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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM8">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:13"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>4</sub></font> <a href="jordan12.html#R2" target="_self">are_in_general_position</a> 
 ;<br/>

<a NAME="E2:13"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>2</sub></font> <a href="jordan12.html#R1" target="_self">is_in_general_position_wrt</a> <font color="Maroon">c<sub>3</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>4</sub></font> <a href="jordan12.html#R1" target="_self">is_in_general_position_wrt</a> <font color="Maroon">c<sub>1</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="jordan12.html#E1:13"><i><font color="Green">E12</font></i></a>, <a class="ref" href="jordan12.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E3:13"/><b>then </b><i><font color="Green">E14</font></i>: 
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan12.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E4:13"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal8.html#T10">TOPREAL8:10</a></i>;<br/>
<a NAME="E5:13"/><b>then </b><i><font color="Green">E15</font></i>: 
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E3:13"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E6:13"/><i><font color="Green">E16</font></i>: 
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan12.html#E2:13"><i><font color="Green">E13</font></i></a>, <a class="ref" href="jordan12.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E7:13"/>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan12.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E8:13"/><b>then </b><i><font color="Green">E17</font></i>: 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan12.html#E6:13"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<div><i><font color="Green">E18</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:13_1"/><i><font color="Green">E19</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 ;<br/><a NAME="E2:13_1"/>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan12.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E3:13_1"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><a NAME="E4:13_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> is <a href="realset1.html#V1">trivial</a>
 <b>by </b><i><a class="txt" href="jordan12.html#E2:13"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jordan12.html#E1:13_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="jordan12.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E5:13_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> is <a href="realset1.html#V1">trivial</a>
 <b>by </b><i><a class="txt" href="jordan12.html#E3:13_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="sprect_1.html#T1">SPRECT_1:1</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:13_2"/><i><font color="Green">E19</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><a NAME="E2:13_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="topreal8.html#T7">TOPREAL8:7</a></i>;<br/><a NAME="E3:13_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E1:13_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:13_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> is <a href="realset1.html#V1">trivial</a>
 <b>by </b><i><a class="txt" href="jordan12.html#E2:13"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jordan12.html#E1:13_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="jordan12.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E5:13_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> is <a href="realset1.html#V1">trivial</a>
 <b>by </b><i><a class="txt" href="jordan12.html#E1:13_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="topreal8.html#T28">TOPREAL8:28</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:13"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>2</sub></font> <a href="jordan12.html#R1" target="_self">is_in_general_position_wrt</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="jordan12.html#R1" target="_self">is_in_general_position_wrt</a> <font color="Maroon">c<sub>1</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="jordan12.html#E5:13"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan12.html#E8:13"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan12.html#E9:13"><i><font color="Green">E18</font></i></a>, <a class="ref" href="jordan12.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E12:13"/>
<font color="Maroon">c<sub>1</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="jordan12.html#R2" target="_self">are_in_general_position</a> 
 <b>by </b><i><a class="ref" href="jordan12.html#D2" target="_self">Def2</a></i>;<br/>


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