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<b>let </b><font color="Maroon">PM</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="metric_1.html#L1">MetrStruct</a> ;<br/><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>reconsider </b><font color="Maroon">r'</font> = <font color="Maroon">r</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>assume </b><a NAME="E2:35"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">PM</font> is  non <a href="struct_0.html#V3">empty</a> <a href="metric_1.html#NM1">MetrSpace</a>
 ;<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/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>now </b></a><div class="add"><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>assume </b><a NAME="E1:35_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">PM</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K10">cl_Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span>
 ;<br/><a NAME="E2:35_1"/><b>then </b><i><font color="Green">E57</font></i>: 
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K10">cl_Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>set </b><font color="Maroon">r1</font> = <span class="p1">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r'</font>;<br/><a NAME="E3:35_1"/>
<font color="Maroon">r'</font> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r'</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r'</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="metric_1.html#T13">METRIC_1:13</a></i>;<br/><a NAME="E4:35_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">r'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r'</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r'</font></span>)</span> <a href="real_1.html#K5">-</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T23">XREAL_1:23</a></i>;<br/><a NAME="E5:35_1"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">y</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r'</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">PM</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K10">cl_Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:35_1_1"/>
not  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">y</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r'</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">PM</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K10">cl_Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:35_1_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">y</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r'</font></span>)</span> &amp; not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">PM</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K10">cl_Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> <b>by </b><i/>;<br/>
<a NAME="E5:35_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">PM</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">y</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r'</font></span>)</span> &amp; (  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">PM</font> or <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K10">cl_Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> ) )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E6:35_1_1"/><b>then </b>
(  <a href="metric_1.html#K2">dist</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r'</font> &amp;  <a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r'</font> )
 
<b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a>, <a class="ref" href="metric_1.html#T13">METRIC_1:13</a>, <a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E7:35_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r'</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">r'</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E8:35_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">z</font>,<font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r'</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">r'</font>
 
<b>by </b><i>, <a class="ref" href="metric_1.html#T3">METRIC_1:3</a></i>;<br/>
<b>hence </b><a NAME="E9:35_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="metric_1.html#T4">METRIC_1:4</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:35_1"/>
 ex <font color="Olive">r2</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r2</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="Olive">r2</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">PM</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K10">cl_Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:35"/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">PM</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K10">cl_Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <font color="Maroon">PM</font>
 <b>by </b><i><a class="ref" href="pcomps_1.html#D5">PCOMPS_1:def 5</a></i>;<br/>


</div>
