>by </b><i><a class="ref" href="toprns_1.html#T35" title="TOPRNS_1:th.35">TOPRNS_1:35</a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E6:79_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c8">q</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:79_1"/><b>then </b>
( <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c8">q</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> )
 <b>by </b><i><a class="ref" href="toprns_1.html#T28" title="TOPRNS_1:th.28">TOPRNS_1:28</a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E8:79_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="ref" href="toprns_1.html#T28" title="TOPRNS_1:th.28">TOPRNS_1:28</a></i>;<br/><b>hence </b><a NAME="E9:79_1"/>
not <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T7" title="TOPREAL9:th.7">TOPREAL9:7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><a NAME="E10:79_1"/>
( <font color="Maroon" title="c8">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies ( <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topgen_5.html#K1" title="TOPGEN_5:func.1">y=0-line</a>  &amp;  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<font color="Maroon" title="c6">r1</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Maroon" title="c6">r1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topgen_5.html#K1" title="TOPGEN_5:func.1">y=0-line</a>  ) )
 <b>by </b><i><a class="txt" href="topgen_5.html#E18:79"><i><font color="Green" title="E65">A6</font></i></a>, <a class="ref" href="topgen_5.html#T25" target="_self" title="TOPGEN_5:th.25">Th25</a></i>;<br/><b>hence </b><a NAME="E11:79_1"/>
<font color="Maroon" title="c8">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:79_1"><i><font color="Green" title="E67">A8</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E22:79"/><b>then </b>
<span class="p1">(<span class="default"><a href="topgen_5.html#K4" title="TOPGEN_5:func.4">+</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="topgen_5.html#E2:79"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="topgen_5.html#E13:79"><i><font color="Green" title="E63">A4</font></i></a>, <a class="txt" href="topgen_5.html#E18:79"><i><font color="Green" title="E65">A6</font></i></a>, <a class="txt" href="topgen_5.html#E20:79"><i><font color="Green" title="E66">A7</font></i></a>, <a class="ref" href="topgen_5.html#D5" target="_self" title="TOPGEN_5:def.5">Def5</a>, <a class="ref" href="setwiseo.html#T6" title="SETWISEO:th.6">SETWISEO:6</a></i>;<br/>
<a NAME="E23:79"/><b>then </b>
<span class="p1">(<span class="default"><a href="topgen_5.html#K4" title="TOPGEN_5:func.4">+</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E24:79"/>
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K4" title="TOPGEN_5:func.4">+</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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