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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:70"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">A</font> is  <a href="tbsp_1.html#V6">bounded</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="jordan2c.html#D2">JORDAN2C:def 2</a><br/>

<b>consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">x</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E3:70"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E4:70"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="metric_6.html#D10">METRIC_6:def 10</a></i>;<br/>
<a NAME="E5:70"/><i><font color="Green">E58</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K10">cl_Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T15">METRIC_1:15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">D</font> =  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B</font> =  <a href="metric_1.html#K10">cl_Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<a NAME="E8:70"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">B</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i/>;<br/>
<a NAME="E9:70"/><i><font color="Green">E66</font></i>: 
 <a href="metric_1.html#K10">cl_Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> is <a href="tbsp_1.html#V6">bounded</a>
 
<b>by </b><i/>;<br/>
<a NAME="E10:70"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:70"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T49">PRE_TOPC:49</a></i>;<br/>
<a NAME="E12:70"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/>
<a NAME="E13:70"/><b>then </b>
<font color="Maroon">D</font> is <a href="tbsp_1.html#V6">bounded</a>
 
<b>by </b><i>, <a class="ref" href="tbsp_1.html#T21">TBSP_1:21</a></i>;<br/>
<b>hence </b><a NAME="E14:70"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> is <a href="jordan2c.html#V1">Bounded</a>
 <b>by </b><i><a class="ref" href="jordan2c.html#D2">JORDAN2C:def 2</a></i>;<br/>


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