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<b>let </b><font color="Maroon">X</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">f</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:9"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">f</font> <a href="nagata_1.html#R1">is_a_pseudometric_of</a> <font color="Maroon">X</font>
 ;<br/>

<div><i><font color="Green">E60</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">xy</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:9_1"/>
<font color="Maroon">xy</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:9_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">xy</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font>, <font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/><a NAME="E5:9_1"/>
<font color="Maroon">f</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, <a class="ref" href="nagata_1.html#T29">NAGATA_1:29</a></i>;<br/><a NAME="E6:9_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">xy</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; 0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> 1 )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:9_1"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">xy</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> 1
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/><b>consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:9_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/><a NAME="E4:9_2"/>
 <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">f</font>,<font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E5:9_2"/><b>then </b><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">f</font>,<font color="Maroon">x</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</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">f</font>,<font color="Maroon">x</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">rn</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:9_2_1"/>
<font color="Maroon">rn</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">f</font>,<font color="Maroon">x</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font>
 ;<br/><b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:9_2_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">y</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">f</font>,<font color="Maroon">x</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">rn</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">f</font>,<font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/><a NAME="E5:9_2_1"/>
( <font color="Maroon">f</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">rn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> )
 <b>by </b><i>, , , <a class="ref" href="nagata_1.html#T29">NAGATA_1:29</a></i>;<br/><b>hence </b><a NAME="E6:9_2_1"/>
<font color="Maroon">rn</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:9_2"/>
( not <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">f</font>,<font color="Maroon">x</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</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">f</font>,<font color="Maroon">x</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font> is <a href="seq_4.html#V2">bounded_below</a> )
 <b>by </b><i>, <a class="ref" href="seq_4.html#D2">SEQ_4:def 2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:9"/>
( <font color="Maroon">f</font> is <a href="seq_2.html#V2">bounded_below</a> &amp; ( for <font color="Olive">A</font> being non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><br/> for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> holds <br/> ( not <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">f</font>,<font color="Olive">x</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Olive">A</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">f</font>,<font color="Olive">x</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Olive">A</font> is <a href="seq_4.html#V2">bounded_below</a> ) ) )
 <b>by </b><i>, <a class="ref" href="seq_2.html#D2">SEQ_2:def 2</a></i>;<br/>


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