<?xml version="1.0"?>
<div class="add">

<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>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>





<b>assume </b><a NAME="E1:103"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span>  )
 ;<br/>

<a NAME="E2:103"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span>  <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">c<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></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="E1:103_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:103_1"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>
<b>thus </b><a NAME="E4:103_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</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>
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:103_1"><i><font color="Green">E96</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> </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">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>5</sub></font> = <font color="Maroon">c<sub>4</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/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</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> ;<br/>
<a NAME="E6:103"/><i><font color="Green">E96</font></i>: 
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">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>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></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">c<sub>7</sub></font> =  <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>6</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E8:103"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></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="E1:103_2"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></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">c<sub>1</sub></font></span>)</span> ;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:103_2"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:103_2"><i><font color="Green">E98</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:103_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:103_2_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:103"><i><font color="Green">E95</font></i></a></i>;<br/><b>hence </b><a NAME="E3:103_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:103_2"><i><font color="Green">E99</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:103_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</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> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>2</sub></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="E7:103_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:103"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="toprns_1.html#T24">TOPRNS_1:24</a></i>;<br/>
<a NAME="E10:103"/><b>then </b><i><font color="Green">E98</font></i>: 
 <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E1:103"><i><font color="Green">E95</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>2</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> <b>by </b><i><a class="txt" href="jordan2c.html#E8:103"><i><font color="Green">E97</font></i></a></i>;<br/>
<a NAME="E12:103"/><i><font color="Green">E99</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>8</sub></font> is <a href="struct_0.html#V3">empty</a>
 
;<br/>
<a NAME="E13:103"/><i><font color="Green">E100</font></i>: 
 <a href="connsp_3.html#K4">Down</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="jordan2c.html#E8:103"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan2c.html#E10:103"><i><font color="Green">E98</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E14:103"/><i><font color="Green">E101</font></i>: 
 <a href="connsp_3.html#K4">Down</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E8:103"><i><font color="Green">E97</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E15:103"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T79" target="_self">Th79</a></i>;<br/>
<a NAME="E16:103"/><b>then </b>
<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="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="E17:103"/><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">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>2</sub></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">c<sub>6</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></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="txt" href="jordan2c.html#E8:103"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan2c.html#E14:103"><i><font color="Green">E101</font></i></a>, <a class="ref" href="pre_topc.html#T28">PRE_TOPC:28</a></i>;<br/>
<a NAME="E18:103"/><b>then </b><i><font color="Green">E102</font></i>: 
 <a href="connsp_3.html#K4">Down</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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="E19:103"/><b>then </b><i><font color="Green">E103</font></i>: 
 <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>6</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </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">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>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E12:103"><i><font color="Green">E99</font></i></a>, <a class="txt" href="jordan2c.html#E13:103"><i><font color="Green">E100</font></i></a>, <a class="ref" href="connsp_3.html#T9">CONNSP_3:9</a></i>;<br/>
<a NAME="E20:103"/><b>then </b><i><font color="Green">E104</font></i>: 
 <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>6</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> 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/>
<a NAME="E21:103"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</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>6</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E10:103"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan2c.html#E12:103"><i><font color="Green">E99</font></i></a>, <a class="txt" href="jordan2c.html#E14:103"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jordan2c.html#E18:103"><i><font color="Green">E102</font></i></a>, <a class="ref" href="connsp_3.html#T13">CONNSP_3:13</a></i>;<br/>
<a NAME="E22:103"/>
 <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>6</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> 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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_3"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> =  <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>6</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jordan2c.html#E6:103"><i><font color="Green">E96</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</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>11</sub></font> = <font color="Maroon">c<sub>10</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> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:103_3_1"/>
not <font color="Maroon">c<sub>9</sub></font> is <a href="jordan2c.html#V1" target="_self">Bounded</a>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:103_3_1"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>12</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="jordan2c.html#T40" target="_self">Th40</a></i>;<br/><b>set </b><font color="Maroon">c<sub>13</sub></font> =  <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E4:103_3_1"/><i><font color="Green">E107</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:103"><i><font color="Green">E95</font></i></a>, <a class="ref" href="toprns_1.html#T24">TOPRNS_1:24</a></i>;<br/><a NAME="E5:103_3_1"/><i><font color="Green">E108</font></i>: 
 <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:103"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan2c.html#E3:103_3_1"><i><font color="Green">E106</font></i></a>, <a class="ref" href="toprns_1.html#T24">TOPRNS_1:24</a></i>;<br/><a NAME="E6:103_3_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> 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">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>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 ;<br/><a NAME="E7:103_3_1"/><b>then </b><i><font color="Green">E109</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></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/><a NAME="E8:103_3_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><a NAME="E9:103_3_1"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="connsp_3.html#K4">Down</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>2</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> <b>by </b><i><a class="txt" href="jordan2c.html#E8:103"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan2c.html#E10:103"><i><font color="Green">E98</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <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>14</sub></font> as  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a> ;<br/><a NAME="E12:103_3_1"/>
<font color="Maroon">c<sub>15</sub></font> is <a href="connsp_2.html#V1">locally_connected</a>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:103"><i><font color="Green">E95</font></i></a>, <a class="ref" href="jordan2c.html#T90" target="_self">Th90</a></i>;<br/><a NAME="E13:103_3_1"/><b>then </b>
 <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>6</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="jordan2c.html#E19:103"><i><font color="Green">E103</font></i></a>, <a class="ref" href="connsp_2.html#T21">CONNSP_2:21</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>16</sub></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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E15:103_3_1"/><i><font color="Green">E111</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="connsp_3.html#K4">Down</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="subset_1.html#K9">/\</a> <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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E9:103_3_1"><i><font color="Green">E110</font></i></a>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/><a NAME="E16:103_3_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E9:103_3_1"><i><font color="Green">E110</font></i></a>, <a class="txt" href="jordan2c.html#E15:103_3_1"><i><font color="Green">E111</font></i></a>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/><a NAME="E17:103_3_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:103"><i><font color="Green">E95</font></i></a>, <a class="ref" href="jordan2c.html#T88" target="_self">Th88</a></i>;<br/><a NAME="E18:103_3_1"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> is <a href="connsp_1.html#V2">connected</a> &amp; <font color="Maroon">c<sub>9</sub></font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E20:103"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan2c.html#E15:103_3_1"><i><font color="Green">E111</font></i></a>, <a class="txt" href="jordan2c.html#E16:103_3_1"><i><font color="Green">E112</font></i></a>, <a class="ref" href="connsp_1.html#T24">CONNSP_1:24</a>, <a class="ref" href="tops_1.html#T38">TOPS_1:38</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <a href="topmetr.html#K5">I[01]</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><b> such that </b><br/><a NAME="E20:103_3_1"/><i><font color="Green">E113</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E10:103"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan2c.html#E21:103"><i><font color="Green">E105</font></i></a>, <a class="txt" href="jordan2c.html#E3:103_3_1"><i><font color="Green">E106</font></i></a>, <a class="txt" href="jordan2c.html#E5:103_3_1"><i><font color="Green">E108</font></i></a>, <a class="ref" href="jordan2c.html#T87" target="_self">Th87</a></i>;<br/><a NAME="E21:103_3_1"/><i><font color="Green">E114</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><a href="jordan2c.html#K6" target="_self">pi</a> <font color="Maroon">c<sub>17</sub></font>,0</span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:103_3_1"><i><font color="Green">E107</font></i></a>, <a class="txt" href="jordan2c.html#E20:103_3_1"><i><font color="Green">E113</font></i></a>, <a class="ref" href="jordan2c.html#D10" target="_self">Def10</a>, <a class="ref" href="borsuk_1.html#D17">BORSUK_1:def 17</a></i>;<br/><a NAME="E22:103_3_1"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><a href="jordan2c.html#K6" target="_self">pi</a> <font color="Maroon">c<sub>17</sub></font>,1</span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:103_3_1"><i><font color="Green">E106</font></i></a>, <a class="txt" href="jordan2c.html#E20:103_3_1"><i><font color="Green">E113</font></i></a>, <a class="ref" href="jordan2c.html#D10" target="_self">Def10</a>, <a class="ref" href="borsuk_1.html#D18">BORSUK_1:def 18</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E24:103_3_1"/><i><font color="Green">E115</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><a href="jordan2c.html#K6" target="_self">pi</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E20:103_3_1"><i><font color="Green">E113</font></i></a>, <a class="txt" href="jordan2c.html#E21:103_3_1"><i><font color="Green">E114</font></i></a>, <a class="ref" href="jordan2c.html#T94" target="_self">Th94</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></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">c<sub>1</sub></font></span>)</span> ;<br/><a NAME="E26:103_3_1"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>19</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E24:103_3_1"><i><font color="Green">E115</font></i></a>, <a class="ref" href="jordan2c.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E27:103_3_1"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:103"><i><font color="Green">E95</font></i></a></i>;<br/><a NAME="E28:103_3_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> 
 ;<br/><a NAME="E29:103_3_1"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E30:103_3_1"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E31:103_3_1"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E20:103_3_1"><i><font color="Green">E113</font></i></a></i>;<br/><a NAME="E32:103_3_1"/><b>then </b>
<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>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="subset_1.html#K1">{}</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>
 <b>by </b><i><a class="txt" href="jordan2c.html#E7:103_3_1"><i><font color="Green">E109</font></i></a>, <a class="txt" href="jordan2c.html#E27:103_3_1"><i><font color="Green">E116</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E33:103_3_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E34:103_3_1"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:103_3"/><b>then </b>
<font color="Maroon">c<sub>9</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>
 
<b>by </b><i><a class="ref" href="jordan2c.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E6:103_3"/>
 <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>6</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> 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>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E23:103"/><b>then </b><i><font color="Green">E106</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E19:103"><i><font color="Green">E103</font></i></a>, <a class="ref" href="jordan2c.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E24:103"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></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="E1:103_4"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E2:103_4"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan2c.html#E23:103"><i><font color="Green">E106</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:103_4"/>
<font color="Maroon">c<sub>9</sub></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<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:103_4"><i><font color="Green">E108</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_5"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:103_5_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 or <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_5_1_1"><b>case </b></a><a NAME="E1:103_5_1_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 ;<br/><div class="add"><a NAME="E2:103_5_1_1"/>
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_5_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></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="E1:103_5_1_1_1"/>
<font color="Maroon">c<sub>9</sub></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<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:103_5_1_1_1"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">c<sub>2</sub></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">c<sub>11</sub></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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E5:103_5_1_1_1"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:103_5_1_1_1"><i><font color="Green">E109</font></i></a></i>;<br/>
<a NAME="E6:103_5_1_1_1"/><i><font color="Green">E111</font></i>: 
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">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>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></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>consider </b><font color="Maroon">c<sub>12</sub></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">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>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span><b> such that </b><br/><a NAME="E8:103_5_1_1_1"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>12</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>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <font color="Maroon">c<sub>12</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> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E5:103_5_1_1_1"><i><font color="Green">E110</font></i></a>, <a class="ref" href="jordan2c.html#T17" target="_self">Th17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>9</sub></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">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>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <b>by </b><i><a class="txt" href="jordan2c.html#E3:103_5_1_1_1"><i><font color="Green">E109</font></i></a>, <a class="txt" href="jordan2c.html#E5:103_5_1_1_1"><i><font color="Green">E110</font></i></a>, <a class="txt" href="jordan2c.html#E8:103_5_1_1_1"><i><font color="Green">E112</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>2</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> <b>by </b><i><a class="txt" href="jordan2c.html#E8:103"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan2c.html#E10:103"><i><font color="Green">E98</font></i></a></i>;<br/>
<a NAME="E11:103_5_1_1_1"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> 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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 
;<br/>
<a NAME="E12:103_5_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</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> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E6:103_5_1_1_1"><i><font color="Green">E111</font></i></a></i>;<br/>
<a NAME="E13:103_5_1_1_1"/><b>then </b><i><font color="Green">E114</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</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> &amp; not <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>13</sub></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">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jordan2c.html#E6:103_5_1_1_1"><i><font color="Green">E111</font></i></a>, <a class="txt" href="jordan2c.html#E11:103_5_1_1_1"><i><font color="Green">E113</font></i></a></i>;<br/>
<a NAME="E15:103_5_1_1_1"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E1:103"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan2c.html#E13:103_5_1_1_1"><i><font color="Green">E114</font></i></a></i>;<br/>
<a NAME="E16:103_5_1_1_1"/><b>then </b><i><font color="Green">E115</font></i>: 
( <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> or <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_5_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_5_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:103_5_1_1_1_1_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span>  or <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="jordan2c.html#E16:103_5_1_1_1"><i><font color="Green">E115</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_5_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:103_5_1_1_1_1_1_1"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 ;<br/><div class="add"><a NAME="E2:103_5_1_1_1_1_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_5_1_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></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="E1:103_5_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>17</sub></font> being   <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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:103_5_1_1_1_1_1_1_1"/><i><font color="Green">E117</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>17</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_5_1_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:103_5_1_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:103_5_1_1_1_1_1_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:103"><i><font color="Green">E95</font></i></a></i>;<br/><b>hence </b><a NAME="E3:103_5_1_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:103_5_1_1_1_1_1_1_1"><i><font color="Green">E117</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:103_5_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</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> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>2</sub></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="E6:103_5_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:103_5_1_1_1_1_1_1_1"><i><font color="Green">E117</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span>  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>2</sub></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">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</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>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> ;<br/><a NAME="E5:103_5_1_1_1_1_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span>  <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">c<sub>1</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_5_1_1_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></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="E1:103_5_1_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>19</sub></font> being   <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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:103_5_1_1_1_1_1_1_2"/><i><font color="Green">E117</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>19</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>
<b>thus </b><a NAME="E4:103_5_1_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</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>
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:103_5_1_1_1_1_1_1_2"><i><font color="Green">E117</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> </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">c<sub>1</sub></font></span>)</span> ;<br/><a NAME="E7:103_5_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>18</sub></font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:103_5_1_1"><i><font color="Green">E108</font></i></a>, <a class="ref" href="jordan2c.html#T59" target="_self">Th59</a></i>;<br/><a NAME="E8:103_5_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E117</font></i>: 
<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>18</sub></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="E9:103_5_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>18</sub></font> is   <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>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> </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">c<sub>1</sub></font></span>)</span> ;<br/><a NAME="E11:103_5_1_1_1_1_1_1"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">c<sub>18</sub></font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:103_5_1_1"><i><font color="Green">E108</font></i></a>, <a class="ref" href="jordan2c.html#T59" target="_self">Th59</a></i>;<br/><a NAME="E12:103_5_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/><a NAME="E13:103_5_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E119</font></i>: 
not <font color="Maroon">c<sub>19</sub></font> is <a href="tbsp_1.html#V6">bounded</a>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:103_5_1_1"><i><font color="Green">E108</font></i></a>, <a class="ref" href="jordan2c.html#T69" target="_self">Th69</a></i>;<br/><div><i><font color="Green">E120</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="103_5_1_1_1_1_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:103_5_1_1_1_1_1_1_3"/>
<font color="Maroon">c<sub>19</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>20</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:103_5_1_1_1_1_1_1_3"/><i><font color="Green">E121</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E4:103_5_1_1_1_1_1_1_3"/><i><font color="Green">E122</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:103_5_1_1_1_1_1_1_3"><i><font color="Green">E121</font></i></a></i>;<br/><a NAME="E5:103_5_1_1_1_1_1_1_3"/>
ex <font color="Olive">b<sub>1</sub></font> being  <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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:103"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan2c.html#E3:103_5_1_1_1_1_1_1_3"><i><font color="Green">E121</font></i></a></i>;<br/><b>hence </b><a NAME="E6:103_5_1_1_1_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:103_5_1_1_1_1_1_1_3"><i><font color="Green">E122</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E15:103_5_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E16:103_5_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#T32">SUBSET_1:32</a></i>;<br/><a NAME="E17:103_5_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><a NAME="E18:103_5_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E122</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><a NAME="E19:103_5_1_1_1_1_1_1"/>
<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>18</sub></font> <a href="hidden.html#R1">=</a> <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>2</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>17</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E17:103_5_1_1_1_1_1_1"><i><font color="Green">E121</font></i></a>, <a class="ref" href="pre_topc.html#T28">PRE_TOPC:28</a></i>;<br/><a NAME="E20:103_5_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">c<sub>17</sub></font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="txt" href="jordan2c.html#E8:103_5_1_1_1_1_1_1"><i><font color="Green">E117</font></i></a>, <a class="ref" href="connsp_1.html#D3">CONNSP_1:def 3</a></i>;<br/><a NAME="E21:103_5_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="connsp_3.html#K4">Down</a> <font color="Maroon">c<sub>18</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E18:103_5_1_1_1_1_1_1"><i><font color="Green">E122</font></i></a></i>;<br/><a NAME="E22:103_5_1_1_1_1_1_1"/><b>then </b>
 <a href="connsp_3.html#K5">Up</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K1">skl</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="jordan2c.html#R2" target="_self">is_outside_component_of</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E11:103_5_1_1_1_1_1_1"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan2c.html#E13:103_5_1_1_1_1_1_1"><i><font color="Green">E119</font></i></a>, <a class="txt" href="jordan2c.html#E14:103_5_1_1_1_1_1_1"><i><font color="Green">E120</font></i></a>, <a class="ref" href="jordan2c.html#T71" target="_self">Th71</a></i>;<br/><a NAME="E23:103_5_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E124</font></i>: 
 <a href="connsp_3.html#K1">skl</a> <font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T27" target="_self">Th27</a></i>;<br/><a NAME="E24:103_5_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a>  <a href="connsp_3.html#K1">skl</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E20:103_5_1_1_1_1_1_1"><i><font color="Green">E123</font></i></a>, <a class="ref" href="connsp_3.html#T1">CONNSP_3:1</a></i>;<br/><a NAME="E25:103_5_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E125</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="connsp_3.html#K1">skl</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:103_5_1_1_1_1_1_1"><i><font color="Green">E116</font></i></a></i>;<br/><a NAME="E26:103_5_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E5:103_5_1_1_1"><i><font color="Green">E110</font></i></a>, <a class="ref" href="jordan2c.html#T26" target="_self">Th26</a></i>;<br/><a NAME="E27:103_5_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></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">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:103_5_1_1_1"><i><font color="Green">E109</font></i></a>, <a class="txt" href="jordan2c.html#E5:103_5_1_1_1"><i><font color="Green">E110</font></i></a>, <a class="txt" href="jordan2c.html#E23:103_5_1_1_1_1_1_1"><i><font color="Green">E124</font></i></a>, <a class="txt" href="jordan2c.html#E25:103_5_1_1_1_1_1_1"><i><font color="Green">E125</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E28:103_5_1_1_1_1_1_1"/><b>then </b>
 <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T4">XBOOLE_0:4</a></i>;<br/><b>hence </b><a NAME="E29:103_5_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T28" target="_self">Th28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_5_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:103_5_1_1_1_1_1_2"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 ;<br/><div class="add"><a NAME="E2:103_5_1_1_1_1_1_2"/>
 <a href="connsp_3.html#K4">Down</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="tarski.html#R1">c=</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>6</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E18:103"><i><font color="Green">E102</font></i></a>, <a class="ref" href="connsp_3.html#T1">CONNSP_3:1</a></i>;<br/><b>hence </b><a NAME="E3:103_5_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E14:103"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jordan2c.html#E1:103_5_1_1_1_1_1_2"><i><font color="Green">E116</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E18:103_5_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>


</div><b>end;</b></div><a NAME="E3:103_5_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="jordan2c.html#E24:103"><i><font color="Green">E107</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E4:103_5_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>hence </b><a NAME="E5:103_5_1_1"/>
ex <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#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E23:103"><i><font color="Green">E106</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_5_1_2"><b>case </b></a><a NAME="E1:103_5_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2
 ;<br/><div class="add"><a NAME="E2:103_5_1_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K23">+</a> 1
 ;<br/><a NAME="E3:103_5_1_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:103_5_1_2"/><b>then </b><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:103"><i><font color="Green">E95</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E5:103_5_1_2"/>
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_5_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></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="E1:103_5_1_2_1"/>
<font color="Maroon">c<sub>9</sub></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<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:103_5_1_2_1"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">c<sub>2</sub></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">c<sub>11</sub></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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E5:103_5_1_2_1"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:103_5_1_2_1"><i><font color="Green">E109</font></i></a></i>;<br/>
<a NAME="E6:103_5_1_2_1"/><i><font color="Green">E111</font></i>: 
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">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>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></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>consider </b><font color="Maroon">c<sub>12</sub></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">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>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span><b> such that </b><br/><a NAME="E8:103_5_1_2_1"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>12</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>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <font color="Maroon">c<sub>12</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> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E5:103_5_1_2_1"><i><font color="Green">E110</font></i></a>, <a class="ref" href="jordan2c.html#T17" target="_self">Th17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>9</sub></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">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>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <b>by </b><i><a class="txt" href="jordan2c.html#E3:103_5_1_2_1"><i><font color="Green">E109</font></i></a>, <a class="txt" href="jordan2c.html#E5:103_5_1_2_1"><i><font color="Green">E110</font></i></a>, <a class="txt" href="jordan2c.html#E8:103_5_1_2_1"><i><font color="Green">E112</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>2</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> <b>by </b><i><a class="txt" href="jordan2c.html#E8:103"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan2c.html#E10:103"><i><font color="Green">E98</font></i></a></i>;<br/>
<a NAME="E11:103_5_1_2_1"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> 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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 
;<br/>
<a NAME="E12:103_5_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</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> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E6:103_5_1_2_1"><i><font color="Green">E111</font></i></a></i>;<br/>
<a NAME="E13:103_5_1_2_1"/><b>then </b><i><font color="Green">E114</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</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> &amp; not <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>13</sub></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">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jordan2c.html#E6:103_5_1_2_1"><i><font color="Green">E111</font></i></a>, <a class="txt" href="jordan2c.html#E11:103_5_1_2_1"><i><font color="Green">E113</font></i></a></i>;<br/>
<a NAME="E15:103_5_1_2_1"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E1:103"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan2c.html#E13:103_5_1_2_1"><i><font color="Green">E114</font></i></a></i>;<br/>
<a NAME="E16:103_5_1_2_1"/><b>then </b><i><font color="Green">E115</font></i>: 
( <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> or <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_5_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_5_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:103_5_1_2_1_1_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span>  or <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="jordan2c.html#E16:103_5_1_2_1"><i><font color="Green">E115</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_5_1_2_1_1_1_1"><b>case </b></a><a NAME="E1:103_5_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>16</sub></font> being   <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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:103_5_1_2_1_1_1_1"/><i><font color="Green">E116</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><a NAME="E4:103_5_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>16</sub></font> is    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/><a NAME="E5:103_5_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:103_5_1_2_1_1_1_1"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>17</sub></font></span><a href="binarith.html#K13">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:103_5_1_2"><i><font color="Green">E108</font></i></a>, <a class="ref" href="finseq_2.html#T117">FINSEQ_2:117</a></i>;<br/><a NAME="E8:103_5_1_2_1_1_1_1"/><i><font color="Green">E118</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E7:103_5_1_2_1_1_1_1"><i><font color="Green">E117</font></i></a>, <a class="ref" href="jordan2c.html#T95" target="_self">Th95</a></i>;<br/><div><i><font color="Green">E119</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="103_5_1_2_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_5_1_2_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:103_5_1_2_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_5_1_2_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:103_5_1_2_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:103_5_1_2_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E3:103_5_1_2_1_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  or <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:103_5_1_2_1_1_1_1"><i><font color="Green">E116</font></i></a>, <a class="txt" href="jordan2c.html#E7:103_5_1_2_1_1_1_1"><i><font color="Green">E117</font></i></a>, <a class="txt" href="jordan2c.html#E8:103_5_1_2_1_1_1_1"><i><font color="Green">E118</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_5_1_2_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:103_5_1_2_1_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:103_5_1_2_1_1_1_1_1_1_2"/><b>then </b>
 <a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:103_5_1_2_1_1_1_1"><i><font color="Green">E116</font></i></a>, <a class="txt" href="jordan2c.html#E8:103_5_1_2_1_1_1_1"><i><font color="Green">E118</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E3:103_5_1_2_1_1_1_1_1_1_2"/><b>then </b>
 <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><b>hence </b><a NAME="E4:103_5_1_2_1_1_1_1_1_1_2"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  or <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:103_5_1_2_1_1_1_1"><i><font color="Green">E116</font></i></a>, <a class="txt" href="jordan2c.html#E7:103_5_1_2_1_1_1_1"><i><font color="Green">E117</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_5_1_2_1_1_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_5_1_2_1_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:103_5_1_2_1_1_1_1_2_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  or <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="jordan2c.html#E9:103_5_1_2_1_1_1_1"><i><font color="Green">E119</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_5_1_2_1_1_1_1_2_1_1"><b>case </b></a><a NAME="E1:103_5_1_2_1_1_1_1_2_1_1"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 ;<br/><div class="add"><a NAME="E2:103_5_1_2_1_1_1_1_2_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_5_1_2_1_1_1_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></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="E1:103_5_1_2_1_1_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>19</sub></font> being   <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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:103_5_1_2_1_1_1_1_2_1_1_1"/><i><font color="Green">E121</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> ) )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:103_5_1_2_1_1_1_1_2_1_1_1"/><i><font color="Green">E122</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:103_5_1_2_1_1_1_1_2_1_1_1"><i><font color="Green">E121</font></i></a></i>;<br/>
<a NAME="E6:103_5_1_2_1_1_1_1_2_1_1_1"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="jordan2c.html#E1:103"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan2c.html#E5:103_5_1_2_1_1_1_1_2_1_1_1"><i><font color="Green">E122</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="jordan2c.html#E4:103_5_1_2"><i><font color="Green">E108</font></i></a></i>;<br/>
<i><font color="Green">E124</font></i>: <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>19</sub></font></span><a href="toprns_1.html#K5">.|</a></span> = 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="txt" href="jordan2c.html#E5:103_5_1_2_1_1_1_1_2_1_1_1"><i><font color="Green">E122</font></i></a>, <a class="ref" href="jgraph_1.html#D5">JGRAPH_1:def 5</a></i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="euclid.html#K11">sqr</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span>

;<br/>
<a NAME="E9:103_5_1_2_1_1_1_1_2_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E10:103_5_1_2_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>21</sub></font> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> 1
 
<b>by </b><i><a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E11:103_5_1_2_1_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E125</font></i>: 
 <a href="jordan2b.html#K1">Proj</a> <font color="Maroon">c<sub>19</sub></font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="goboard1.html#K1">.</a> 1
 
<b>by </b><i><a class="txt" href="jordan2c.html#E5:103_5_1_2_1_1_1_1_2_1_1_1"><i><font color="Green">E122</font></i></a>, <a class="ref" href="jordan2b.html#D1">JORDAN2B:def 1</a></i>;<br/>
<a NAME="E12:103_5_1_2_1_1_1_1_2_1_1_1"/><i><font color="Green">E126</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="euclid.html#K11">sqr</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="jordan2c.html#E4:103_5_1_2"><i><font color="Green">E108</font></i></a>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E13:103_5_1_2_1_1_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="euclid.html#K11">sqr</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan2b.html#K1">Proj</a> <font color="Maroon">c<sub>19</sub></font>,1</span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan2c.html#E11:103_5_1_2_1_1_1_1_2_1_1_1"><i><font color="Green">E125</font></i></a>, <a class="ref" href="rvsum_1.html#T79">RVSUM_1:79</a></i>;<br/>
<a NAME="E14:103_5_1_2_1_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E127</font></i>: 
 <a href="euclid.html#K11">sqr</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan2b.html#K1">Proj</a> <font color="Maroon">c<sub>19</sub></font>,1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span><a href="binarith.html#K13">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E12:103_5_1_2_1_1_1_1_2_1_1_1"><i><font color="Green">E126</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan2b.html#K1">Proj</a> <font color="Maroon">c<sub>19</sub></font>,1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="jordan2b.html#K1">Proj</a> <font color="Maroon">c<sub>19</sub></font>,1</span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T158">COMPLEX1:158</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="txt" href="jordan2c.html#E11:103_5_1_2_1_1_1_1_2_1_1_1"><i><font color="Green">E125</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/>
<b>then </b><i><font color="Green">E128</font></i>: <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>19</sub></font></span><a href="toprns_1.html#K5">.|</a></span> = 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="txt" href="jordan2c.html#E8:103_5_1_2_1_1_1_1_2_1_1_1"><i><font color="Green">E124</font></i></a>, <a class="txt" href="jordan2c.html#E14:103_5_1_2_1_1_1_1_2_1_1_1"><i><font color="Green">E127</font></i></a>, <a class="ref" href="finsop_1.html#T12">FINSOP_1:12</a></i>
<br/>.= 
<font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="txt" href="jordan2c.html#E6:103_5_1_2_1_1_1_1_2_1_1_1"><i><font color="Green">E123</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_5_1_2_1_1_1_1_2_1_1_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:103_5_1_2_1_1_1_1_2_1_1_1_4"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:103_5_1_2_1_1_1_1_2_1_1_1_4"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:103"><i><font color="Green">E95</font></i></a></i>;<br/><b>hence </b><a NAME="E3:103_5_1_2_1_1_1_1_2_1_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E5:103_5_1_2_1_1_1_1_2_1_1_1"><i><font color="Green">E122</font></i></a>, <a class="txt" href="jordan2c.html#E16:103_5_1_2_1_1_1_1_2_1_1_1"><i><font color="Green">E128</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E18:103_5_1_2_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</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> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>2</sub></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="E19:103_5_1_2_1_1_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:103_5_1_2_1_1_1_1_2_1_1_1"><i><font color="Green">E121</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  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>2</sub></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">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</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>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> ;<br/><a NAME="E5:103_5_1_2_1_1_1_1_2_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span>  <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">c<sub>1</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_5_1_2_1_1_1_1_2_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>20</sub></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="E1:103_5_1_2_1_1_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>21</sub></font> being   <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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:103_5_1_2_1_1_1_1_2_1_1_2"/><i><font color="Green">E121</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>
<b>thus </b><a NAME="E4:103_5_1_2_1_1_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</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>
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:103_5_1_2_1_1_1_1_2_1_1_2"><i><font color="Green">E121</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> </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">c<sub>1</sub></font></span>)</span> ;<br/><a NAME="E7:103_5_1_2_1_1_1_1_2_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  <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">c<sub>1</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_5_1_2_1_1_1_1_2_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>21</sub></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="E1:103_5_1_2_1_1_1_1_2_1_1_3"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>22</sub></font> being   <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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:103_5_1_2_1_1_1_1_2_1_1_3"/><i><font color="Green">E121</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> ) )
 ;<br/>
<b>thus </b><a NAME="E4:103_5_1_2_1_1_1_1_2_1_1_3"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</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>
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:103_5_1_2_1_1_1_1_2_1_1_3"><i><font color="Green">E121</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>21</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> ) </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">c<sub>1</sub></font></span>)</span> ;<br/><a NAME="E9:103_5_1_2_1_1_1_1_2_1_1"/>
<font color="Maroon">c<sub>20</sub></font> is   <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>then reconsider </b><font color="Maroon">c<sub>22</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> </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">c<sub>1</sub></font></span>)</span> ;<br/><b>reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>21</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/><a NAME="E12:103_5_1_2_1_1_1_1_2_1_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/><a NAME="E13:103_5_1_2_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">c<sub>21</sub></font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:103_5_1_2"><i><font color="Green">E108</font></i></a>, <a class="ref" href="jordan2c.html#T67" target="_self">Th67</a></i>;<br/><a NAME="E14:103_5_1_2_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E122</font></i>: 
<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>21</sub></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="E15:103_5_1_2_1_1_1_1_2_1_1"/><i><font color="Green">E123</font></i>: 
not <font color="Maroon">c<sub>23</sub></font> is <a href="tbsp_1.html#V6">bounded</a>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:103_5_1_2"><i><font color="Green">E108</font></i></a>, <a class="ref" href="jordan2c.html#T67" target="_self">Th67</a></i>;<br/><div><i><font color="Green">E124</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="103_5_1_2_1_1_1_1_2_1_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:103_5_1_2_1_1_1_1_2_1_1_4"/><i><font color="Green">E125</font></i>: 
not <font color="Maroon">c<sub>22</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>consider </b><font color="Maroon">c<sub>24</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>22</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E3:103_5_1_2_1_1_1_1_2_1_1_4"/><i><font color="Green">E126</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:103_5_1_2_1_1_1_1_2_1_1_4"><i><font color="Green">E125</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:103_5_1_2_1_1_1_1_2_1_1_4"/><b>then </b><i><font color="Green">E127</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><a NAME="E5:103_5_1_2_1_1_1_1_2_1_1_4"/>
ex <font color="Olive">b<sub>1</sub></font> being  <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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:103"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan2c.html#E3:103_5_1_2_1_1_1_1_2_1_1_4"><i><font color="Green">E126</font></i></a></i>;<br/><b>hence </b><a NAME="E6:103_5_1_2_1_1_1_1_2_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:103_5_1_2_1_1_1_1_2_1_1_4"><i><font color="Green">E127</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E17:103_5_1_2_1_1_1_1_2_1_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>22</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_5_1_2_1_1_1_1_2_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>24</sub></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="E1:103_5_1_2_1_1_1_1_2_1_1_5"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>25</sub></font> being   <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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:103_5_1_2_1_1_1_1_2_1_1_5"/><i><font color="Green">E125</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> ) )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>26</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:103_5_1_2_1_1_1_1_2_1_1_5"/><i><font color="Green">E126</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>26</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:103_5_1_2_1_1_1_1_2_1_1_5"><i><font color="Green">E125</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>27</sub></font> = <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>26</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="jordan2c.html#E4:103_5_1_2"><i><font color="Green">E108</font></i></a></i>;<br/>
<i><font color="Green">E127</font></i>: <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>25</sub></font></span><a href="toprns_1.html#K5">.|</a></span> = 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>27</sub></font></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="txt" href="jordan2c.html#E5:103_5_1_2_1_1_1_1_2_1_1_5"><i><font color="Green">E126</font></i></a>, <a class="ref" href="jgraph_1.html#D5">JGRAPH_1:def 5</a></i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="euclid.html#K11">sqr</a> <font color="Maroon">c<sub>27</sub></font></span>)</span></span>)</span>

;<br/>
<a NAME="E8:103_5_1_2_1_1_1_1_2_1_1_5"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E9:103_5_1_2_1_1_1_1_2_1_1_5"/><b>then </b>
<font color="Maroon">c<sub>27</sub></font> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> <a href="finseq_4.html#K4">/.</a> 1
 
<b>by </b><i><a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E10:103_5_1_2_1_1_1_1_2_1_1_5"/><b>then </b><i><font color="Green">E128</font></i>: 
 <a href="jordan2b.html#K1">Proj</a> <font color="Maroon">c<sub>25</sub></font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> <a href="goboard1.html#K1">.</a> 1
 
<b>by </b><i><a class="txt" href="jordan2c.html#E5:103_5_1_2_1_1_1_1_2_1_1_5"><i><font color="Green">E126</font></i></a>, <a class="ref" href="jordan2b.html#D1">JORDAN2B:def 1</a></i>;<br/>
<a NAME="E11:103_5_1_2_1_1_1_1_2_1_1_5"/><i><font color="Green">E129</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="euclid.html#K11">sqr</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="jordan2c.html#E4:103_5_1_2"><i><font color="Green">E108</font></i></a>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E12:103_5_1_2_1_1_1_1_2_1_1_5"/>
<span class="p1">(<span class="default"><a href="euclid.html#K11">sqr</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan2b.html#K1">Proj</a> <font color="Maroon">c<sub>25</sub></font>,1</span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan2c.html#E10:103_5_1_2_1_1_1_1_2_1_1_5"><i><font color="Green">E128</font></i></a>, <a class="ref" href="rvsum_1.html#T79">RVSUM_1:79</a></i>;<br/>
<a NAME="E13:103_5_1_2_1_1_1_1_2_1_1_5"/><b>then </b><i><font color="Green">E130</font></i>: 
 <a href="euclid.html#K11">sqr</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan2b.html#K1">Proj</a> <font color="Maroon">c<sub>25</sub></font>,1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span><a href="binarith.html#K13">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E11:103_5_1_2_1_1_1_1_2_1_1_5"><i><font color="Green">E129</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan2b.html#K1">Proj</a> <font color="Maroon">c<sub>25</sub></font>,1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="jordan2b.html#K1">Proj</a> <font color="Maroon">c<sub>25</sub></font>,1</span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T158">COMPLEX1:158</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="jordan2c.html#E10:103_5_1_2_1_1_1_1_2_1_1_5"><i><font color="Green">E128</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/>
<a NAME="E15:103_5_1_2_1_1_1_1_2_1_1_5"/><b>then </b><i><font color="Green">E131</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>25</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>26</sub></font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E7:103_5_1_2_1_1_1_1_2_1_1_5"><i><font color="Green">E127</font></i></a>, <a class="txt" href="jordan2c.html#E13:103_5_1_2_1_1_1_1_2_1_1_5"><i><font color="Green">E130</font></i></a>, <a class="ref" href="finsop_1.html#T12">FINSOP_1:12</a></i>;<br/>
<a NAME="E16:103_5_1_2_1_1_1_1_2_1_1_5"/>
<font color="Maroon">c<sub>26</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="jordan2c.html#E1:103"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan2c.html#E5:103_5_1_2_1_1_1_1_2_1_1_5"><i><font color="Green">E126</font></i></a></i>;<br/>
<a NAME="E17:103_5_1_2_1_1_1_1_2_1_1_5"/><b>then </b>
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>26</sub></font>
 
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<b>hence </b><a NAME="E18:103_5_1_2_1_1_1_1_2_1_1_5"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:103_5_1_2_1_1_1_1_2_1_1_5"><i><font color="Green">E125</font></i></a>, <a class="txt" href="jordan2c.html#E5:103_5_1_2_1_1_1_1_2_1_1_5"><i><font color="Green">E126</font></i></a>, <a class="txt" href="jordan2c.html#E15:103_5_1_2_1_1_1_1_2_1_1_5"><i><font color="Green">E131</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E18:103_5_1_2_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E125</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>22</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><a NAME="E19:103_5_1_2_1_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>23</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</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#E16:103_5_1_2_1_1_1_1_2_1_1"><i><font color="Green">E124</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><a NAME="E20:103_5_1_2_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E21:103_5_1_2_1_1_1_1_2_1_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="jordan2c.html#E16:103_5_1_2_1_1_1_1_2_1_1"><i><font color="Green">E124</font></i></a>, <a class="txt" href="jordan2c.html#E18:103_5_1_2_1_1_1_1_2_1_1"><i><font color="Green">E125</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><a NAME="E22:103_5_1_2_1_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>23</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#T32">SUBSET_1:32</a></i>;<br/><a NAME="E23:103_5_1_2_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E127</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><a NAME="E24:103_5_1_2_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E128</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><a NAME="E25:103_5_1_2_1_1_1_1_2_1_1"/>
<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>21</sub></font> <a href="hidden.html#R1">=</a> <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>2</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>19</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E23:103_5_1_2_1_1_1_1_2_1_1"><i><font color="Green">E127</font></i></a>, <a class="ref" href="pre_topc.html#T28">PRE_TOPC:28</a></i>;<br/><a NAME="E26:103_5_1_2_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E129</font></i>: 
<font color="Maroon">c<sub>19</sub></font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="txt" href="jordan2c.html#E14:103_5_1_2_1_1_1_1_2_1_1"><i><font color="Green">E122</font></i></a>, <a class="ref" href="connsp_1.html#D3">CONNSP_1:def 3</a></i>;<br/><a NAME="E27:103_5_1_2_1_1_1_1_2_1_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="connsp_3.html#K4">Down</a> <font color="Maroon">c<sub>21</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E24:103_5_1_2_1_1_1_1_2_1_1"><i><font color="Green">E128</font></i></a></i>;<br/><a NAME="E28:103_5_1_2_1_1_1_1_2_1_1"/><b>then </b>
 <a href="connsp_3.html#K5">Up</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K1">skl</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="jordan2c.html#R2" target="_self">is_outside_component_of</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E13:103_5_1_2_1_1_1_1_2_1_1"><i><font color="Green">E121</font></i></a>, <a class="txt" href="jordan2c.html#E15:103_5_1_2_1_1_1_1_2_1_1"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jordan2c.html#E20:103_5_1_2_1_1_1_1_2_1_1"><i><font color="Green">E126</font></i></a>, <a class="ref" href="jordan2c.html#T71" target="_self">Th71</a></i>;<br/><a NAME="E29:103_5_1_2_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E130</font></i>: 
 <a href="connsp_3.html#K1">skl</a> <font color="Maroon">c<sub>19</sub></font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T27" target="_self">Th27</a></i>;<br/><a NAME="E30:103_5_1_2_1_1_1_1_2_1_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="tarski.html#R1">c=</a>  <a href="connsp_3.html#K1">skl</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E26:103_5_1_2_1_1_1_1_2_1_1"><i><font color="Green">E129</font></i></a>, <a class="ref" href="connsp_3.html#T1">CONNSP_3:1</a></i>;<br/><a NAME="E31:103_5_1_2_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E131</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="connsp_3.html#K1">skl</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:103_5_1_2_1_1_1_1_2_1_1"><i><font color="Green">E120</font></i></a></i>;<br/><a NAME="E32:103_5_1_2_1_1_1_1_2_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E5:103_5_1_2_1"><i><font color="Green">E110</font></i></a>, <a class="ref" href="jordan2c.html#T26" target="_self">Th26</a></i>;<br/><a NAME="E33:103_5_1_2_1_1_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:103_5_1_2_1"><i><font color="Green">E109</font></i></a>, <a class="txt" href="jordan2c.html#E5:103_5_1_2_1"><i><font color="Green">E110</font></i></a>, <a class="txt" href="jordan2c.html#E29:103_5_1_2_1_1_1_1_2_1_1"><i><font color="Green">E130</font></i></a>, <a class="txt" href="jordan2c.html#E31:103_5_1_2_1_1_1_1_2_1_1"><i><font color="Green">E131</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E34:103_5_1_2_1_1_1_1_2_1_1"/><b>then </b>
 <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E35:103_5_1_2_1_1_1_1_2_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T28" target="_self">Th28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_5_1_2_1_1_1_1_2_1_2"><b>case </b></a><a NAME="E1:103_5_1_2_1_1_1_1_2_1_2"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 ;<br/><div class="add"><a NAME="E2:103_5_1_2_1_1_1_1_2_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_5_1_2_1_1_1_1_2_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></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="E1:103_5_1_2_1_1_1_1_2_1_2_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>19</sub></font> being   <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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:103_5_1_2_1_1_1_1_2_1_2_1"/><i><font color="Green">E121</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>3</sub></font> ) )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:103_5_1_2_1_1_1_1_2_1_2_1"/><i><font color="Green">E122</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:103_5_1_2_1_1_1_1_2_1_2_1"><i><font color="Green">E121</font></i></a></i>;<br/>
<a NAME="E6:103_5_1_2_1_1_1_1_2_1_2_1"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> 0
 
<b>by </b><i><a class="txt" href="jordan2c.html#E1:103"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan2c.html#E5:103_5_1_2_1_1_1_1_2_1_2_1"><i><font color="Green">E122</font></i></a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="jordan2c.html#E4:103_5_1_2"><i><font color="Green">E108</font></i></a></i>;<br/>
<i><font color="Green">E124</font></i>: <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>19</sub></font></span><a href="toprns_1.html#K5">.|</a></span> = 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="txt" href="jordan2c.html#E5:103_5_1_2_1_1_1_1_2_1_2_1"><i><font color="Green">E122</font></i></a>, <a class="ref" href="jgraph_1.html#D5">JGRAPH_1:def 5</a></i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="euclid.html#K11">sqr</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span>

;<br/>
<a NAME="E9:103_5_1_2_1_1_1_1_2_1_2_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E10:103_5_1_2_1_1_1_1_2_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>21</sub></font> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> 1
 
<b>by </b><i><a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E11:103_5_1_2_1_1_1_1_2_1_2_1"/><b>then </b><i><font color="Green">E125</font></i>: 
 <a href="jordan2b.html#K1">Proj</a> <font color="Maroon">c<sub>19</sub></font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="goboard1.html#K1">.</a> 1
 
<b>by </b><i><a class="txt" href="jordan2c.html#E5:103_5_1_2_1_1_1_1_2_1_2_1"><i><font color="Green">E122</font></i></a>, <a class="ref" href="jordan2b.html#D1">JORDAN2B:def 1</a></i>;<br/>
<a NAME="E12:103_5_1_2_1_1_1_1_2_1_2_1"/><i><font color="Green">E126</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="euclid.html#K11">sqr</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="jordan2c.html#E4:103_5_1_2"><i><font color="Green">E108</font></i></a>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E13:103_5_1_2_1_1_1_1_2_1_2_1"/>
<span class="p1">(<span class="default"><a href="euclid.html#K11">sqr</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan2b.html#K1">Proj</a> <font color="Maroon">c<sub>19</sub></font>,1</span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan2c.html#E11:103_5_1_2_1_1_1_1_2_1_2_1"><i><font color="Green">E125</font></i></a>, <a class="ref" href="rvsum_1.html#T79">RVSUM_1:79</a></i>;<br/>
<a NAME="E14:103_5_1_2_1_1_1_1_2_1_2_1"/><b>then </b><i><font color="Green">E127</font></i>: 
 <a href="euclid.html#K11">sqr</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan2b.html#K1">Proj</a> <font color="Maroon">c<sub>19</sub></font>,1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span><a href="binarith.html#K13">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E12:103_5_1_2_1_1_1_1_2_1_2_1"><i><font color="Green">E126</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan2b.html#K1">Proj</a> <font color="Maroon">c<sub>19</sub></font>,1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="jordan2b.html#K1">Proj</a> <font color="Maroon">c<sub>19</sub></font>,1</span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T158">COMPLEX1:158</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="txt" href="jordan2c.html#E11:103_5_1_2_1_1_1_1_2_1_2_1"><i><font color="Green">E125</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/>
<b>then </b><i><font color="Green">E128</font></i>: <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>19</sub></font></span><a href="toprns_1.html#K5">.|</a></span> = 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="txt" href="jordan2c.html#E8:103_5_1_2_1_1_1_1_2_1_2_1"><i><font color="Green">E124</font></i></a>, <a class="txt" href="jordan2c.html#E14:103_5_1_2_1_1_1_1_2_1_2_1"><i><font color="Green">E127</font></i></a>, <a class="ref" href="finsop_1.html#T12">FINSOP_1:12</a></i>
<br/>.= 
 <a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="txt" href="jordan2c.html#E6:103_5_1_2_1_1_1_1_2_1_2_1"><i><font color="Green">E123</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_5_1_2_1_1_1_1_2_1_2_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:103_5_1_2_1_1_1_1_2_1_2_1_4"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:103_5_1_2_1_1_1_1_2_1_2_1_4"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:103"><i><font color="Green">E95</font></i></a></i>;<br/><b>hence </b><a NAME="E3:103_5_1_2_1_1_1_1_2_1_2_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E5:103_5_1_2_1_1_1_1_2_1_2_1"><i><font color="Green">E122</font></i></a>, <a class="txt" href="jordan2c.html#E16:103_5_1_2_1_1_1_1_2_1_2_1"><i><font color="Green">E128</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E18:103_5_1_2_1_1_1_1_2_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</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> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>2</sub></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="E19:103_5_1_2_1_1_1_1_2_1_2_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:103_5_1_2_1_1_1_1_2_1_2_1"><i><font color="Green">E121</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  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>2</sub></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">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</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>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> ;<br/><a NAME="E5:103_5_1_2_1_1_1_1_2_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span>  <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">c<sub>1</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_5_1_2_1_1_1_1_2_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>20</sub></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="E1:103_5_1_2_1_1_1_1_2_1_2_2"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>21</sub></font> being   <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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:103_5_1_2_1_1_1_1_2_1_2_2"/><i><font color="Green">E121</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>
<b>thus </b><a NAME="E4:103_5_1_2_1_1_1_1_2_1_2_2"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</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>
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:103_5_1_2_1_1_1_1_2_1_2_2"><i><font color="Green">E121</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> </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">c<sub>1</sub></font></span>)</span> ;<br/><a NAME="E7:103_5_1_2_1_1_1_1_2_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  <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">c<sub>1</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_5_1_2_1_1_1_1_2_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>21</sub></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="E1:103_5_1_2_1_1_1_1_2_1_2_3"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>22</sub></font> being   <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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:103_5_1_2_1_1_1_1_2_1_2_3"/><i><font color="Green">E121</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>3</sub></font> ) )
 ;<br/>
<b>thus </b><a NAME="E4:103_5_1_2_1_1_1_1_2_1_2_3"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</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>
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:103_5_1_2_1_1_1_1_2_1_2_3"><i><font color="Green">E121</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>21</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>3</sub></font> ) </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">c<sub>1</sub></font></span>)</span> ;<br/><a NAME="E9:103_5_1_2_1_1_1_1_2_1_2"/>
<font color="Maroon">c<sub>20</sub></font> is   <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>then reconsider </b><font color="Maroon">c<sub>22</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> </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">c<sub>1</sub></font></span>)</span> ;<br/><b>reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>21</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/><a NAME="E12:103_5_1_2_1_1_1_1_2_1_2"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/><a NAME="E13:103_5_1_2_1_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">c<sub>21</sub></font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:103_5_1_2"><i><font color="Green">E108</font></i></a>, <a class="ref" href="jordan2c.html#T68" target="_self">Th68</a></i>;<br/><a NAME="E14:103_5_1_2_1_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E122</font></i>: 
<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>21</sub></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="E15:103_5_1_2_1_1_1_1_2_1_2"/><i><font color="Green">E123</font></i>: 
not <font color="Maroon">c<sub>23</sub></font> is <a href="tbsp_1.html#V6">bounded</a>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:103_5_1_2"><i><font color="Green">E108</font></i></a>, <a class="ref" href="jordan2c.html#T68" target="_self">Th68</a></i>;<br/><div><i><font color="Green">E124</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="103_5_1_2_1_1_1_1_2_1_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:103_5_1_2_1_1_1_1_2_1_2_4"/><i><font color="Green">E125</font></i>: 
not <font color="Maroon">c<sub>22</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>consider </b><font color="Maroon">c<sub>24</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>22</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E3:103_5_1_2_1_1_1_1_2_1_2_4"/><i><font color="Green">E126</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:103_5_1_2_1_1_1_1_2_1_2_4"><i><font color="Green">E125</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:103_5_1_2_1_1_1_1_2_1_2_4"/><b>then </b><i><font color="Green">E127</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><a NAME="E5:103_5_1_2_1_1_1_1_2_1_2_4"/>
ex <font color="Olive">b<sub>1</sub></font> being  <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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:103"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan2c.html#E3:103_5_1_2_1_1_1_1_2_1_2_4"><i><font color="Green">E126</font></i></a></i>;<br/><b>hence </b><a NAME="E6:103_5_1_2_1_1_1_1_2_1_2_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:103_5_1_2_1_1_1_1_2_1_2_4"><i><font color="Green">E127</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E17:103_5_1_2_1_1_1_1_2_1_2"/>
<font color="Maroon">c<sub>23</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>22</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_5_1_2_1_1_1_1_2_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>24</sub></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="E1:103_5_1_2_1_1_1_1_2_1_2_5"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>25</sub></font> being   <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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:103_5_1_2_1_1_1_1_2_1_2_5"/><i><font color="Green">E125</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>3</sub></font> ) )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>26</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:103_5_1_2_1_1_1_1_2_1_2_5"/><i><font color="Green">E126</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>26</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:103_5_1_2_1_1_1_1_2_1_2_5"><i><font color="Green">E125</font></i></a></i>;<br/>
<a NAME="E6:103_5_1_2_1_1_1_1_2_1_2_5"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> 0
 
<b>by </b><i><a class="txt" href="jordan2c.html#E1:103"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan2c.html#E5:103_5_1_2_1_1_1_1_2_1_2_5"><i><font color="Green">E126</font></i></a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E7:103_5_1_2_1_1_1_1_2_1_2_5"/><i><font color="Green">E128</font></i>: 
 <a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>26</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E5:103_5_1_2_1_1_1_1_2_1_2_5"><i><font color="Green">E126</font></i></a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>27</sub></font> = <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>26</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="jordan2c.html#E4:103_5_1_2"><i><font color="Green">E108</font></i></a></i>;<br/>
<i><font color="Green">E129</font></i>: <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>25</sub></font></span><a href="toprns_1.html#K5">.|</a></span> = 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>27</sub></font></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="txt" href="jordan2c.html#E5:103_5_1_2_1_1_1_1_2_1_2_5"><i><font color="Green">E126</font></i></a>, <a class="ref" href="jgraph_1.html#D5">JGRAPH_1:def 5</a></i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="euclid.html#K11">sqr</a> <font color="Maroon">c<sub>27</sub></font></span>)</span></span>)</span>

;<br/>
<a NAME="E10:103_5_1_2_1_1_1_1_2_1_2_5"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E11:103_5_1_2_1_1_1_1_2_1_2_5"/><b>then </b>
<font color="Maroon">c<sub>27</sub></font> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> <a href="finseq_4.html#K4">/.</a> 1
 
<b>by </b><i><a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E12:103_5_1_2_1_1_1_1_2_1_2_5"/><b>then </b><i><font color="Green">E130</font></i>: 
 <a href="jordan2b.html#K1">Proj</a> <font color="Maroon">c<sub>25</sub></font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> <a href="goboard1.html#K1">.</a> 1
 
<b>by </b><i><a class="txt" href="jordan2c.html#E5:103_5_1_2_1_1_1_1_2_1_2_5"><i><font color="Green">E126</font></i></a>, <a class="ref" href="jordan2b.html#D1">JORDAN2B:def 1</a></i>;<br/>
<a NAME="E13:103_5_1_2_1_1_1_1_2_1_2_5"/><i><font color="Green">E131</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="euclid.html#K11">sqr</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="jordan2c.html#E4:103_5_1_2"><i><font color="Green">E108</font></i></a>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E14:103_5_1_2_1_1_1_1_2_1_2_5"/>
<span class="p1">(<span class="default"><a href="euclid.html#K11">sqr</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan2b.html#K1">Proj</a> <font color="Maroon">c<sub>25</sub></font>,1</span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan2c.html#E12:103_5_1_2_1_1_1_1_2_1_2_5"><i><font color="Green">E130</font></i></a>, <a class="ref" href="rvsum_1.html#T79">RVSUM_1:79</a></i>;<br/>
<a NAME="E15:103_5_1_2_1_1_1_1_2_1_2_5"/><b>then </b><i><font color="Green">E132</font></i>: 
 <a href="euclid.html#K11">sqr</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan2b.html#K1">Proj</a> <font color="Maroon">c<sub>25</sub></font>,1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span><a href="binarith.html#K13">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E13:103_5_1_2_1_1_1_1_2_1_2_5"><i><font color="Green">E131</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan2b.html#K1">Proj</a> <font color="Maroon">c<sub>25</sub></font>,1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="jordan2b.html#K1">Proj</a> <font color="Maroon">c<sub>25</sub></font>,1</span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T158">COMPLEX1:158</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="jordan2c.html#E12:103_5_1_2_1_1_1_1_2_1_2_5"><i><font color="Green">E130</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/>
<a NAME="E17:103_5_1_2_1_1_1_1_2_1_2_5"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>25</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>26</sub></font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E9:103_5_1_2_1_1_1_1_2_1_2_5"><i><font color="Green">E129</font></i></a>, <a class="txt" href="jordan2c.html#E15:103_5_1_2_1_1_1_1_2_1_2_5"><i><font color="Green">E132</font></i></a>, <a class="ref" href="finsop_1.html#T12">FINSOP_1:12</a></i>;<br/>
<a NAME="E18:103_5_1_2_1_1_1_1_2_1_2_5"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>25</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E6:103_5_1_2_1_1_1_1_2_1_2_5"><i><font color="Green">E127</font></i></a>, <a class="txt" href="jordan2c.html#E7:103_5_1_2_1_1_1_1_2_1_2_5"><i><font color="Green">E128</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<b>hence </b><a NAME="E19:103_5_1_2_1_1_1_1_2_1_2_5"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:103_5_1_2_1_1_1_1_2_1_2_5"><i><font color="Green">E125</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E18:103_5_1_2_1_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E125</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>22</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><a NAME="E19:103_5_1_2_1_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>23</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</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#E16:103_5_1_2_1_1_1_1_2_1_2"><i><font color="Green">E124</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><a NAME="E20:103_5_1_2_1_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E21:103_5_1_2_1_1_1_1_2_1_2"/>
<font color="Maroon">c<sub>23</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="jordan2c.html#E16:103_5_1_2_1_1_1_1_2_1_2"><i><font color="Green">E124</font></i></a>, <a class="txt" href="jordan2c.html#E18:103_5_1_2_1_1_1_1_2_1_2"><i><font color="Green">E125</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><a NAME="E22:103_5_1_2_1_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>23</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#T32">SUBSET_1:32</a></i>;<br/><a NAME="E23:103_5_1_2_1_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E127</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><a NAME="E24:103_5_1_2_1_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E128</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><a NAME="E25:103_5_1_2_1_1_1_1_2_1_2"/>
<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>21</sub></font> <a href="hidden.html#R1">=</a> <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>2</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>19</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E23:103_5_1_2_1_1_1_1_2_1_2"><i><font color="Green">E127</font></i></a>, <a class="ref" href="pre_topc.html#T28">PRE_TOPC:28</a></i>;<br/><a NAME="E26:103_5_1_2_1_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E129</font></i>: 
<font color="Maroon">c<sub>19</sub></font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="txt" href="jordan2c.html#E14:103_5_1_2_1_1_1_1_2_1_2"><i><font color="Green">E122</font></i></a>, <a class="ref" href="connsp_1.html#D3">CONNSP_1:def 3</a></i>;<br/><a NAME="E27:103_5_1_2_1_1_1_1_2_1_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="connsp_3.html#K4">Down</a> <font color="Maroon">c<sub>21</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E24:103_5_1_2_1_1_1_1_2_1_2"><i><font color="Green">E128</font></i></a></i>;<br/><a NAME="E28:103_5_1_2_1_1_1_1_2_1_2"/><b>then </b>
 <a href="connsp_3.html#K5">Up</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K1">skl</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="jordan2c.html#R2" target="_self">is_outside_component_of</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E13:103_5_1_2_1_1_1_1_2_1_2"><i><font color="Green">E121</font></i></a>, <a class="txt" href="jordan2c.html#E15:103_5_1_2_1_1_1_1_2_1_2"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jordan2c.html#E20:103_5_1_2_1_1_1_1_2_1_2"><i><font color="Green">E126</font></i></a>, <a class="ref" href="jordan2c.html#T71" target="_self">Th71</a></i>;<br/><a NAME="E29:103_5_1_2_1_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E130</font></i>: 
 <a href="connsp_3.html#K1">skl</a> <font color="Maroon">c<sub>19</sub></font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T27" target="_self">Th27</a></i>;<br/><a NAME="E30:103_5_1_2_1_1_1_1_2_1_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="tarski.html#R1">c=</a>  <a href="connsp_3.html#K1">skl</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E26:103_5_1_2_1_1_1_1_2_1_2"><i><font color="Green">E129</font></i></a>, <a class="ref" href="connsp_3.html#T1">CONNSP_3:1</a></i>;<br/><a NAME="E31:103_5_1_2_1_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E131</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="connsp_3.html#K1">skl</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:103_5_1_2_1_1_1_1_2_1_2"><i><font color="Green">E120</font></i></a></i>;<br/><a NAME="E32:103_5_1_2_1_1_1_1_2_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E5:103_5_1_2_1"><i><font color="Green">E110</font></i></a>, <a class="ref" href="jordan2c.html#T26" target="_self">Th26</a></i>;<br/><a NAME="E33:103_5_1_2_1_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></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">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:103_5_1_2_1"><i><font color="Green">E109</font></i></a>, <a class="txt" href="jordan2c.html#E5:103_5_1_2_1"><i><font color="Green">E110</font></i></a>, <a class="txt" href="jordan2c.html#E29:103_5_1_2_1_1_1_1_2_1_2"><i><font color="Green">E130</font></i></a>, <a class="txt" href="jordan2c.html#E31:103_5_1_2_1_1_1_1_2_1_2"><i><font color="Green">E131</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E34:103_5_1_2_1_1_1_1_2_1_2"/><b>then </b>
 <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E35:103_5_1_2_1_1_1_1_2_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T28" target="_self">Th28</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E11:103_5_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_5_1_2_1_1_1_2"><b>case </b></a><a NAME="E1:103_5_1_2_1_1_1_2"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 ;<br/><div class="add"><a NAME="E2:103_5_1_2_1_1_1_2"/>
 <a href="connsp_3.html#K4">Down</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="tarski.html#R1">c=</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>6</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E18:103"><i><font color="Green">E102</font></i></a>, <a class="ref" href="connsp_3.html#T1">CONNSP_3:1</a></i>;<br/><b>hence </b><a NAME="E3:103_5_1_2_1_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E14:103"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jordan2c.html#E1:103_5_1_2_1_1_1_2"><i><font color="Green">E116</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E18:103_5_1_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>


</div><b>end;</b></div><a NAME="E6:103_5_1_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="jordan2c.html#E24:103"><i><font color="Green">E107</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E7:103_5_1_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>hence </b><a NAME="E8:103_5_1_2"/>
ex <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#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E23:103"><i><font color="Green">E106</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E26:103"/>
ex <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#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>


</div>
