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

<b>let </b><font color="Maroon" title="c1">p1</font>, <font color="Maroon" title="c2">p2</font>, <font color="Maroon" title="c3">p3</font>, <font color="Maroon" title="c4">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>







<b>assume </b><a NAME="E1:71"/><i><font color="Green" title="E34">A1</font></i>: 
<font color="Maroon" title="c2">p2</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c1">p1</font> <a href="euclid_3.html#R1" title="EUCLID_3:pred.1">are_lindependent2</a> 
 ;<br/>

<b>thus </b><a NAME="E2:71"/>
(  not <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_3.html#K8" title="EUCLID_3:func.8">outside_of_triangle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font> or  <a href="euclid_3.html#K10" title="EUCLID_3:func.10">tricord1</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="euclid_3.html#K11" title="EUCLID_3:func.11">tricord2</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="euclid_3.html#K12" title="EUCLID_3:func.12">tricord3</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:71_1"/>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_3.html#K8" title="EUCLID_3:func.8">outside_of_triangle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">p0</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:71_1"/><i><font color="Green" title="E35">A2</font></i>: 
( <font color="Maroon" title="c5">p0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> &amp;  ex <font color="Olive" title="b1">a1</font>, <font color="Olive" title="b2">a2</font>, <font color="Olive" title="b3">a3</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">a1</font> or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">a2</font> or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b3">a3</font> ) &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">a1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b2">a2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b3">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c5">p0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">a1</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">a2</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">a3</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p3</font></span>)</span> ) )
 ;<br/>
<a NAME="E4:71_1"/>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
;<br/>
<a NAME="E5:71_1"/><b>then </b>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2
 
<b>by </b><i><a class="ref" href="euclid.html#T25" title="EUCLID:th.25">EUCLID:25</a></i>;<br/>
<a NAME="E6:71_1"/><b>then </b>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_3.html#K9" title="EUCLID_3:func.9">plane</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>
 
<b>by </b><i><a class="txt" href="euclid_3.html#E1:71"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="euclid_3.html#T63" target="_self" title="EUCLID_3:th.63">Th63</a></i>;<br/>
<b>hence </b><a NAME="E7:71_1"/>
(  <a href="euclid_3.html#K10" title="EUCLID_3:func.10">tricord1</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="euclid_3.html#K11" title="EUCLID_3:func.11">tricord2</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="euclid_3.html#K12" title="EUCLID_3:func.12">tricord3</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="euclid_3.html#E1:71"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="euclid_3.html#E3:71_1"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="euclid_3.html#D11" target="_self" title="EUCLID_3:def.11">Def11</a>, <a class="ref" href="euclid_3.html#D12" target="_self" title="EUCLID_3:def.12">Def12</a>, <a class="ref" href="euclid_3.html#D13" target="_self" title="EUCLID_3:def.13">Def13</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:71"/><i><font color="Green" title="E35">A3</font></i>: 
(  <a href="euclid_3.html#K10" title="EUCLID_3:func.10">tricord1</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="euclid_3.html#K11" title="EUCLID_3:func.11">tricord2</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="euclid_3.html#K12" title="EUCLID_3:func.12">tricord3</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/>

<a NAME="E4:71"/>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
;<br/>
<a NAME="E5:71"/><b>then </b>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2
 
<b>by </b><i><a class="ref" href="euclid.html#T25" title="EUCLID:th.25">EUCLID:25</a></i>;<br/>
<a NAME="E6:71"/><b>then </b><i><font color="Green" title="E36">A4</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_3.html#K9" title="EUCLID_3:func.9">plane</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>
 
<b>by </b><i><a class="txt" href="euclid_3.html#E1:71"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="euclid_3.html#T63" target="_self" title="EUCLID_3:th.63">Th63</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">i1</font> =  <a href="euclid_3.html#K10" title="EUCLID_3:func.10">tricord1</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font>;<br/>
<b>set </b><font color="Maroon" title="c6">i2</font> =  <a href="euclid_3.html#K11" title="EUCLID_3:func.11">tricord2</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font>;<br/>
<b>set </b><font color="Maroon" title="c7">i3</font> =  <a href="euclid_3.html#K12" title="EUCLID_3:func.12">tricord3</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font>;<br/>
<b>consider </b><font color="Maroon" title="c8">a2</font>, <font color="Maroon" title="c9">a3</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E8:71"/><i><font color="Green" title="E37">A5</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K10" title="EUCLID_3:func.10">tricord1</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">a2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c9">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid_3.html#K10" title="EUCLID_3:func.10">tricord1</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">a2</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a3</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p3</font></span>)</span> )
 <b>by </b><i><a class="txt" href="euclid_3.html#E1:71"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="euclid_3.html#E6:71"><i><font color="Green" title="E36">A4</font></i></a>, <a class="ref" href="euclid_3.html#D11" target="_self" title="EUCLID_3:def.11">Def11</a></i>;<br/>
<a NAME="E9:71"/><i><font color="Green" title="E38">A6</font></i>: 
<font color="Maroon" title="c8">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K11" title="EUCLID_3:func.11">tricord2</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font>
 
<b>by </b><i><a class="txt" href="euclid_3.html#E1:71"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="euclid_3.html#E6:71"><i><font color="Green" title="E36">A4</font></i></a>, <a class="txt" href="euclid_3.html#E8:71"><i><font color="Green" title="E37">A5</font></i></a>, <a class="ref" href="euclid_3.html#D12" target="_self" title="EUCLID_3:def.12">Def12</a></i>;<br/>
<a NAME="E10:71"/>
<font color="Maroon" title="c9">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K12" title="EUCLID_3:func.12">tricord3</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font>
 
<b>by </b><i><a class="txt" href="euclid_3.html#E1:71"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="euclid_3.html#E6:71"><i><font color="Green" title="E36">A4</font></i></a>, <a class="txt" href="euclid_3.html#E8:71"><i><font color="Green" title="E37">A5</font></i></a>, <a class="ref" href="euclid_3.html#D13" target="_self" title="EUCLID_3:def.13">Def13</a></i>;<br/>
<b>hence </b><a NAME="E11:71"/>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_3.html#K8" title="EUCLID_3:func.8">outside_of_triangle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>
 <b>by </b><i><a class="txt" href="euclid_3.html#E3:71"><i><font color="Green" title="E35">A3</font></i></a>, <a class="txt" href="euclid_3.html#E8:71"><i><font color="Green" title="E37">A5</font></i></a>, <a class="txt" href="euclid_3.html#E9:71"><i><font color="Green" title="E38">A6</font></i></a></i>;<br/>


</div>
