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<div class="add">

<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="E1:81"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &amp; <font color="Maroon">A</font> is <a href="jordan2c.html#V1" target="_self">Bounded</a> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">C</font> = <font color="Maroon">A</font> as  <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> <b>by </b><i/>;<br/>
<a NAME="E3:81"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_1"/>
( <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1"><b>suppose </b></a><a NAME="E1:81_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon">x0</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>;<br/><a NAME="E3:81_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">q1</font> = <font color="Maroon">x0</font> as   <a href="pre_topc.html#NM2">Point</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>reconsider </b><font color="Maroon">x0</font> = <font color="Maroon">x0</font> as   <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>by </b><i/>;<br/><b>consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:81_1_1"/><i><font color="Green">E59</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</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>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> holds <br/> <a href="metric_1.html#K4">dist</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> ) )
 <b>by </b><i><a class="ref" href="tbsp_1.html#D9">TBSP_1:def 9</a></i>;<br/><b>reconsider </b><font color="Maroon">o</font> =  <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font> as   <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> ;<br/><b>set </b><font color="Maroon">R0</font> = <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">o</font>,<font color="Maroon">x0</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1;<br/><b>reconsider </b><font color="Maroon">W</font> = <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">o</font>,<font color="Maroon">x0</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1 </span>}</span>  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> ;<br/><b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">W</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/><b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P</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> ;<br/><a NAME="E12:81_1_1"/>
( <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">o</font>,<font color="Maroon">x0</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1 </span>}</span>  &amp; <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> )
 ;<br/><a NAME="E13:81_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
( <font color="Maroon">P</font> is <a href="connsp_1.html#V2">connected</a> &amp; not <font color="Maroon">W</font> is <a href="tbsp_1.html#V6">bounded</a> )
 <b>by </b><i>, </i>;<br/><a NAME="E14:81_1_1"/>
the <a href="struct_0.html#U1">carrier</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> <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="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><b>then reconsider </b><font color="Maroon">P1</font> =  <a href="connsp_3.html#K1">Component_of</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K4">Down</a> <font color="Maroon">P</font>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> 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/><div><i><font color="Green">E61</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:81_1_1_1"/>
<font color="Maroon">W</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font>
 ;<br/><b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:81_1_1_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E4:81_1_1_1"/><i><font color="Green">E64</font></i>: 
not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">o</font>,<font color="Maroon">x0</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1 </span>}</span> 
 <b>by </b><i><a class="ref" href="jordan2c.html#T11" target="_self">Th11</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</font> as   <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>by </b><i><a class="ref" href="jordan2c.html#T11" target="_self">Th11</a></i>;<br/><b>reconsider </b><font color="Maroon">q1</font> = <font color="Maroon">z</font> as   <a href="pre_topc.html#NM2">Point</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="E7:81_1_1_1"/><i><font color="Green">E65</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">o</font>,<font color="Maroon">x0</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1
 <b>by </b><i><a class="ref" href="jordan2c.html#T12" target="_self">Th12</a></i>;<br/><a NAME="E8:81_1_1_1"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">o</font>,<font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="jgraph_1.html#T45">JGRAPH_1:45</a></i>;<br/><a NAME="E9:81_1_1_1"/><b>then </b><i><font color="Green">E66</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">o</font>,<font color="Maroon">z</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">o</font>,<font color="Maroon">x0</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1
 <b>by </b><i>, </i>;<br/><a NAME="E10:81_1_1_1"/><i><font color="Green">E67</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x0</font>,<font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="jordan2c.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E11:81_1_1_1"/><i><font color="Green">E130</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">o</font>,<font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">o</font>,<font color="Maroon">x0</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x0</font>,<font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="metric_1.html#T4">METRIC_1:4</a></i>;<br/><a NAME="E12:81_1_1_1"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">o</font>,<font color="Maroon">x0</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x0</font>,<font color="Maroon">z</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">o</font>,<font color="Maroon">x0</font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E13:81_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">o</font>,<font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">o</font>,<font color="Maroon">x0</font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T15" target="_self">Th15</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E14:81_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">o</font>,<font color="Maroon">x0</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">o</font>,<font color="Maroon">x0</font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T13" target="_self">Th13</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E15:81_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">o</font>,<font color="Maroon">x0</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">o</font>,<font color="Maroon">x0</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">o</font>,<font color="Maroon">x0</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">o</font>,<font color="Maroon">x0</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>hence </b><a NAME="E16:81_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E17:81_1_1"/><b>then </b><i><font color="Green">E131</font></i>: 
<font color="Maroon">P1</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="E18:81_1_1"/><b>then </b><i><font color="Green">E152</font></i>: 
<font color="Maroon">P1</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/><a NAME="E19:81_1_1"/>
 <a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_2"><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="E1:81_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:81_1_1_2"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</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" target="_self">is_outside_component_of</a> <font color="Maroon">A</font> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:81_1_1_2"/><i><font color="Green">E154</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <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" target="_self">is_outside_component_of</a> <font color="Maroon">A</font> </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">B</font> being   <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> such that </b><br/><a NAME="E6:81_1_1_2"/><i><font color="Green">E157</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="jordan2c.html#R2" target="_self">is_outside_component_of</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">C2</font> being   <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">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> such that </b><br/><a NAME="E8:81_1_1_2"/><i><font color="Green">E158</font></i>: 
( <font color="Maroon">C2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">C2</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> &amp; <font color="Maroon">C2</font> is  not  <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> )
 <b>by </b><i><a class="ref" href="jordan2c.html#T13" target="_self">Th13</a>, </i>;<br/>
<b>consider </b><font color="Maroon">D2</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">n</font></span>)</span><b> such that </b><br/><a NAME="E10:81_1_1_2"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">D2</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">o</font>,<font color="Maroon">x0</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1 </span>}</span> 
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">D2</font> = <font color="Maroon">D2</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> ;<br/>
<a NAME="E12:81_1_1_2"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">W</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">D2</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T15" target="_self">Th15</a>, <a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E13:81_1_1_2"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">C2</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i>, <a class="ref" href="connsp_1.html#D5">CONNSP_1:def 5</a></i>;<br/>
<div><i><font color="Green">E162</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:81_1_1_2_1"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">W</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:81_1_1_2_1"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">C2</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">o</font>,<font color="Maroon">x0</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1 </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x8</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="E1:81_1_1_2_1_1"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">x8</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C2</font>
 ;<br/>

<a NAME="E2:81_1_1_2_1_1"/><b>then </b><i><font color="Green">E166</font></i>: 
<font color="Maroon">x8</font> <a href="hidden.html#R2">in</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a>, </i>;<br/>
<b>assume </b><a NAME="E3:81_1_1_2_1_1"/>
not <font color="Maroon">x8</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">o</font>,<font color="Maroon">x0</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1 </span>}</span> 
 ;<br/>

<a NAME="E4:81_1_1_2_1_1"/><b>then </b>
<font color="Maroon">x8</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E5:81_1_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan2c.html#D2" target="_self">Def2</a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">C2</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/><a NAME="E4:81_1_1_2_1"/><i><font color="Green">E167</font></i>: 
not <font color="Maroon">D</font> is <a href="tbsp_1.html#V6">bounded</a>
 <b>by </b><i/>;<br/><a NAME="E5:81_1_1_2_1"/>
<font color="Maroon">D2</font> is <a href="tbsp_1.html#V6">bounded</a>
 <b>by </b><i><a class="ref" href="jordan2c.html#T15" target="_self">Th15</a>, </i>;<br/><b>hence </b><a NAME="E6:81_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan2c.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan2c.html#T16" target="_self">Th16</a>, , <a class="ref" href="tbsp_1.html#T21">TBSP_1:21</a></i>;<br/></div><b>end;</b></div>
<a NAME="E15:81_1_1_2"/><i><font color="Green">E168</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_2_2"><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="E1:81_1_1_2_2"/><i><font color="Green">E169</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">q2</font> = <font color="Maroon">z</font> as   <a href="pre_topc.html#NM2">Point</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>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">q2</font> as   <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>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<a NAME="E4:81_1_1_2_2"/><i><font color="Green">E170</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q2</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q1</font></span>)</span> <a href="euclid.html#K17">+</a> <font color="Maroon">q1</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T52">EUCLID:52</a></i>;<br/>
<a NAME="E5:81_1_1_2_2"/><i><font color="Green">E171</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q2</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q1</font></span>)</span> <a href="euclid.html#K17">+</a> <font color="Maroon">q1</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q2</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q1</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K9">+</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i><a class="ref" href="toprns_1.html#T30">TOPRNS_1:30</a></i>;<br/>
<a NAME="E6:81_1_1_2_2"/><i><font color="Green">E172</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q2</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q1</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">x1</font>,<font color="Maroon">x0</font>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T45">JGRAPH_1:45</a></i>;<br/>
<a NAME="E7:81_1_1_2_2"/><i><font color="Green">E205</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x1</font>,<font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="jordan2c.html#T16" target="_self">Th16</a></i>;<br/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span> = 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i/>
<br/>.= 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x0</font>,<font color="Maroon">o</font>
<b>by </b><i><a class="ref" href="jgraph_1.html#T45">JGRAPH_1:45</a></i>
;<br/>
<a NAME="E9:81_1_1_2_2"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q2</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q1</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K9">+</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">o</font>,<font color="Maroon">x0</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="jordan2c.html#D3" target="_self">Def3</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E10:81_1_1_2_2"/><b>then </b><i><font color="Green">E206</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">o</font>,<font color="Maroon">x0</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:81_1_1_2_2"/>
<font color="Maroon">r</font> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">o</font>,<font color="Maroon">x0</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">o</font>,<font color="Maroon">x0</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E12:81_1_1_2_2"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">o</font>,<font color="Maroon">x0</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E13:81_1_1_2_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T15" target="_self">Th15</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:81_1_1_2"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> <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">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<a NAME="E17:81_1_1_2"/><b>then </b>
<font color="Maroon">D2</font> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</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">n</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_1.html#T34">XBOOLE_1:34</a></i>;<br/>
<a NAME="E18:81_1_1_2"/><b>then </b><i><font color="Green">E207</font></i>: 
<font color="Maroon">D2</font> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 
;<br/>
<a NAME="E19:81_1_1_2"/><i><font color="Green">E208</font></i>: 
<font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D2</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i>, <a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/>
<a NAME="E20:81_1_1_2"/><i><font color="Green">E209</font></i>: 
<font color="Maroon">P</font> <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="jordan2c.html#T9" target="_self">Th9</a>, <a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/>
<a NAME="E21:81_1_1_2"/><b>then </b><i><font color="Green">E210</font></i>: 
 <a href="connsp_3.html#K4">Down</a> <font color="Maroon">P</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E22:81_1_1_2"/>
<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">P</font> is <a href="connsp_1.html#V1">connected</a>
 
<b>by </b><i>, <a class="ref" href="connsp_1.html#D3">CONNSP_1:def 3</a></i>;<br/>
<a NAME="E23:81_1_1_2"/><b>then </b>
<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> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K4">Down</a> <font color="Maroon">P</font>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> is <a href="connsp_1.html#V1">connected</a>
 
<b>by </b><i>, , <a class="ref" href="pre_topc.html#T28">PRE_TOPC:28</a></i>;<br/>
<a NAME="E24:81_1_1_2"/><b>then </b><i><font color="Green">E211</font></i>: 
 <a href="connsp_3.html#K4">Down</a> <font color="Maroon">P</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="ref" href="connsp_1.html#D3">CONNSP_1:def 3</a></i>;<br/>
<a NAME="E25:81_1_1_2"/><i><font color="Green">E212</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">D2</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E26:81_1_1_2"/>
<font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">D2</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> <a href="subset_1.html#K9">/\</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="jordan2c.html#T16" target="_self">Th16</a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/>
<a NAME="E27:81_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">D2</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/>
<a NAME="E28:81_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="connsp_3.html#K4">Down</a> <font color="Maroon">P</font>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E29:81_1_1_2"/><b>then </b><i><font color="Green">E213</font></i>: 
 <a href="connsp_3.html#K4">Down</a> <font color="Maroon">P</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C2</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">G</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>, , </i>;<br/>
<a NAME="E31:81_1_1_2"/>
<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">G</font> is  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>
 
;<br/>
<a NAME="E32:81_1_1_2"/><b>then </b>
<font color="Maroon">C2</font> <a href="tarski.html#R1">c=</a>  <a href="connsp_3.html#K1">Component_of</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K4">Down</a> <font color="Maroon">P</font>,<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="jordan2c.html#D3" target="_self">Def3</a>, <a class="ref" href="jordan2c.html#T17" target="_self">Th17</a>, <a class="ref" href="connsp_3.html#T16">CONNSP_3:16</a></i>;<br/>
<b>hence </b><a NAME="E33:81_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font>
 <b>by </b><i>, <a class="ref" href="jordan2c.html#T13" target="_self">Th13</a>, </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E20:81_1_1"/>
 <a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">A</font> <a href="jordan2c.html#R2" target="_self">is_outside_component_of</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T11" target="_self">Th11</a>, <a class="ref" href="jordan2c.html#T12" target="_self">Th12</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_2"><b>suppose </b></a><a NAME="E1:81_1_2"/><i><font color="Green">E214</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:81_1_2"/><b>then </b><i><font color="Green">E215</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</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>
 ;<br/><a NAME="E3:81_1_2"/>
 <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">W</font> =  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</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> ;<br/><b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">W</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/><b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P</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> ;<br/><a NAME="E7:81_1_2"/><i><font color="Green">E216</font></i>: 
<font color="Maroon">P</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i/>;<br/><a NAME="E8:81_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E9:81_1_2"/><b>then </b><i><font color="Green">E217</font></i>: 
not <font color="Maroon">W</font> is <a href="tbsp_1.html#V6">bounded</a>
 <b>by </b><i/>;<br/><a NAME="E10:81_1_2"/>
<font color="Maroon">W</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/><a NAME="E11:81_1_2"/><b>then </b><i><font color="Green">E218</font></i>: 
<font color="Maroon">W</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E12:81_1_2"/>
the <a href="struct_0.html#U1">carrier</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> <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="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><b>then reconsider </b><font color="Maroon">P1</font> =  <a href="connsp_3.html#K1">Component_of</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K4">Down</a> <font color="Maroon">P</font>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> 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="E14:81_1_2"/><i><font color="Green">E219</font></i>: 
<font color="Maroon">P1</font> <a href="jordan2c.html#R2" target="_self">is_outside_component_of</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="jordan2c.html#T9" target="_self">Th9</a>, </i>;<br/><a NAME="E15:81_1_2"/><i><font color="Green">E220</font></i>: 
 <a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E16:81_1_2"/><i><font color="Green">E221</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="hidden.html#R1">=</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/><a NAME="E17:81_1_2"/><i><font color="Green">E222</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <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"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</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="tsep_1.html#T3">TSEP_1:3</a></i>;<br/><a NAME="E18:81_1_2"/>
<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="E19:81_1_2"/><b>then </b><i><font color="Green">E223</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <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"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <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"><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>
 <b>by </b><i/>;<br/><a NAME="E20:81_1_2"/>
<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</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/>;<br/><a NAME="E21:81_1_2"/><b>then </b>
<font color="Maroon">P1</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="jordan2c.html#T13" target="_self">Th13</a>, , <a class="ref" href="connsp_3.html#T7">CONNSP_3:7</a></i>;<br/><b>hence </b><a NAME="E22:81_1_2"/>
 <a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">A</font> <a href="jordan2c.html#R2" target="_self">is_outside_component_of</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T11" target="_self">Th11</a>, <a class="ref" href="jordan2c.html#T12" target="_self">Th12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
