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<b>let </b><font color="Maroon">X</font> be    <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="E1:58"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">f</font> <a href="nagata_1.html#R1" target="_self">is_a_pseudometric_of</a> <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>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>

<a NAME="E2:58"/>
( <font color="Maroon">f</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">y</font>,<font color="Maroon">x</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="nagata_1.html#E1:58"><i><font color="Green">E33</font></i></a>, <a class="txt" href="nagata_1.html#E1"><i><font color="Green">Lemma29</font></i></a></i>;<br/>
<a NAME="E3:58"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="nagata_1.html#E1:58"><i><font color="Green">E33</font></i></a>, <a class="txt" href="nagata_1.html#E1"><i><font color="Green">Lemma29</font></i></a></i>;<br/>
<a NAME="E4:58"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="real_2.html#T125">REAL_2:125</a></i>;<br/>
<b>hence </b><a NAME="E5:58"/>
<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
 ;<br/>


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