<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">r</font> be  non <a href="xxreal_0.html#V3">negative</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">s</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">t</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:9"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">t</font>
 <b>and </b><br/><a NAME="E2:9"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">s</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span>
 <b>and </b><br/><a NAME="E3:9"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">s</font> is  not   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span>
 ;<br/>

<a NAME="E4:9"/><i><font color="Green">E63</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="toprealb.html#T9">TOPREALB:9</a></i>;<br/>
<a NAME="E5:9"/><i><font color="Green">E64</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" target="_self">Tdisk</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:9"/><i><font color="Green">E65</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font>
 
<b>by </b><i>, , <a class="ref" href="topreal9.html#T9">TOPREAL9:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">S</font> = <font color="Maroon">s</font>, <font color="Maroon">T</font> = <font color="Maroon">t</font>, <font color="Maroon">X</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<b>set </b><font color="Maroon">a</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K20">-</a> <font color="Maroon">s</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">s</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon">T</font> <a href="euclid.html#K8">-</a> <font color="Maroon">S</font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K20">-</a> <font color="Maroon">s</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">s</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="euclid.html#K8">-</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">T</font> <a href="euclid.html#K8">-</a> <font color="Maroon">S</font></span>)</span></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E8:9"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i>, , <a class="ref" href="topreal9.html#T8">TOPREAL9:8</a></i>;<br/>
<a NAME="E9:9"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E10:9"/><b>then </b>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="topreal9.html#T7">TOPREAL9:7</a></i>;<br/>
<b>then consider </b><font color="Maroon">e1</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E12:9"/><i><font color="Green">E68</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e1</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal9.html#K4">halfline</a> <font color="Maroon">s</font>,<font color="Maroon">t</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span>
 <b>and </b><br/><a NAME="E13:9"/>
<font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K20">-</a> <font color="Maroon">s</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p0">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">T</font> <a href="euclid.html#K8">-</a> <font color="Maroon">S</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K20">-</a> <font color="Maroon">s</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">S</font> <a href="euclid.html#K8">-</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="euclid.html#K8">-</a> <font color="Maroon">S</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">s</font></span>)</span> <a href="euclid.html#K17">+</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"><a href="xcmplx_0.html#K4">-</a> <span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p0"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K20">-</a> <font color="Maroon">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default"><a href="quin_1.html#K1">delta</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="euclid.html#K8">-</a> <font color="Maroon">S</font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K20">-</a> <font color="Maroon">s</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="euclid.html#K8">-</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon">T</font> <a href="euclid.html#K8">-</a> <font color="Maroon">S</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">t</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="topreal9.html#T37">TOPREAL9:37</a></i>;<br/>
<a NAME="E14:9"/>
<font color="Maroon">e1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e1</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E15:9"/><b>then </b>
<font color="Maroon">e1</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E16:9"/>
 ex <font color="Olive">e</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span> st <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">e</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal9.html#K4">halfline</a> <font color="Maroon">s</font>,<font color="Maroon">t</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span>
 <b>by </b><i>, </i>;<br/>


</div>
