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<b>let </b><font color="Maroon">T</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">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">m</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">T</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</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="E1:60"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E2:60"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">m</font> <a href="pcomps_1.html#R1">is_metric_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 ;<br/>

<a NAME="E3:60"/>
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> holds <br/> ( <font color="Maroon">m</font> <a href="metric_1.html#K1">.</a> <font color="Olive">a</font>,<font color="Olive">a</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">m</font> <a href="metric_1.html#K1">.</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="metric_1.html#K1">.</a> <font color="Olive">b</font>,<font color="Olive">a</font> &amp; <font color="Maroon">m</font> <a href="metric_1.html#K1">.</a> <font color="Olive">a</font>,<font color="Olive">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="metric_1.html#K1">.</a> <font color="Olive">a</font>,<font color="Olive">b</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="metric_1.html#K1">.</a> <font color="Olive">b</font>,<font color="Olive">c</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="pcomps_1.html#D7">PCOMPS_1:def 7</a></i>;<br/>
<a NAME="E4:60"/><b>then </b>
<font color="Maroon">m</font> <a href="nagata_1.html#R1" target="_self">is_a_pseudometric_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="txt" href="nagata_1.html#E1"><i><font color="Green">Lemma29</font></i></a></i>;<br/>
<a NAME="E5:60"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="nagata_1.html#K5" target="_self">min</a> <font color="Maroon">r</font>,<font color="Maroon">m</font> <a href="nagata_1.html#R1" target="_self">is_a_pseudometric_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="txt" href="nagata_1.html#E2"><i><font color="Green">Lemma34</font></i></a>, <a class="txt" href="nagata_1.html#E1:60"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>, <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="pcomps_1.html#D7">PCOMPS_1:def 7</a><br/>

<b>thus </b><a NAME="E6:60"/>
( <span class="p1">(<span class="default"><a href="nagata_1.html#K5" target="_self">min</a> <font color="Maroon">r</font>,<font color="Maroon">m</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 iff <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>proof </b></a><div class="add">

<a NAME="E1:60_1"/>
( <span class="p1">(<span class="default"><a href="nagata_1.html#K5" target="_self">min</a> <font color="Maroon">r</font>,<font color="Maroon">m</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:60_1_1"/>
<span class="p1">(<span class="default"><a href="nagata_1.html#K5" target="_self">min</a> <font color="Maroon">r</font>,<font color="Maroon">m</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:60_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="xxreal_0.html#K3">min</a> <font color="Maroon">r</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E3:60_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="txt" href="nagata_1.html#E2"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="xxreal_0.html#D8">XXREAL_0:def 8</a></i>;<br/>
<a NAME="E4:60_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#D8">XXREAL_0:def 8</a></i>;<br/>
<b>hence </b><a NAME="E5:60_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="pcomps_1.html#D7">PCOMPS_1:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:60_1"/>
( <span class="p1">(<span class="default"><a href="nagata_1.html#K5" target="_self">min</a> <font color="Maroon">r</font>,<font color="Maroon">m</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 iff <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, <a class="txt" href="nagata_1.html#E1"><i><font color="Green">Lemma29</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E7:60"/>
( <span class="p1">(<span class="default"><a href="nagata_1.html#K5" target="_self">min</a> <font color="Maroon">r</font>,<font color="Maroon">m</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="nagata_1.html#K5" target="_self">min</a> <font color="Maroon">r</font>,<font color="Maroon">m</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font> &amp; <span class="p1">(<span class="default"><a href="nagata_1.html#K5" target="_self">min</a> <font color="Maroon">r</font>,<font color="Maroon">m</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nagata_1.html#K5" target="_self">min</a> <font color="Maroon">r</font>,<font color="Maroon">m</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nagata_1.html#K5" target="_self">min</a> <font color="Maroon">r</font>,<font color="Maroon">m</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span> )
 <b>by </b><i>, <a class="txt" href="nagata_1.html#E1"><i><font color="Green">Lemma29</font></i></a></i>;<br/>


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