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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</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"><font color="Maroon">c<sub>1</sub></font>,<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><a NAME="E1:11"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="nagata_1.html#R1">is_a_pseudometric_of</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E2:11"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E4:11"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="metric_1.html#V2">Reflexive</a>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:11"><i><font color="Green">E11</font></i></a>, <a class="ref" href="nagata_1.html#D10">NAGATA_1:def 10</a></i>;<br/>
<a NAME="E5:11"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</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>4</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="metric_1.html#D3">METRIC_1:def 3</a></i>;<br/>
<a NAME="E6:11"/><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>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>4</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>4</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="nagata_2.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E7:11"/><b>then </b>
( 0 <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>4</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>5</sub></font> &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>4</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>5</sub></font> 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>4</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>5</sub></font> is <a href="seq_4.html#V2">bounded_below</a> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:11"><i><font color="Green">E11</font></i></a>, <a class="txt" href="nagata_2.html#E2:11"><i><font color="Green">E12</font></i></a>, <a class="txt" href="nagata_2.html#E8"><i><font color="Green">Lemma8</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E8:11"/><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>4</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 &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#E1:11"><i><font color="Green">E11</font></i></a>, <a class="txt" href="nagata_2.html#E8"><i><font color="Green">Lemma8</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>
<a NAME="E9:11"/><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>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0 &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>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:11"><i><font color="Green">E11</font></i></a>, <a class="ref" href="nagata_2.html#D3" target="_self">Def3</a>, <a class="ref" href="nagata_2.html#T5" target="_self">Th5</a></i>;<br/>
<b>hence </b><a NAME="E10:11"/>
<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="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>


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