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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> 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="E1:74"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>1</sub></font> <a href="euclid_3.html#R1" target="_self">are_lindependent2</a> 
 ;<br/>

<a NAME="E2:74"/><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="hidden.html#R1">=</a> 1
 
;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="euclid_3.html#K10" target="_self">tricord1</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="euclid_3.html#K11" target="_self">tricord2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="euclid_3.html#K12" target="_self">tricord3</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>;<br/>
<a NAME="E3:74"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <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="E4:74"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <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="E5:74"/><b>then </b><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="euclid_3.html#K9" target="_self">plane</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="euclid_3.html#E1:74"><i><font color="Green">E38</font></i></a>, <a class="ref" href="euclid_3.html#T63" target="_self">Th63</a></i>;<br/>
<a NAME="E6:74"/><b>then </b><i><font color="Green">E41</font></i>: 
1 <a href="real_1.html#K6">/</a> 3 <a href="hidden.html#R1">=</a>  <a href="euclid_3.html#K10" target="_self">tricord1</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="euclid_3.html#E1:74"><i><font color="Green">E38</font></i></a>, <a class="txt" href="euclid_3.html#E2:74"><i><font color="Green">E39</font></i></a>, <a class="ref" href="euclid_3.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E7:74"/><i><font color="Green">E42</font></i>: 
1 <a href="real_1.html#K6">/</a> 3 <a href="hidden.html#R1">=</a>  <a href="euclid_3.html#K11" target="_self">tricord2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="euclid_3.html#E1:74"><i><font color="Green">E38</font></i></a>, <a class="txt" href="euclid_3.html#E2:74"><i><font color="Green">E39</font></i></a>, <a class="txt" href="euclid_3.html#E5:74"><i><font color="Green">E40</font></i></a>, <a class="ref" href="euclid_3.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E8:74"/>
1 <a href="real_1.html#K6">/</a> 3 <a href="hidden.html#R1">=</a>  <a href="euclid_3.html#K12" target="_self">tricord3</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="euclid_3.html#E1:74"><i><font color="Green">E38</font></i></a>, <a class="txt" href="euclid_3.html#E2:74"><i><font color="Green">E39</font></i></a>, <a class="txt" href="euclid_3.html#E5:74"><i><font color="Green">E40</font></i></a>, <a class="ref" href="euclid_3.html#D13" target="_self">Def13</a></i>;<br/>
<b>hence </b><a NAME="E9:74"/>
not  <a href="euclid_3.html#K7" target="_self">inside_of_triangle</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="euclid_3.html#E1:74"><i><font color="Green">E38</font></i></a>, <a class="txt" href="euclid_3.html#E6:74"><i><font color="Green">E41</font></i></a>, <a class="txt" href="euclid_3.html#E7:74"><i><font color="Green">E42</font></i></a>, <a class="ref" href="euclid_3.html#T67" target="_self">Th67</a></i>;<br/>


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