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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>, <font color="Maroon">c<sub>3</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>let </b><font color="Maroon">c<sub>4</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>let </b><font color="Maroon">c<sub>5</sub></font> be   <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>;<br/>







<b>assume </b><a NAME="E1:76"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="connsp_1.html#V2">connected</a> &amp; not <font color="Maroon">c<sub>5</sub></font> is <a href="tbsp_1.html#V6">bounded</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="connsp_3.html#K1">skl</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K4">Down</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<a NAME="E2:76"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
;<br/>
<a NAME="E3:76"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="jordan2c.html#E1:76"><i><font color="Green">E70</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:76_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:76_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
 ;<br/><a NAME="E3:76_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K2">[#]</a> the <a href="struct_0.html#U1">carrier</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/><a NAME="E4:76_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</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>hence </b><a NAME="E5:76_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:76"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan2c.html#E2:76"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan2c.html#E3:76"><i><font color="Green">E72</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font></span>)</span> ;<br/>
<a NAME="E6:76"/><i><font color="Green">E73</font></i>: 
not <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>6</sub></font> is <a href="struct_0.html#V3">empty</a>
 
;<br/>
<i><font color="Green">E74</font></i>:  <a href="connsp_3.html#K4">Down</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> = 
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>

<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="subset_1.html#T32">SUBSET_1:32</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="jordan2c.html#E1:76"><i><font color="Green">E70</font></i></a>, <a class="ref" href="xboole_1.html#T83">XBOOLE_1:83</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM2">Subset</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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> =  <a href="connsp_3.html#K1">skl</a> <font color="Maroon">c<sub>7</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="txt" href="jordan2c.html#E1:76"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan2c.html#E7:76"><i><font color="Green">E74</font></i></a>, <a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<a NAME="E10:76"/>
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="txt" href="jordan2c.html#E1:76"><i><font color="Green">E70</font></i></a>, <a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/>
<a NAME="E11:76"/><b>then </b><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T28">PRE_TOPC:28</a></i>;<br/>
<a NAME="E12:76"/>
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font> is <a href="connsp_1.html#V1">connected</a>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E1:76"><i><font color="Green">E70</font></i></a>, <a class="ref" href="connsp_1.html#D3">CONNSP_1:def 3</a></i>;<br/>
<a NAME="E13:76"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E11:76"><i><font color="Green">E75</font></i></a>, <a class="ref" href="connsp_1.html#D3">CONNSP_1:def 3</a></i>;<br/>
<a NAME="E14:76"/><b>then </b><i><font color="Green">E77</font></i>: 
 <a href="connsp_3.html#K1">skl</a> <font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E1:76"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan2c.html#E2:76"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan2c.html#E6:76"><i><font color="Green">E73</font></i></a>, <a class="ref" href="connsp_3.html#T9">CONNSP_3:9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:76_3"/>
for <font color="Olive">b<sub>1</sub></font> being  <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> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="tbsp_1.html#V6">bounded</a>
 ;<br/><a NAME="E2:76_3"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is <a href="tbsp_1.html#V6">bounded</a>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:76"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan2c.html#E7:76"><i><font color="Green">E74</font></i></a></i>;<br/><a NAME="E3:76_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:76"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan2c.html#E13:76"><i><font color="Green">E76</font></i></a>, <a class="ref" href="connsp_3.html#T1">CONNSP_3:1</a></i>;<br/><b>hence </b><a NAME="E4:76_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:76"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan2c.html#E2:76_3"><i><font color="Green">E78</font></i></a>, <a class="ref" href="tbsp_1.html#T21">TBSP_1:21</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E16:76"/>
<font color="Maroon">c<sub>3</sub></font> <a href="jordan2c.html#R2" target="_self">is_outside_component_of</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:76"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan2c.html#E7:76"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan2c.html#E14:76"><i><font color="Green">E77</font></i></a>, <a class="ref" href="jordan2c.html#T18" target="_self">Th18</a></i>;<br/>


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