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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:15"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="pcomps_1.html#R1">is_metric_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:15"/><i><font color="Green">E16</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> holds  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> : <span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 </span>}</span> 
 ;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E3:15"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> holds <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K4">dist</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>

<a NAME="E1:15_1"/>
 <a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#G1">MetrStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> #)
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:15"><i><font color="Green">E15</font></i></a>, <a class="ref" href="pcomps_1.html#D8">PCOMPS_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E2:15_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:15"/><i><font color="Green">E18</font></i>: 
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:15_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="nagata_2.html#E1:15"><i><font color="Green">E15</font></i></a>, <a class="txt" href="nagata_2.html#E1:15_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="pcomps_2.html#T8">PCOMPS_2:8</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_2_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2_1_1"><b>suppose </b></a><a NAME="E1:15_2_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:15_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E3:15_2_1_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp;  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="tops_1.html#T8">TOPS_1:8</a></i>;<br/><a NAME="E4:15_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="tops_1.html#T21">TOPS_1:21</a></i>;<br/><a NAME="E5:15_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:15"><i><font color="Green">E15</font></i></a>, <a class="ref" href="pcomps_2.html#T8">PCOMPS_2:8</a></i>;<br/><b>hence </b><a NAME="E6:15_2_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="pcomps_1.html#T34">PCOMPS_1:34</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2_1_2"><b>suppose </b></a><a NAME="E1:15_2_1_2"/><i><font color="Green">E20</font></i>: 
not <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:15_2_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:15_2_1_2_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="nagata_2.html#E1:15"><i><font color="Green">E15</font></i></a>, <a class="ref" href="pcomps_2.html#T8">PCOMPS_2:8</a></i>;<br/>
<a NAME="E3:15_2_1_2_1"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:15_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E4:15_2_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T30">TOPS_1:30</a></i>;<br/>
<a NAME="E5:15_2_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/>
<a NAME="E6:15_2_1_2_1"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> : <span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 </span>}</span> 
 
<b>by </b><i><a class="txt" href="nagata_2.html#E2:15"><i><font color="Green">E16</font></i></a>, <a class="txt" href="nagata_2.html#E1:15_2_1_2"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E7:15_2_1_2_1"/><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_2_1_2_1_1"/><i><font color="Green">E24</font></i>: 
<span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/>

<a NAME="E2:15_2_1_2_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="nagata_1.html#R1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:15"><i><font color="Green">E15</font></i></a>, <a class="ref" href="nagata_2.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E3:15_2_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:15_2_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="nagata_2.html#E1:15_2_1_2_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="nagata_2.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E4:15_2_1_2_1_1"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:15_2_1_2_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="nagata_2.html#E6:15_2_1_2_1"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E5:15_2_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E6:15_2_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>;<br/>

<a NAME="E8:15_2_1_2_1"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:15_2_1_2_1_2"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font> &amp; not <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="nagata_2.html#E1:15"><i><font color="Green">E15</font></i></a>, <a class="ref" href="pcomps_2.html#T8">PCOMPS_2:8</a></i>;<br/>
<a NAME="E4:15_2_1_2_1_2"/><i><font color="Green">E25</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:15_2_1_2_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<a NAME="E5:15_2_1_2_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:15_2_1_2_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E6:15_2_1_2_1_2"/><b>then </b>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a>  &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:15_2_1_2_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="nagata_1.html#R1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:15"><i><font color="Green">E15</font></i></a>, <a class="ref" href="nagata_2.html#T9" target="_self">Th9</a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E8:15_2_1_2_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; not <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> is <a href="xboole_0.html#V1">empty</a> &amp; <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> is <a href="seq_4.html#V2">bounded_below</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V2">bounded_below</a> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:15_2_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="nagata_2.html#E8"><i><font color="Green">Lemma8</font></i></a></i>;<br/>
<a NAME="E9:15_2_1_2_1_2"/><b>then </b>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> &amp; <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>12</sub></font> &amp;  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="ref" href="nagata_2.html#D2" target="_self">Def2</a>, <a class="ref" href="nagata_2.html#D3" target="_self">Def3</a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>
<b>hence </b><a NAME="E10:15_2_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="nagata_2.html#E3:15"><i><font color="Green">E17</font></i></a>, <a class="txt" href="nagata_2.html#E4:15_2_1_2_1_2"><i><font color="Green">E25</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:15_2_1_2_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E7:15_2_1_2_1"><i><font color="Green">E23</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:15_2_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="pcomps_1.html#D5">PCOMPS_1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E5:15"/>
 <a href="pcomps_1.html#K3">Family_open_set</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:15_3"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="nagata_2.html#E1:15"><i><font color="Green">E15</font></i></a>, <a class="ref" href="pcomps_2.html#T8">PCOMPS_2:8</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_3_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_3_1_1"><b>suppose </b></a><a NAME="E1:15_3_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:15_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E3:15_3_1_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp;  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="tops_1.html#T8">TOPS_1:8</a></i>;<br/><a NAME="E4:15_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="tops_1.html#T21">TOPS_1:21</a></i>;<br/><b>hence </b><a NAME="E5:15_3_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_3_1_2"><b>suppose </b></a><a NAME="E1:15_3_1_2"/><i><font color="Green">E20</font></i>: 
not <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:15_3_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> iff ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_3_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<a NAME="E1:15_3_1_2_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_3_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_3_1_2_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="nagata_2.html#E1:15"><i><font color="Green">E15</font></i></a>, <a class="txt" href="nagata_2.html#E1:15_3_1_2_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="pcomps_2.html#T8">PCOMPS_2:8</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:15_3_1_2_1_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:15_3"><i><font color="Green">E19</font></i></a>, <a class="txt" href="nagata_2.html#E1:15_3_1_2_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="pcomps_1.html#D5">PCOMPS_1:def 5</a></i>;<br/>
<a NAME="E6:15_3_1_2_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:15"><i><font color="Green">E15</font></i></a>, <a class="ref" href="pcomps_1.html#D7">PCOMPS_1:def 7</a></i>;<br/>
<a NAME="E7:15_3_1_2_1_1"/><b>then </b><i><font color="Green">E23</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E3:15"><i><font color="Green">E17</font></i></a>, <a class="txt" href="nagata_2.html#E5:15_3_1_2_1_1"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="nagata_2.html#E1:15"><i><font color="Green">E15</font></i></a>, <a class="ref" href="pcomps_2.html#T8">PCOMPS_2:8</a></i>;<br/>
<a NAME="E9:15_3_1_2_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K3">`</a>  &amp; ex <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> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a>  )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:15_3_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="nagata_2.html#E5:15_3_1_2_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="subset_1.html#T31">SUBSET_1:31</a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:15_3_1_2_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K3">`</a> 
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="nagata_2.html#E11:15_3_1_2_1_1"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> : <span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 </span>}</span> ;<br/>
<a NAME="E13:15_3_1_2_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> : <span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_3_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:15_3_1_2_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> : <span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_3_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:15_3_1_2_1_1_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K3">`</a> 
 ;<br/>

<a NAME="E2:15_3_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="nagata_1.html#R1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:15"><i><font color="Green">E15</font></i></a>, <a class="ref" href="nagata_2.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E3:15_3_1_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:15_3_1_2_1_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="nagata_2.html#T6" target="_self">Th6</a></i>;<br/>
<b>hence </b><a NAME="E4:15_3_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> : <span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 </span>}</span> 
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:15_3_1_2_1_1_1_1"><i><font color="Green">E25</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:15_3_1_2_1_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> : <span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:15_3_1_2_1_1_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="nagata_2.html#E1:15"><i><font color="Green">E15</font></i></a>, <a class="ref" href="pcomps_2.html#T8">PCOMPS_2:8</a></i>;<br/>
<b>assume </b><a NAME="E6:15_3_1_2_1_1_1"/>
not <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K3">`</a> 
 ;<br/>

<a NAME="E7:15_3_1_2_1_1_1"/><b>then </b>
not <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E4:15_3_1_2_1_1_1"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E8:15_3_1_2_1_1_1"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E9:15_3_1_2_1_1_1"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pcomps_1.html#T33">PCOMPS_1:33</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E11:15_3_1_2_1_1_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E8:15_3_1_2_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="pcomps_1.html#D5">PCOMPS_1:def 5</a></i>;<br/>
<a NAME="E12:15_3_1_2_1_1_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K3">`</a>  &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E11:15_3_1_2_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E13:15_3_1_2_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="nagata_1.html#R1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:15"><i><font color="Green">E15</font></i></a>, <a class="ref" href="nagata_2.html#T9" target="_self">Th9</a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E14:15_3_1_2_1_1_1"/><b>then </b><i><font color="Green">E28</font></i>: 
( not <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> is <a href="xboole_0.html#V1">empty</a> &amp; <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> is <a href="seq_4.html#V2">bounded_below</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V2">bounded_below</a> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E11:15_3_1_2_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="nagata_2.html#E8"><i><font color="Green">Lemma8</font></i></a></i>;<br/>
<a NAME="E15:15_3_1_2_1_1_1"/>
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="nagata_2.html#E4:15_3_1_2_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="nagata_2.html#D3" target="_self">Def3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>18</sub></font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E17:15_3_1_2_1_1_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E11:15_3_1_2_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="nagata_2.html#E14:15_3_1_2_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>19</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E19:15_3_1_2_1_1_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E17:15_3_1_2_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="nagata_2.html#E19:15_3_1_2_1_1_1"><i><font color="Green">E30</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>19</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="nagata_2.html#E1:15"><i><font color="Green">E15</font></i></a>, <a class="txt" href="nagata_2.html#E19:15_3_1_2_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="pcomps_2.html#T8">PCOMPS_2:8</a></i>;<br/>
<a NAME="E22:15_3_1_2_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E17:15_3_1_2_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="nagata_2.html#E19:15_3_1_2_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="nagata_2.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E23:15_3_1_2_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E3:15"><i><font color="Green">E17</font></i></a></i>;<br/>
<a NAME="E24:15_3_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<a NAME="E25:15_3_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K3">`</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E11:15_3_1_2_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="nagata_2.html#E19:15_3_1_2_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E26:15_3_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:15_3_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E2:15"><i><font color="Green">E16</font></i></a>, <a class="txt" href="nagata_2.html#E11:15_3_1_2_1_1"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E15:15_3_1_2_1_1"/><b>then </b>
( <font color="Maroon">c<sub>10</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E7:15_3_1_2_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a>, <a class="ref" href="tops_1.html#T30">TOPS_1:30</a></i>;<br/>
<b>hence </b><a NAME="E16:15_3_1_2_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E5:15_3_1_2_1_1"><i><font color="Green">E22</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:15_3_1_2_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> iff ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>


</div><b>end;</b></div><a NAME="E3:15_3_1_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T57">TOPS_1:57</a></i>;<br/><b>hence </b><a NAME="E4:15_3_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E6:15"/><b>then </b>
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E4:15"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E7:15"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="pcomps_1.html#V3">metrizable</a>
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:15"><i><font color="Green">E15</font></i></a>, <a class="ref" href="pcomps_1.html#D9">PCOMPS_1:def 9</a></i>;<br/>


</div>
