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<b>let </b><font color="Maroon">n</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">o</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>, <font color="Maroon">p</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">r</font> be  <a href="xxreal_0.html#V2">positive</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>





<b>assume </b><a NAME="E1:102"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>
 ;<br/>

<b>set </b><font color="Maroon">f</font> =  <a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font>;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="abian.html#D5">ABIAN:def 5</a><br/>

<b>assume </b><a NAME="E2:102"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="abian.html#D3">ABIAN:def 3</a><br/>

<b>set </b><font color="Maroon">S</font> =  <a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>;<br/>
<a NAME="E3:102"/><i><font color="Green">E74</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> 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">o</font>,<font color="Maroon">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">y</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="E5:102"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E6:102"/><i><font color="Green">E79</font></i>: 
<span class="p1">(<span class="default"><a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="brouwer.html#K3">HC</a> <font color="Maroon">y</font>,<font color="Maroon">p</font>,<font color="Maroon">o</font>,<font color="Maroon">r</font>
 <b>by </b><i>, , <a class="ref" href="jordan.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E7:102"/><i><font color="Green">E80</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">o</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="toprealb.html#T9">TOPREALB:9</a></i>;<br/>
<a NAME="E8:102"/>
 <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="topreal9.html#T17">TOPREAL9:17</a></i>;<br/>
<a NAME="E9:102"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">y</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="jordan.html#E5:102"><i><font color="Green">E78</font></i></a>, <a class="txt" href="jordan.html#E6:102"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan.html#E9:102"><i><font color="Green">E81</font></i></a>, <a class="ref" href="brouwer.html#T3">BROUWER:3</a></i>;<br/>
<a NAME="E10:102"/>
 <a href="topreal9.html#K1">Ball</a> <font color="Maroon">o</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="topreal9.html#T16">TOPREAL9:16</a></i>;<br/>
<a NAME="E11:102"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="brouwer.html#T3">BROUWER:3</a></i>;<br/>
<a NAME="E12:102"/>
 <a href="topreal9.html#K1">Ball</a> <font color="Maroon">o</font>,<font color="Maroon">r</font> <a href="subset_1.html#R1">misses</a>  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="topreal9.html#T19">TOPREAL9:19</a></i>;<br/>
<a NAME="E13:102"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="txt" href="jordan.html#E6:102"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan.html#E9:102"><i><font color="Green">E81</font></i></a>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E14:102"/>
<span class="p1">(<span class="default"><a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="jordan.html#E6:102"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan.html#E7:102"><i><font color="Green">E80</font></i></a>, <a class="txt" href="jordan.html#E11:102"><i><font color="Green">E82</font></i></a>, , <a class="ref" href="brouwer.html#D3">BROUWER:def 3</a></i>;<br/>


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