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<b>let </b><font color="Maroon">PM</font> be    <a href="metric_1.html#L1">MetrStruct</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">p</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font>;<br/><b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font>;<br/>











<b>assume </b><a NAME="E1:33"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">PM</font> is <a href="metric_1.html#V9">triangle</a>
 ;<br/>

<b>assume </b><a NAME="E2:33"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">z</font>,<font color="Maroon">p</font></span>)</span>
 ;<br/>

<a NAME="E3:33"/><b>then </b><i><font color="Green">E27</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">z</font>,<font color="Maroon">p</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">s</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:33"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">y</font>,<font color="Maroon">s</font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> )
 <b>by </b><i><a class="ref" href="pcomps_1.html#T1" target="_self">Th1</a>, </i>;<br/>
<b>consider </b><font color="Maroon">t</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:33"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">t</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">y</font>,<font color="Maroon">t</font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">z</font>,<font color="Maroon">p</font> )
 <b>by </b><i><a class="ref" href="pcomps_1.html#T1" target="_self">Th1</a>, <a class="txt" href="pcomps_1.html#E1:33"><i><font color="Green">E26</font></i></a>, </i>;<br/>
<b>take </b><font color="Maroon">q</font> =  <a href="square_1.html#K1">min</a> <font color="Maroon">s</font>,<font color="Maroon">t</font>;<br/>

<a NAME="E8:33"/>
( <font color="Maroon">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">q</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="pcomps_1.html#E3:33"><i><font color="Green">E27</font></i></a>, <a class="txt" href="pcomps_1.html#E5:33"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a>, <a class="ref" href="xxreal_0.html#T15">XXREAL_0:15</a></i>;<br/>
<a NAME="E9:33"/><b>then </b>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">y</font>,<font color="Maroon">q</font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">y</font>,<font color="Maroon">s</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E10:33"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">y</font>,<font color="Maroon">q</font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E3:33"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>

<a NAME="E11:33"/>
( <font color="Maroon">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> &amp; <font color="Maroon">q</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="pcomps_1.html#E3:33"><i><font color="Green">E27</font></i></a>, <a class="txt" href="pcomps_1.html#E5:33"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a>, <a class="ref" href="xxreal_0.html#T15">XXREAL_0:15</a></i>;<br/>
<a NAME="E12:33"/><b>then </b>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">y</font>,<font color="Maroon">q</font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">y</font>,<font color="Maroon">t</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E13:33"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">y</font>,<font color="Maroon">q</font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">z</font>,<font color="Maroon">p</font>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E5:33"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


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