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<b>let </b><font color="Maroon">s</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">r1</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">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">s1</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">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>set </b><font color="Maroon">p</font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">r1</font></span><a href="euclid.html#K23">]|</a></span>;<br/>
<b>set </b><font color="Maroon">q</font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">s1</font></span><a href="euclid.html#K23">]|</a></span>;<br/>
<b>set </b><font color="Maroon">p3</font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s1</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span>;<br/>
<b>assume </b><a NAME="E1:24"/>
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">r1</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">r</font> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">s1</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">r</font> )
 ;<br/>

<a NAME="E2:24"/><b>then </b><i><font color="Green">E56</font></i>: 
(  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">r1</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">s1</font></span><a href="euclid.html#K23">]|</a></span> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">u</font>,<font color="Maroon">r</font> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">r1</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">r1</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">s1</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">s1</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> &amp; 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<b>then </b><i><font color="Green">E57</font></i>: <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s1</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">r1</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">s1</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">r1</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">s1</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">r1</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">s1</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">r1</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">s1</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i/>
;<br/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s1</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">r1</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">s1</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">r1</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">s1</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">r1</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">s1</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">r1</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">s1</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i/>
;<br/>
<a NAME="E5:24"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s1</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">r1</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">s1</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:24"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s1</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">lambda</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">r1</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Olive">lambda</font> <a href="euclid.html#K18">*</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">s1</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> where <font color="Olive">lambda</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">lambda</font> &amp; <font color="Olive">lambda</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
;<br/>
<a NAME="E7:24"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s1</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">r1</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">s1</font></span><a href="euclid.html#K23">]|</a></span>
 
;<br/>
<b>hence </b><a NAME="E8:24"/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s1</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></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">r</font>
 <b>by </b><i/>;<br/>


</div>
