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<span class="kw">let </span><font color="Maroon" title="c7">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> ; <a class="txt" 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="c7">n</font></span>)</span><br/>  for <font color="Olive" title="b2">r</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</a>   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds   <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">r</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">r</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c8">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="c7">n</font></span>)</span>; <a class="txt" 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   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</a>   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds   <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c8">x</font>,<font color="Olive" title="b1">r</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c8">x</font>,<font color="Olive" title="b1">r</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c9">r</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</a>   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">r</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">r</font>)</span><br/>





<span class="kw">set </span><font color="Maroon" title="c10">P</font> =  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">r</font>);<br/>
<span class="kw">thus </span> <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">r</font>)</span>)</span> = 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p2">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">r</font>)</span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">r</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T42" title="TOPS_1:th.42">TOPS_1:42</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">r</font>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">r</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan.html#T23" target="_self" title="JORDAN:th.23">Th23</a></span>
<br/>.= 
 <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">r</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan.html#T19" target="_self" title="JORDAN:th.19">Th19</a></span>
; <a class="txt" 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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