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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="anproj_2.html#V7">up-3-dimensional</a> <a href="anproj_2.html#NM1">CollProjectiveSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>











<b>assume </b><a NAME="E1:12"/><i><font color="Green">E10</font></i>: 
( not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:12"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="ref" href="projdes1.html#D1" target="_self">Def1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:12"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="projdes1.html#E1:12"><i><font color="Green">E10</font></i></a>, <a class="ref" href="projdes1.html#D1" target="_self">Def1</a></i>;<br/>
<div><i><font color="Green">E13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:12_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="anproj_2.html#D7">ANPROJ_2:def 7</a></i>;<br/><b>hence </b><a NAME="E3:12_1"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a> 
 <b>by </b><i><a class="ref" href="projdes1.html#T10" target="_self">Th10</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:12_2"/><b>then </b><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="projdes1.html#E3:12"><i><font color="Green">E11</font></i></a>, <a class="txt" href="projdes1.html#E5:12"><i><font color="Green">E12</font></i></a>, <a class="ref" href="collsp.html#T11">COLLSP:11</a></i>;<br/><a NAME="E3:12_2"/>
( <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="projdes1.html#E3:12"><i><font color="Green">E11</font></i></a>, <a class="txt" href="projdes1.html#E5:12"><i><font color="Green">E12</font></i></a>, <a class="ref" href="projdes1.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E4:12_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="projdes1.html#E1:12_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="collsp.html#T11">COLLSP:11</a></i>;<br/><a NAME="E5:12_2"/><b>then </b><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="projdes1.html#E2:12_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="projdes1.html#T1" target="_self">Th1</a></i>;<br/><div><i><font color="Green">E17</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E2:12_2_1"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="projdes1.html#E3:12"><i><font color="Green">E11</font></i></a>, <a class="txt" href="projdes1.html#E5:12"><i><font color="Green">E12</font></i></a>, <a class="ref" href="projdes1.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E3:12_2_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="projdes1.html#E1:12"><i><font color="Green">E10</font></i></a>, <a class="txt" href="projdes1.html#E3:12"><i><font color="Green">E11</font></i></a>, <a class="ref" href="collsp.html#T11">COLLSP:11</a></i>;<br/><a NAME="E4:12_2_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="projdes1.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E5:12_2_1"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a> 
 <b>by </b><i><a class="txt" href="projdes1.html#E3:12"><i><font color="Green">E11</font></i></a>, <a class="ref" href="projdes1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2_2"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E2:12_2_2"/>
( <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="projdes1.html#E3:12"><i><font color="Green">E11</font></i></a>, <a class="txt" href="projdes1.html#E5:12"><i><font color="Green">E12</font></i></a>, <a class="ref" href="projdes1.html#T1" target="_self">Th1</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:12_2_2"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="ref" href="anproj_2.html#D9">ANPROJ_2:def 9</a></i>;<br/><a NAME="E5:12_2_2"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="projdes1.html#E5:12_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="projdes1.html#E1:12_2_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="projdes1.html#E4:12_2_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="anproj_2.html#D8">ANPROJ_2:def 8</a></i>;<br/><b>hence </b><a NAME="E6:12_2_2"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a> 
 <b>by </b><i><a class="txt" href="projdes1.html#E4:12_2_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="projdes1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:12_2"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a> 
 <b>by </b><i><a class="txt" href="projdes1.html#E6:12_2"><i><font color="Green">E17</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:12"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a> 
 <b>by </b><i><a class="txt" href="projdes1.html#E6:12"><i><font color="Green">E13</font></i></a></i>;<br/>


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