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

<b>let </b><font color="Maroon">p1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">p2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">p3</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</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="E1:73"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">p2</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p1</font>,<font color="Maroon">p3</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p1</font> <a href="euclid_3.html#R1" target="_self">are_lindependent2</a> 
 ;<br/>

<a NAME="E2:73"/><i><font color="Green">E49</font></i>: 
 <a href="euclid_3.html#K7" target="_self">inside_of_triangle</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p0</font> where <font color="Olive">p0</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> :  ex <font color="Olive">a1</font>, <font color="Olive">a2</font>, <font color="Olive">a3</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">a1</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">a2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">a3</font> &amp; <span class="p1">(<span class="default"><font color="Olive">a1</font> <a href="real_1.html#K3">+</a> <font color="Olive">a2</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Olive">a3</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">p0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">a1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Olive">a2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Olive">a3</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p3</font></span>)</span> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:73_1"/>
 <a href="euclid_3.html#K7" target="_self">inside_of_triangle</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">p0</font> where <font color="Olive">p0</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> :  ex <font color="Olive">a1</font>, <font color="Olive">a2</font>, <font color="Olive">a3</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">a1</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">a2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">a3</font> &amp; <span class="p1">(<span class="default"><font color="Olive">a1</font> <a href="real_1.html#K3">+</a> <font color="Olive">a2</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Olive">a3</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">p0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">a1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Olive">a2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Olive">a3</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p3</font></span>)</span> ) </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:73_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="euclid_3.html#K7" target="_self">inside_of_triangle</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>
 ;<br/>

<a NAME="E2:73_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="euclid_3.html#K6" target="_self">closed_inside_of_triangle</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="euclid_3.html#K5" target="_self">Triangle</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">p0</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> 2</span>)</span><b> such that </b><br/><a NAME="E4:73_1_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">p0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  ex <font color="Olive">a1</font>, <font color="Olive">a2</font>, <font color="Olive">a3</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a1</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a3</font> &amp; <span class="p1">(<span class="default"><font color="Olive">a1</font> <a href="real_1.html#K3">+</a> <font color="Olive">a2</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Olive">a3</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">p0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">a1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Olive">a2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Olive">a3</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p3</font></span>)</span> ) )
 ;<br/>
<b>consider </b><font color="Maroon">a1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">a2</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">a3</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:73_1_1"/><i><font color="Green">E71</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a3</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">a2</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">a3</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">p0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p3</font></span>)</span> )
 <b>by </b><i><a class="txt" href="euclid_3.html#E1:73"><i><font color="Green">E41</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">i01</font> =  <a href="euclid_3.html#K10" target="_self">tricord1</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p0</font>;<br/>
<b>set </b><font color="Maroon">i02</font> =  <a href="euclid_3.html#K11" target="_self">tricord2</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p0</font>;<br/>
<b>set </b><font color="Maroon">i03</font> =  <a href="euclid_3.html#K12" target="_self">tricord3</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p0</font>;<br/>
<a NAME="E7:73_1_1"/>
<font color="Maroon">p0</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> 2</span>)</span>
 
;<br/>
<a NAME="E8:73_1_1"/><b>then </b>
<font color="Maroon">p0</font> <a href="hidden.html#R2">in</a>  <a href="euclid.html#K1">REAL</a> 2
 
<b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<a NAME="E9:73_1_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">p0</font> <a href="hidden.html#R2">in</a>  <a href="euclid_3.html#K9" target="_self">plane</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:73_1_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="euclid_3.html#K10" target="_self">tricord1</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p0</font>
 
<b>by </b><i>, <a class="ref" href="euclid_3.html#T6" target="_self">Th6</a>, </i>;<br/>
<a NAME="E11:73_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="euclid_3.html#K11" target="_self">tricord2</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p0</font>
 
<b>by </b><i>, <a class="ref" href="euclid_3.html#T6" target="_self">Th6</a>, <a class="ref" href="euclid_3.html#T7" target="_self">Th7</a>, </i>;<br/>
<a NAME="E12:73_1_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">a3</font> <a href="hidden.html#R1">=</a>  <a href="euclid_3.html#K12" target="_self">tricord3</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p0</font>
 
<b>by </b><i>, <a class="ref" href="euclid_3.html#T6" target="_self">Th6</a>, <a class="ref" href="euclid_3.html#T7" target="_self">Th7</a>, </i>;<br/>
<a NAME="E13:73_1_1"/><b>then </b>
(  <a href="euclid_3.html#K10" target="_self">tricord1</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp;  <a href="euclid_3.html#K11" target="_self">tricord2</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp;  <a href="euclid_3.html#K12" target="_self">tricord3</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i>, <a class="ref" href="euclid_3.html#T2" target="_self">Th2</a>, <a class="txt" href="euclid_3.html#E1:73"><i><font color="Green">E41</font></i></a>, <a class="ref" href="euclid_3.html#T6" target="_self">Th6</a>, <a class="ref" href="euclid_3.html#T8" target="_self">Th8</a>, <a class="ref" href="euclid_3.html#T9" target="_self">Th9</a>, </i>;<br/>
<b>hence </b><a NAME="E14:73_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p8</font> where <font color="Olive">p8</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> :  ex <font color="Olive">a1</font>, <font color="Olive">a2</font>, <font color="Olive">a3</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">a1</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">a2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">a3</font> &amp; <span class="p1">(<span class="default"><font color="Olive">a1</font> <a href="real_1.html#K3">+</a> <font color="Olive">a2</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Olive">a3</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">p8</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">a1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Olive">a2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Olive">a3</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p3</font></span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="euclid_3.html#E1:73"><i><font color="Green">E41</font></i></a>, <a class="ref" href="euclid_3.html#T6" target="_self">Th6</a>, <a class="ref" href="euclid_3.html#T8" target="_self">Th8</a>, <a class="ref" href="euclid_3.html#T9" target="_self">Th9</a>, <a class="ref" href="euclid_3.html#T10" target="_self">Th10</a></i>;<br/>


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

<b>thus </b><a NAME="E2:73_1"/>
<span class="p1">{<span class="default"> <font color="Olive">p0</font> where <font color="Olive">p0</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> :  ex <font color="Olive">a1</font>, <font color="Olive">a2</font>, <font color="Olive">a3</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">a1</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">a2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">a3</font> &amp; <span class="p1">(<span class="default"><font color="Olive">a1</font> <a href="real_1.html#K3">+</a> <font color="Olive">a2</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Olive">a3</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">p0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">a1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Olive">a2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Olive">a3</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p3</font></span>)</span> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="euclid_3.html#K7" target="_self">inside_of_triangle</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:73_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p0</font> where <font color="Olive">p0</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> :  ex <font color="Olive">a1</font>, <font color="Olive">a2</font>, <font color="Olive">a3</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">a1</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">a2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">a3</font> &amp; <span class="p1">(<span class="default"><font color="Olive">a1</font> <a href="real_1.html#K3">+</a> <font color="Olive">a2</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Olive">a3</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">p0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">a1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Olive">a2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Olive">a3</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p3</font></span>)</span> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">p0</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> 2</span>)</span><b> such that </b><br/><a NAME="E3:73_1_2"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p0</font> &amp;  ex <font color="Olive">a1</font>, <font color="Olive">a2</font>, <font color="Olive">a3</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">a1</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">a2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">a3</font> &amp; <span class="p1">(<span class="default"><font color="Olive">a1</font> <a href="real_1.html#K3">+</a> <font color="Olive">a2</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Olive">a3</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">p0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">a1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Olive">a2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Olive">a3</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p3</font></span>)</span> ) )
 ;<br/>
<b>consider </b><font color="Maroon">a01</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">a02</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">a03</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:73_1_2"/><i><font color="Green">E89</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a01</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a02</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a03</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">a01</font> <a href="real_1.html#K3">+</a> <font color="Maroon">a02</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">a03</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">p0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a01</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">a02</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">a03</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p3</font></span>)</span> )
 <b>by </b><i><a class="ref" href="euclid_3.html#T2" target="_self">Th2</a></i>;<br/>
<b>set </b><font color="Maroon">i01</font> =  <a href="euclid_3.html#K10" target="_self">tricord1</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p0</font>;<br/>
<b>set </b><font color="Maroon">i02</font> =  <a href="euclid_3.html#K11" target="_self">tricord2</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p0</font>;<br/>
<b>set </b><font color="Maroon">i03</font> =  <a href="euclid_3.html#K12" target="_self">tricord3</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p0</font>;<br/>
<a NAME="E6:73_1_2"/>
<font color="Maroon">p0</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> 2</span>)</span>
 
;<br/>
<a NAME="E7:73_1_2"/><b>then </b>
<font color="Maroon">p0</font> <a href="hidden.html#R2">in</a>  <a href="euclid.html#K1">REAL</a> 2
 
<b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<a NAME="E8:73_1_2"/><b>then </b><i><font color="Green">E103</font></i>: 
<font color="Maroon">p0</font> <a href="hidden.html#R2">in</a>  <a href="euclid_3.html#K9" target="_self">plane</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:73_1_2"/><b>then </b><i><font color="Green">E104</font></i>: 
<font color="Maroon">a01</font> <a href="hidden.html#R1">=</a>  <a href="euclid_3.html#K10" target="_self">tricord1</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p0</font>
 
<b>by </b><i>, <a class="txt" href="euclid_3.html#E1:73"><i><font color="Green">E41</font></i></a>, </i>;<br/>
<a NAME="E10:73_1_2"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">a02</font> <a href="hidden.html#R1">=</a>  <a href="euclid_3.html#K11" target="_self">tricord2</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p0</font>
 
<b>by </b><i>, <a class="txt" href="euclid_3.html#E1:73"><i><font color="Green">E41</font></i></a>, <a class="ref" href="euclid_3.html#T6" target="_self">Th6</a>, </i>;<br/>
<a NAME="E11:73_1_2"/>
<font color="Maroon">a03</font> <a href="hidden.html#R1">=</a>  <a href="euclid_3.html#K12" target="_self">tricord3</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p0</font>
 
<b>by </b><i>, <a class="txt" href="euclid_3.html#E1:73"><i><font color="Green">E41</font></i></a>, <a class="ref" href="euclid_3.html#T6" target="_self">Th6</a>, </i>;<br/>
<a NAME="E12:73_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="euclid_3.html#K6" target="_self">closed_inside_of_triangle</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="euclid_3.html#K5" target="_self">Triangle</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font> )
 
<b>by </b><i>, <a class="ref" href="euclid_3.html#T2" target="_self">Th2</a>, <a class="txt" href="euclid_3.html#E1:73"><i><font color="Green">E41</font></i></a>, <a class="ref" href="euclid_3.html#T7" target="_self">Th7</a>, <a class="ref" href="euclid_3.html#T8" target="_self">Th8</a>, </i>;<br/>
<b>hence </b><a NAME="E13:73_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="euclid_3.html#K7" target="_self">inside_of_triangle</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


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


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:73"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="euclid_3.html#K7" target="_self">inside_of_triangle</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font> implies (  <a href="euclid_3.html#K10" target="_self">tricord1</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="euclid_3.html#K11" target="_self">tricord2</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="euclid_3.html#K12" target="_self">tricord3</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:73_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="euclid_3.html#K7" target="_self">inside_of_triangle</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>
 ;<br/>

<a NAME="E2:73_2"/><b>then </b><i><font color="Green">E106</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="euclid_3.html#K6" target="_self">closed_inside_of_triangle</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font> &amp; not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="euclid_3.html#K5" target="_self">Triangle</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">p0</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> 2</span>)</span><b> such that </b><br/><a NAME="E4:73_2"/><i><font color="Green">E129</font></i>: 
( <font color="Maroon">p0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp;  ex <font color="Olive">a1</font>, <font color="Olive">a2</font>, <font color="Olive">a3</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a1</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a3</font> &amp; <span class="p1">(<span class="default"><font color="Olive">a1</font> <a href="real_1.html#K3">+</a> <font color="Olive">a2</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Olive">a3</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">p0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">a1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Olive">a2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Olive">a3</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p3</font></span>)</span> ) )
 ;<br/>
<a NAME="E5:73_2"/><i><font color="Green">E130</font></i>: 
(  not  <a href="euclid_3.html#K10" target="_self">tricord1</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p0</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or  not  <a href="euclid_3.html#K11" target="_self">tricord2</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p0</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or  not  <a href="euclid_3.html#K12" target="_self">tricord3</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p0</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or ( not  <a href="euclid_3.html#K10" target="_self">tricord1</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p0</font> <a href="hidden.html#R1">=</a> 0 &amp; not  <a href="euclid_3.html#K11" target="_self">tricord2</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p0</font> <a href="hidden.html#R1">=</a> 0 &amp; not  <a href="euclid_3.html#K12" target="_self">tricord3</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p0</font> <a href="hidden.html#R1">=</a> 0 ) )
 
<b>by </b><i>, <a class="ref" href="euclid_3.html#T2" target="_self">Th2</a>, <a class="txt" href="euclid_3.html#E1:73"><i><font color="Green">E41</font></i></a>, </i>;<br/>
<a NAME="E6:73_2"/>
<font color="Maroon">p</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> 2</span>)</span>
 
;<br/>
<a NAME="E7:73_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="euclid.html#K1">REAL</a> 2
 
<b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<a NAME="E8:73_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="euclid_3.html#K9" target="_self">plane</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E9:73_2"/>
(  <a href="euclid_3.html#K10" target="_self">tricord1</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="euclid_3.html#K11" target="_self">tricord2</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="euclid_3.html#K12" target="_self">tricord3</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i>, <a class="txt" href="euclid_3.html#E1:73"><i><font color="Green">E41</font></i></a>, <a class="ref" href="euclid_3.html#T6" target="_self">Th6</a>, , , </i>;<br/>


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

<b>thus </b><a NAME="E4:73"/>
(  <a href="euclid_3.html#K10" target="_self">tricord1</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="euclid_3.html#K11" target="_self">tricord2</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="euclid_3.html#K12" target="_self">tricord3</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 implies <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="euclid_3.html#K7" target="_self">inside_of_triangle</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:73_3"/><i><font color="Green">E131</font></i>: 
(  <a href="euclid_3.html#K10" target="_self">tricord1</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="euclid_3.html#K11" target="_self">tricord2</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="euclid_3.html#K12" target="_self">tricord3</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>

<a NAME="E2:73_3"/>
<font color="Maroon">p</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> 2</span>)</span>
 
;<br/>
<a NAME="E3:73_3"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="euclid.html#K1">REAL</a> 2
 
<b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<a NAME="E4:73_3"/><b>then </b><i><font color="Green">E132</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="euclid_3.html#K9" target="_self">plane</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>
 
<b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">i1</font> =  <a href="euclid_3.html#K10" target="_self">tricord1</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p</font>;<br/>
<b>set </b><font color="Maroon">i2</font> =  <a href="euclid_3.html#K11" target="_self">tricord2</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p</font>;<br/>
<b>set </b><font color="Maroon">i3</font> =  <a href="euclid_3.html#K12" target="_self">tricord3</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p</font>;<br/>
<b>consider </b><font color="Maroon">a2</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">a3</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:73_3"/><i><font color="Green">E133</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K10" target="_self">tricord1</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">a2</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">a3</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</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" target="_self">tricord1</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p3</font></span>)</span> )
 <b>by </b><i>, <a class="txt" href="euclid_3.html#E1:73"><i><font color="Green">E41</font></i></a>, </i>;<br/>
<a NAME="E7:73_3"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="euclid_3.html#K11" target="_self">tricord2</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p</font>
 
<b>by </b><i>, <a class="txt" href="euclid_3.html#E1:73"><i><font color="Green">E41</font></i></a>, <a class="ref" href="euclid_3.html#T6" target="_self">Th6</a>, </i>;<br/>
<a NAME="E8:73_3"/>
<font color="Maroon">a3</font> <a href="hidden.html#R1">=</a>  <a href="euclid_3.html#K12" target="_self">tricord3</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">p</font>
 
<b>by </b><i>, <a class="txt" href="euclid_3.html#E1:73"><i><font color="Green">E41</font></i></a>, <a class="ref" href="euclid_3.html#T6" target="_self">Th6</a>, </i>;<br/>
<b>hence </b><a NAME="E9:73_3"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="euclid_3.html#K7" target="_self">inside_of_triangle</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>
 <b>by </b><i><a class="ref" href="euclid_3.html#T1" target="_self">Th1</a>, <a class="ref" href="euclid_3.html#T2" target="_self">Th2</a>, <a class="ref" href="euclid_3.html#T6" target="_self">Th6</a>, <a class="ref" href="euclid_3.html#T7" target="_self">Th7</a></i>;<br/>


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


</div>
