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<b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">M</font> be  non <a href="struct_0.html#V3">empty</a> <a href="metric_1.html#NM1">MetrSpace</a>;<br/><b>let </b><font color="Maroon">z1</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>, <font color="Maroon">z2</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>, <font color="Maroon">z3</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:6"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">z1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z2</font>
 <b>and </b><br/><a NAME="E2:6"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">z1</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K10">cl_Ball</a> <font color="Maroon">z3</font>,<font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E3:6"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">z2</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K10">cl_Ball</a> <font color="Maroon">z3</font>,<font color="Maroon">r</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:6_1"/><b>then </b>
 <a href="metric_1.html#K10">cl_Ball</a> <font color="Maroon">z3</font>,<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z3</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="topreal6.html#T64">TOPREAL6:64</a></i>;<br/><a NAME="E3:6_1"/><b>then </b>
( <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z3</font> &amp; <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z3</font> )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:6_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:6"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="topreal6.html#T63">TOPREAL6:63</a></i>;<br/>


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