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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/>



<a NAME="E1:35"/><i><font color="Green">E47</font></i>: 
 <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E2:35"/>
 <a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:35"/><b>then </b><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">A</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E4:35"/>
<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">A</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:35_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  ;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">p</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  as  non <a href="xboole_0.html#V1">empty</a> <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/>;<br/>
<a NAME="E5:35_1"/><i><font color="Green">E58</font></i>: 
not <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> <font color="Maroon">B</font> is <a href="struct_0.html#V3">empty</a>
 
;<br/>
<a NAME="E6:35_1"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">q</font> <a href="connsp_1.html#R3">is_a_component_of</a> <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"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="connsp_1.html#T43">CONNSP_1:43</a></i>;<br/>
<a NAME="E7:35_1"/>
 <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">q</font> is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<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="p3">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span></span>)</span>
 
;<br/>
<a NAME="E8:35_1"/><b>then </b>
 <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">q</font> is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">G</font> =  <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">q</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="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E10:35_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="connsp_1.html#T40">CONNSP_1:40</a></i>;<br/>
<a NAME="E11:35_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">G</font> <a href="connsp_1.html#R4">is_a_component_of</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i>, <a class="ref" href="connsp_1.html#D6">CONNSP_1:def 6</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1_1"/>
( <font color="Maroon">G</font> is <a href="jordan2c.html#V1" target="_self">Bounded</a> or  not <font color="Maroon">G</font> is <a href="jordan2c.html#V1" target="_self">Bounded</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1"><b>suppose </b></a><a NAME="E1:35_1_1_1"/>
<font color="Maroon">G</font> is <a href="jordan2c.html#V1" target="_self">Bounded</a>
 ;<br/><div class="add"><a NAME="E2:35_1_1_1"/><b>then </b>
<font color="Maroon">G</font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:35_1_1_1"/><b>then </b>
<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:35_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2"><b>suppose </b></a><a NAME="E1:35_1_1_2"/>
not <font color="Maroon">G</font> is <a href="jordan2c.html#V1" target="_self">Bounded</a>
 ;<br/><div class="add"><a NAME="E2:35_1_1_2"/><b>then </b>
<font color="Maroon">G</font> <a href="jordan2c.html#R2" target="_self">is_outside_component_of</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:35_1_1_2"/><b>then </b>
<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:35_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E5:35"/>
<span class="p1">(<span class="default"><a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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