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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="metric_1.html#NM1">MetrSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>





<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E3:1"/><i><font color="Green">E2</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:1_1"/><i><font color="Green">E3</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 </span><b>by </b><i><a class="txt" href="weierstr.html#E1:1_1"><i><font color="Green">E3</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1"><b>case </b></a><a NAME="E1:1_1_1_1_1"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:1_1_1_1_1"/><i><font color="Green">E5</font></i>: 
not <font color="Maroon">c<sub>1</sub></font> is <a href="struct_0.html#V3">empty</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:1_1_1_1_1_1"/>
not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="weierstr.html#E1:1_1_1_1_1"><i><font color="Green">E4</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a><br/>


</div><b>end;</b></div><a NAME="E3:1_1_1_1_1"/><i><font color="Green">E6</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="weierstr.html#E1:1_1_1_1_1"><i><font color="Green">E4</font></i></a>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/><a NAME="E4:1_1_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>4</sub></font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a>, <a class="ref" href="absvalue.html#T11">ABSVALUE:11</a></i>;<br/><a NAME="E5:1_1_1_1_1"/><b>then </b><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E6:1_1_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="metric_1.html#T5">METRIC_1:5</a></i>;<br/><a NAME="E7:1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="weierstr.html#E5:1_1_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E8:1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="weierstr.html#E3:1_1_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E9:1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E10:1_1_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="weierstr.html#E2:1_1_1_1_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_2"><b>case </b></a><a NAME="E1:1_1_1_1_2"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><a NAME="E2:1_1_1_1_2"/><i><font color="Green">E5</font></i>: 
not <font color="Maroon">c<sub>1</sub></font> is <a href="struct_0.html#V3">empty</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:1_1_1_1_2_1"/>
not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="weierstr.html#E1:1_1_1_1_2"><i><font color="Green">E4</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a><br/>


</div><b>end;</b></div><a NAME="E3:1_1_1_1_2"/><i><font color="Green">E6</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="metric_1.html#T4">METRIC_1:4</a></i>;<br/><a NAME="E4:1_1_1_1_2"/><i><font color="Green">E7</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="weierstr.html#E1:1_1_1_1_2"><i><font color="Green">E4</font></i></a>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/><a NAME="E5:1_1_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a></i>;<br/><a NAME="E6:1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="weierstr.html#E4:1_1_1_1_2"><i><font color="Green">E7</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E7:1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E8:1_1_1_1_2"/><b>then </b><i><font color="Green">E8</font></i>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="weierstr.html#E3:1_1_1_1_2"><i><font color="Green">E6</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E9:1_1_1_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><a NAME="E10:1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 0
 <b>by </b><i><a class="txt" href="weierstr.html#E8:1_1_1_1_2"><i><font color="Green">E8</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E11:1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E12:1_1_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="weierstr.html#E2:1_1_1_1_2"><i><font color="Green">E5</font></i></a>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>thus </b><a NAME="E4:1"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="weierstr.html#E3:1"><i><font color="Green">E2</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


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