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<span class="kw">let </span><font color="Maroon" title="c1">M</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</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">z</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><br/>  for <font color="Olive" title="b2">r</font> being   <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="metric_1.html#K11" title="METRIC_1:func.11">Sphere</a> (<font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">r</font>) is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">z</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>; <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   <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="metric_1.html#K11" title="METRIC_1:func.11">Sphere</a> (<font color="Maroon" title="c2">z</font>,<font color="Olive" title="b1">r</font>) is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K11" title="METRIC_1:func.11">Sphere</a> (<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">r</font>) is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> </span><br/>





<a NAME="E1:70"/>
 <a href="metric_1.html#K11" title="METRIC_1:func.11">Sphere</a> (<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">r</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T15" title="METRIC_1:th.15">METRIC_1:15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:70"/>
 <a href="metric_1.html#K11" title="METRIC_1:func.11">Sphere</a> (<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">r</font>) is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal6.html#T59" target="_self" title="TOPREAL6:th.59">Th67</a>, <a class="ref" href="tbsp_1.html#T14" title="TBSP_1:th.14">TBSP_1:14</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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