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<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>set </b><font color="Maroon">C</font> =  <a href="toprealb.html#K7" target="_self">Tcircle</a> <span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span>,<font color="Maroon">r</font>;<br/>
<b>set </b><font color="Maroon">T</font> =  <a href="topreala.html#K1">Trectangle</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font>;<br/>
<a NAME="E1:39"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7" target="_self">Tcircle</a> <span class="p2">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span>,<font color="Maroon">r</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topreala.html#K1">Trectangle</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<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="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:39_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7" target="_self">Tcircle</a> <span class="p2">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span>,<font color="Maroon">r</font></span>)</span>
 ;<br/>

<a NAME="E2:39_1"/><b>then </b>
not  <a href="toprealb.html#K7" target="_self">Tcircle</a> <span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span>,<font color="Maroon">r</font> is <a href="struct_0.html#V3">empty</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i>, <a class="ref" href="topmetr.html#T2">TOPMETR:2</a></i>;<br/>
<a NAME="E4:39_1"/><i><font color="Green">E66</font></i>: 
 <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : (  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp;  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="jgraph_6.html#D2">JGRAPH_6:def 2</a></i>;<br/>
<a NAME="E5:39_1"/><i><font color="Green">E68</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topreala.html#K1">Trectangle</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="jordan1.html#T1">JORDAN1:1</a></i>;<br/>
<a NAME="E6:39_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7" target="_self">Tcircle</a> <span class="p2">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal9.html#K3">Sphere</a> <span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span>,<font color="Maroon">r</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:39_1"/><b>then </b><i><font color="Green">E69</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">x</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="topreal9.html#T12">TOPREAL9:12</a></i>;<br/>
<a NAME="E8:39_1"/>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">x</font></span><a href="toprns_1.html#K5">.|</a></span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">x</font></span><a href="toprns_1.html#K5">.|</a></span> )
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T50">JGRAPH_1:50</a></i>;<br/>
<a NAME="E9:39_1"/><b>then </b>
(  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">x</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp;  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">x</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> )
 
<b>by </b><i>, <a class="ref" href="absvalue.html#T12">ABSVALUE:12</a></i>;<br/>
<b>hence </b><a NAME="E10:39_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topreala.html#K1">Trectangle</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span>)</span>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:39"/>
 <a href="toprealb.html#K7" target="_self">Tcircle</a> <span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span>,<font color="Maroon">r</font> is    <a href="pre_topc.html#M1">SubSpace</a> of  <a href="topreala.html#K1">Trectangle</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T4">TSEP_1:4</a></i>;<br/>


</div>
