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<b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">d</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">e</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">r3</font> be   <a href="real_1.html#NM1">Real</a>;<br/><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> , <font color="Maroon">PM2</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">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font>;<br/><b>let </b><font color="Maroon">x2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM2</font>;<br/>







<b>assume </b><a NAME="E1:3"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> &amp; <font color="Maroon">PM</font> <a href="hidden.html#R1">=</a>  <a href="topmetr.html#K2">Closed-Interval-MSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">PM2</font> <a href="hidden.html#R1">=</a>  <a href="topmetr.html#K2">Closed-Interval-MSpace</a> <font color="Maroon">d</font>,<font color="Maroon">e</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">x</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">x2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">PM2</font> )
 ;<br/>

<a NAME="E2:3"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E3:3"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:3"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E5:3"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r3</font>
 ;<br/>

<a NAME="E6:3"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> :  <a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r3</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="metric_1.html#T18">METRIC_1:18</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font><b> such that </b><br/><a NAME="E8:3"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp;  <a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r3</font> )
 ;<br/>
<a NAME="E9:3"/><i><font color="Green">E45</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">PM</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T14">TOPMETR:14</a></i>;<br/>
<a NAME="E10:3"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">e</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, , <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E11:3"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">e</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, , <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E12:3"/><b>then </b><i><font color="Green">E47</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">e</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T16">RCOMP_1:16</a></i>;<br/>
<a NAME="E13:3"/><i><font color="Green">E48</font></i>: 
the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">PM</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <a href="metric_1.html#K7">real_dist</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a>, <a class="ref" href="topmetr.html#D1">TOPMETR:def 1</a></i>;<br/>
<a NAME="E14:3"/><i><font color="Green">E49</font></i>: 
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">PM</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/>
<a NAME="E15:3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">y3</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM2</font> <b>by </b><i>, , , <a class="ref" href="topmetr.html#T14">TOPMETR:14</a></i>;<br/>
<a href="metric_1.html#K7">real_dist</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> = 
the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">PM2</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x2</font>,<font color="Maroon">y3</font>
<b>by </b><i>, <a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a>, <a class="ref" href="topmetr.html#D1">TOPMETR:def 1</a></i>
<br/>.= 
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">x2</font>,<font color="Maroon">y3</font>
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>
;<br/>
<a NAME="E18:3"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y2</font> where <font color="Olive">y2</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM2</font> :  <a href="metric_1.html#K2">dist</a> <font color="Maroon">x2</font>,<font color="Olive">y2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r3</font> </span>}</span> 
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E19:3"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x2</font>,<font color="Maroon">r3</font>
 <b>by </b><i><a class="ref" href="metric_1.html#T18">METRIC_1:18</a></i>;<br/>


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