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<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="E1:71"/><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/>

<b>thus </b><a NAME="E2:71"/>
(  not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="euclid_3.html#K8" target="_self">outside_of_triangle</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font> or  <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#R1">&lt;</a> 0 or  <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#R1">&lt;</a> 0 or  <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#R1">&lt;</a> 0 )
 <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">p</font> <a href="hidden.html#R2">in</a>  <a href="euclid_3.html#K8" target="_self">outside_of_triangle</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>
 ;<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="E3:71_1"/><i><font color="Green">E49</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#NR4" target="_self">&gt;</a> <font color="Olive">a1</font> or 0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">a2</font> or 0 <a href="xxreal_0.html#NR4" target="_self">&gt;</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="E4:71_1"/>
<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="E5:71_1"/><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="E6:71_1"/><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="E7:71_1"/>
(  <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#R1">&lt;</a> 0 or  <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#R1">&lt;</a> 0 or  <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#R1">&lt;</a> 0 )
 <b>by </b><i>, , , , </i>;<br/>


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

<b>assume </b><a NAME="E3:71"/><i><font color="Green">E50</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#R1">&lt;</a> 0 or  <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#R1">&lt;</a> 0 or  <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#R1">&lt;</a> 0 )
 ;<br/>

<a NAME="E4:71"/>
<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="E5:71"/><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="E6:71"/><b>then </b><i><font color="Green">E62</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="E8:71"/><i><font color="Green">E71</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>, , </i>;<br/>
<a NAME="E9:71"/><i><font color="Green">E72</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="ref" href="euclid_3.html#T1" target="_self">Th1</a>, </i>;<br/>
<a NAME="E10:71"/>
<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="ref" href="euclid_3.html#T1" target="_self">Th1</a>, </i>;<br/>
<b>hence </b><a NAME="E11:71"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="euclid_3.html#K8" target="_self">outside_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></i>;<br/>


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