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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <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="E1:14"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="pcomps_1.html#R1">is_metric_of</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:14"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Maroon">c<sub>2</sub></font> <a href="metric_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="metric_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> <font color="Maroon">c<sub>2</sub></font> <a href="metric_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="metric_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="metric_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="metric_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:14"><i><font color="Green">E14</font></i></a>, <a class="ref" href="pcomps_1.html#D7">PCOMPS_1:def 7</a></i>;<br/>
<a NAME="E3:14"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> is <a href="metric_1.html#V2">Reflexive</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="metric_1.html#V4">symmetric</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="metric_1.html#V5">triangle</a> )
 
<b>by </b><i><a class="ref" href="metric_1.html#D3">METRIC_1:def 3</a>, <a class="ref" href="metric_1.html#D5">METRIC_1:def 5</a>, <a class="ref" href="metric_1.html#D6">METRIC_1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E4:14"/>
<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>
 <b>by </b><i><a class="ref" href="nagata_1.html#D10">NAGATA_1:def 10</a></i>;<br/>


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