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<b>let </b><font color="Maroon" title="c7">r</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">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">x</font> being  <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> <font color="Olive" title="b1">n</font></span>)</span>  st <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Olive" title="b3">x</font>,<font color="Maroon" title="c7">r</font> holds <br/><span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Olive" title="b3">x</font>,<font color="Olive" title="b2">y</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b3">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Olive" title="b3">x</font>,<font color="Maroon" title="c7">r</font></span><br/><b>let </b><font color="Maroon" title="c8">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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">y</font>, <font color="Olive" title="b2">x</font> being  <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> <font color="Maroon" title="c8">n</font></span>)</span>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Olive" title="b2">x</font>,<font color="Maroon" title="c7">r</font> holds <br/><span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">y</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Olive" title="b2">x</font>,<font color="Maroon" title="c7">r</font></span><br/><b>let </b><font color="Maroon" title="c9">y</font>, <font color="Maroon" title="c10">x</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> <font color="Maroon" title="c8">n</font></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="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c7">r</font> implies <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">y</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c7">r</font> )</span><br/>







<b>assume </b><a NAME="E1:34"/><i><font color="Green" title="E24">A1</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c7">r</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="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">y</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c7">r</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_1"/>
( <font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c7">r</font> <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/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1"><b>suppose </b></a><a NAME="E1:34_1_1"/><i><font color="Green" title="E25">A2</font></i>: 
<font color="Maroon" title="c7">r</font> <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"> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">y</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c7">r</font></span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c11">xe</font> = <font color="Maroon" title="c10">x</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#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c8">n</font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13" title="TOPREAL3:th.13">TOPREAL3:13</a></i>;<br/><a NAME="E3:34_1_1"/>
 <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K11" title="METRIC_1:func.11">Sphere</a> <font color="Maroon" title="c11">xe</font>,<font color="Maroon" title="c7">r</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T15" title="TOPREAL9:th.15">TOPREAL9:15</a></i>;<br/><a NAME="E4:34_1_1"/><b>then </b>
 <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="jordan.html#E1:34_1_1"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="topreal6.html#T62" title="TOPREAL6:th.62">TOPREAL6:62</a></i>;<br/><a NAME="E5:34_1_1"/><b>then </b><i><font color="Green" title="E26">A3</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y</font>
 <b>by </b><i><a class="txt" href="jordan.html#E1:34"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E6:34_1_1"/><i><font color="Green" title="E27">A4</font></i>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="topreal1.html#T7" title="TOPREAL1:th.7">TOPREAL1:7</a></i>;<br/><a NAME="E7:34_1_1"/><i><font color="Green" title="E28">A5</font></i>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c10">x</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#T69" title="ENUMSET1:th.69">ENUMSET1:69</a></i>;<br/><a NAME="E8:34_1_1"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>;<br/><b>hence </b><a NAME="E9:34_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">y</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c7">r</font>
 <b>by </b><i><a class="txt" href="jordan.html#E5:34_1_1"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="jordan.html#E6:34_1_1"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="jordan.html#E7:34_1_1"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2"><b>suppose </b></a><a NAME="E1:34_1_2"/><i><font color="Green" title="E25">A6</font></i>: 
<font color="Maroon" title="c7">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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"> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">y</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c7">r</font></span><br/><div class="add"><b>let </b><font color="Maroon" title="c11">k</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="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">y</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> or <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c7">r</font> )</span><br/><b>assume </b><a NAME="E2:34_1_2"/><i><font color="Green" title="E26">A7</font></i>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">y</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</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="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c7">r</font></span><br/><a NAME="E3:34_1_2"/><b>then </b>
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c10">x</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>then consider </b><font color="Maroon" title="c12">l</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:34_1_2"/><i><font color="Green" title="E27">A8</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">l</font> &amp; <font color="Maroon" title="c12">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>and </b><br/><a NAME="E6:34_1_2"/><i><font color="Green" title="E28">A9</font></i>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">l</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c9">y</font></span>)</span>
 <b>by </b><i><a class="ref" href="sppol_1.html#T21" title="SPPOL_1:th.21">SPPOL_1:21</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">k</font> = <font color="Maroon" title="c11">k</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> <font color="Maroon" title="c8">n</font></span>)</span> <b>by </b><i><a class="txt" href="jordan.html#E6:34_1_2"><i><font color="Green" title="E28">A9</font></i></a></i>;<br/><a NAME="E8:34_1_2"/>
not <font color="Maroon" title="c13">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <b>by </b><i><a class="txt" href="jordan.html#E2:34_1_2"><i><font color="Green" title="E26">A7</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E9:34_1_2"/><b>then </b>
( <font color="Maroon" title="c13">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Maroon" title="c13">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">y</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E10:34_1_2"/><b>then </b>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">l</font> &amp; <font color="Maroon" title="c12">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 <b>by </b><i><a class="txt" href="jordan.html#E6:34_1_2"><i><font color="Green" title="E28">A9</font></i></a>, <a class="ref" href="topreal9.html#T4" title="TOPREAL9:th.4">TOPREAL9:4</a></i>;<br/><a NAME="E11:34_1_2"/><b>then </b><i><font color="Green" title="E29">A10</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">l</font> &amp; <font color="Maroon" title="c12">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 )
 <b>by </b><i><a class="txt" href="jordan.html#E5:34_1_2"><i><font color="Green" title="E27">A8</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><font color="Maroon" title="c13">k</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c10">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">l</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c9">y</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan.html#E6:34_1_2"><i><font color="Green" title="E28">A9</font></i></a>, <a class="ref" href="jordan2c.html#T9" title="JORDAN2C:th.9">JORDAN2C:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">x</font></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c9">y</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T54" title="EUCLID:th.54">EUCLID:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">x</font></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c9">y</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">x</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c9">y</font></span>)</span>
<b>by </b><i><a class="ref" href="topreal9.html#T1" title="TOPREAL9:th.1">TOPREAL9:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">x</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c9">y</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T46" title="EUCLID:th.46">EUCLID:46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c9">y</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">x</font></span>)</span>
<b>by </b><i><a class="ref" href="jordan2c.html#T13" title="JORDAN2C:th.13">JORDAN2C:13</a></i>
<br/>.= 
<font color="Maroon" title="c12">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c10">x</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T53" title="EUCLID:th.53">EUCLID:53</a></i>
;<br/><b>then </b><i><font color="Green" title="E30">A11</font></i>: <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="c13">k</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c10">x</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c12">l</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="c9">y</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c10">x</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#T8" title="TOPRNS_1:th.8">TOPRNS_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c12">l</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="c9">y</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c10">x</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="txt" href="jordan.html#E5:34_1_2"><i><font color="Green" title="E27">A8</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c12">l</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">r</font>
<b>by </b><i><a class="txt" href="jordan.html#E1:34"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="topreal9.html#T9" title="TOPREAL9:th.9">TOPREAL9:9</a></i>
;<br/><a NAME="E14:34_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">r</font>
 <b>by </b><i><a class="txt" href="jordan.html#E1:34"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="topreal9.html#T39" title="TOPREAL9:th.39">TOPREAL9:39</a></i>;<br/><a NAME="E15:34_1_2"/><b>then </b>
<font color="Maroon" title="c12">l</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">r</font>
 <b>by </b><i><a class="txt" href="jordan.html#E1:34_1_2"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="jordan.html#E11:34_1_2"><i><font color="Green" title="E29">A10</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><b>hence </b><a NAME="E16:34_1_2"/>
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c7">r</font>
 <b>by </b><i><a class="txt" href="jordan.html#E13:34_1_2"><i><font color="Green" title="E30">A11</font></i></a>, <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"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
