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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b2">x</font> being   <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> holds   <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> (<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">r</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">r</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <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> holds   <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c2">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> (<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c2">r</font>)</span><br/><span class="kw">let </span><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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">r</font>)</span><br/>





<span class="kw">reconsider </span><font color="Maroon" title="c4">e</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#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal3.html#T8" onmouseover="rs('topreal3/T8')" onmouseout="rh()">TOPREAL3:8</a></span>;<br/>
<a NAME="E2:18"/>
(  <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>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K10" title="METRIC_1:func.10">Sphere</a> (<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c2">r</font>) &amp;  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">r</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">cl_Ball</a> (<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c2">r</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal9.html#T14" target="_self" onmouseover="rs('topreal9/T14')" onmouseout="rh()">Th12</a>, <a class="ref" href="topreal9.html#T15" target="_self" onmouseover="rs('topreal9/T15')" onmouseout="rh()">Th13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:18"/>
 <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>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">r</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T15" onmouseover="rs('metric_1/T15')" onmouseout="rh()">METRIC_1:15</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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