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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:17"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/>

<a NAME="E2:17"/><b>then </b><i><font color="Green">E36</font></i>: 
not  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> is <a href="jordan2c.html#V1">Bounded</a>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T41">JORDAN2C:41</a></i>;<br/>
<b>thus </b><a NAME="E3:17"/>
 <a href="jordan2c.html#K2">UBD</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K1">{}</a> <span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<a NAME="E4:17"/><i><font color="Green">E40</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="connsp_1.html#R3">is_a_component_of</a>  <a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T23">JORDAN2C:23</a></i>;<br/>
<a NAME="E5:17"/>
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T3">TSEP_1:3</a></i>;<br/>
<a NAME="E6:17"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="connsp_1.html#R4">is_a_component_of</a>  <a href="pre_topc.html#K2">[#]</a> <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="connsp_1.html#D6">CONNSP_1:def 6</a></i>;<br/>
<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is   <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> : <font color="Olive">B</font> <a href="jordan2c.html#R2">is_outside_component_of</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> </span>}</span> ;<br/>
<a NAME="E7:17"/>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K1">{}</a> <span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
 
;<br/>
<a NAME="E8:17"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="jordan2c.html#R2">is_outside_component_of</a>  <a href="pre_topc.html#K1">{}</a> <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="jordan2c.html#D4">JORDAN2C:def 4</a></i>;<br/>
<a NAME="E9:17"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is   <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> : <font color="Olive">B</font> <a href="jordan2c.html#R2">is_outside_component_of</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> </span>}</span> 
 
;<br/>
<a NAME="E10:17"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is   <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> : <font color="Olive">B</font> <a href="jordan2c.html#R2">is_outside_component_of</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> </span>}</span> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<b>hence </b><a NAME="E11:17"/>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K2">UBD</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K1">{}</a> <span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan2c.html#D6">JORDAN2C:def 6</a></i>;<br/>


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