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<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">s</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">r2</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">r1</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">u</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:9"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E2:9"/><i><font color="Green">E41</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font>
 <b>and </b><br/><a NAME="E3:9"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">v</font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r2</font></span>)</span>,<font color="Maroon">s</font></span><a href="euclid.html#K23">]|</a></span> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">u</font>,<font color="Maroon">v</font> = 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r2</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>

<br/>.= 
<font color="Maroon">r2</font>
<b>by </b><i>, <a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>
;<br/>
<b>hence </b><a NAME="E6:9"/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r2</font></span>)</span>,<font color="Maroon">s</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">u</font>,<font color="Maroon">r1</font>
 <b>by </b><i>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>


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