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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>let </b><font color="Maroon" title="c3">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:58"/><i><font color="Green" title="E33">A1</font></i>: 
not <font color="Maroon" title="c1">n</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>and </b><br/><a NAME="E2:58"/><i><font color="Green" title="E34">A2</font></i>: 
 <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">r</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/>

<b>assume </b><a NAME="E3:58"/>
<font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c4">r</font> = <font color="Maroon" title="c2">r</font> as  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a> non <a href="xxreal_0.html#V3" title="XXREAL_0:attr.3">negative</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c5">n</font> = <font color="Maroon" title="c1">n</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="topreal9.html#E1:58"><i><font color="Green" title="E33">A1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c3">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">n</font></span>)</span> ;<br/>
<a NAME="E7:58"/>
not  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c4">r</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
;<br/>
<b>hence </b><a NAME="E8:58"/>
contradiction
 <b>by </b><i><a class="txt" href="topreal9.html#E2:58"><i><font color="Green" title="E34">A2</font></i></a></i>;<br/>


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