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

<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">q2</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#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c3">q</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="c1">n</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c4">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> ;<br/>







<b>assume </b><a NAME="E1:2"/><i><font color="Green" title="E7">A1</font></i>: 
<font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">q2</font>
 ;<br/>

<a NAME="E2:2"/><i><font color="Green" title="E8">A2</font></i>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c2">q2</font>,<font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q4</font> where <font color="Olive" title="b1">q4</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span> :  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">q2</font>,<font color="Olive" title="b1">q4</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="metric_1.html#T18" title="METRIC_1:th.18">METRIC_1:18</a></i>;<br/>
<a NAME="E3:2"/><i><font color="Green" title="E9">A3</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">q4</font> where <font color="Olive" title="b1">q4</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span> :  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">q2</font>,<font color="Olive" title="b1">q4</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q3</font> where <font color="Olive" title="b1">q3</font> is   <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="c1">n</font></span>)</span> : <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="c3">q</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Olive" title="b1">q3</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="c4">r</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</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><br/>

<b>assume </b><a NAME="E1:2_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q4</font> where <font color="Olive" title="b1">q4</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span> :  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">q2</font>,<font color="Olive" title="b1">q4</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">q4</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E3:2_1"/><i><font color="Green" title="E10">A4</font></i>: 
( <font color="Maroon" title="c6">q4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> &amp;  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">q2</font>,<font color="Maroon" title="c6">q4</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">q44</font> = <font color="Maroon" title="c6">q4</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="c1">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="E5:2_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">q2</font>,<font color="Maroon" title="c6">q4</font> <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"><font color="Maroon" title="c3">q</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">q44</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E1:2"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="jgraph_1.html#T45" title="JGRAPH_1:th.45">JGRAPH_1:45</a></i>;<br/>
<b>hence </b><a NAME="E6:2_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q3</font> where <font color="Olive" title="b1">q3</font> is   <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="c1">n</font></span>)</span> : <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="c3">q</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Olive" title="b1">q3</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="c4">r</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_2.html#E3:2_1"><i><font color="Green" title="E10">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:2"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">q3</font> where <font color="Olive" title="b1">q3</font> is   <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="c1">n</font></span>)</span> : <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="c3">q</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Olive" title="b1">q3</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="c4">r</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q4</font> where <font color="Olive" title="b1">q4</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span> :  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">q2</font>,<font color="Olive" title="b1">q4</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</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><br/>

<b>assume </b><a NAME="E1:2_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q3</font> where <font color="Olive" title="b1">q3</font> is   <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="c1">n</font></span>)</span> : <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="c3">q</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Olive" title="b1">q3</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="c4">r</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">q3</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="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E3:2_2"/><i><font color="Green" title="E10">A5</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q3</font> &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="c3">q</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c6">q3</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="c4">r</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">q34</font> = <font color="Maroon" title="c6">q3</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="c1">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="E5:2_2"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">q2</font>,<font color="Maroon" title="c7">q34</font> <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"><font color="Maroon" title="c3">q</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c6">q3</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E1:2"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="jgraph_1.html#T45" title="JGRAPH_1:th.45">JGRAPH_1:45</a></i>;<br/>
<b>hence </b><a NAME="E6:2_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q4</font> where <font color="Olive" title="b1">q4</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span> :  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">q2</font>,<font color="Olive" title="b1">q4</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_2.html#E3:2_2"><i><font color="Green" title="E10">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:2"/>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c2">q2</font>,<font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q3</font> where <font color="Olive" title="b1">q3</font> is   <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="c1">n</font></span>)</span> : <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="c3">q</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Olive" title="b1">q3</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="c4">r</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_2.html#E2:2"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="jgraph_2.html#E3:2"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div>
