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<b>let </b><font color="Maroon" title="c1">PM</font> be    <a href="metric_1.html#L1" title="METRIC_1:struct.1">MetrStruct</a> ;<br/><b>let </b><font color="Maroon" title="c2">r</font>, <font color="Maroon" title="c3">p</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">PM</font>;<br/>











<b>assume </b><a NAME="E1:33"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c1">PM</font> is <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>
 ;<br/>

<b>assume </b><a NAME="E2:33"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c2">r</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c3">p</font></span>)</span>
 ;<br/>

<a NAME="E3:33"/><b>then </b><i><font color="Green" title="E26">A2</font></i>: 
( <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c2">r</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c3">p</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:33"/><i><font color="Green" title="E27">A3</font></i>: 
( <font color="Maroon" title="c7">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c7">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c2">r</font> )
 <b>by </b><i><a class="txt" href="pcomps_1.html#E1:33"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="pcomps_1.html#T30" target="_self" title="PCOMPS_1:th.30">Th30</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">t</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E7:33"/><i><font color="Green" title="E28">A4</font></i>: 
( <font color="Maroon" title="c8">t</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c8">t</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c3">p</font> )
 <b>by </b><i><a class="txt" href="pcomps_1.html#E1:33"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="pcomps_1.html#E3:33"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="pcomps_1.html#T30" target="_self" title="PCOMPS_1:th.30">Th30</a></i>;<br/>
<b>take </b><font color="Maroon" title="c9">q</font> =  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c8">t</font>;<br/>

<a NAME="E8:33"/>
( <font color="Maroon" title="c9">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">s</font> &amp; <font color="Maroon" title="c9">q</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="pcomps_1.html#E5:33"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="pcomps_1.html#E7:33"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a>, <a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/>
<a NAME="E9:33"/><b>then </b>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c9">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c7">s</font>
 
<b>by </b><i><a class="ref" href="pcomps_1.html#T1" target="_self" title="PCOMPS_1:th.1">Th1</a></i>;<br/>
<b>hence </b><a NAME="E10:33"/>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c9">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c2">r</font>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E5:33"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>

<a NAME="E11:33"/>
( <font color="Maroon" title="c9">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">t</font> &amp; <font color="Maroon" title="c9">q</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="pcomps_1.html#E5:33"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="pcomps_1.html#E7:33"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a>, <a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/>
<a NAME="E12:33"/><b>then </b>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c9">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c8">t</font>
 
<b>by </b><i><a class="ref" href="pcomps_1.html#T1" target="_self" title="PCOMPS_1:th.1">Th1</a></i>;<br/>
<b>hence </b><a NAME="E13:33"/>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c9">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E7:33"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>


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