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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:70"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></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">c<sub>3</sub></font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></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">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E4:70"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="topreal6.html#E1:70"><i><font color="Green">E36</font></i></a>, <a class="ref" href="metric_6.html#D10">METRIC_6:def 10</a></i>;<br/>
<a NAME="E5:70"/><i><font color="Green">E38</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K10">cl_Ball</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></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">c<sub>5</sub></font> =  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></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">c<sub>6</sub></font> =  <a href="metric_1.html#K10">cl_Ball</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></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">E39</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="topreal6.html#T66" target="_self">Th66</a></i>;<br/>
<a NAME="E9:70"/><i><font color="Green">E40</font></i>: 
 <a href="metric_1.html#K10">cl_Ball</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> is <a href="tbsp_1.html#V6">bounded</a>
 
<b>by </b><i><a class="ref" href="topreal6.html#T67" target="_self">Th67</a></i>;<br/>
<a NAME="E10:70"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="topreal6.html#E4:70"><i><font color="Green">E37</font></i></a>, <a class="txt" href="topreal6.html#E5:70"><i><font color="Green">E38</font></i></a>, <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">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="topreal6.html#E8:70"><i><font color="Green">E39</font></i></a>, <a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/>
<a NAME="E13:70"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> is <a href="tbsp_1.html#V6">bounded</a>
 
<b>by </b><i><a class="txt" href="topreal6.html#E9:70"><i><font color="Green">E40</font></i></a>, <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">c<sub>2</sub></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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