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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>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</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="E1:13"/><i><font color="Green">E11</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>  &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>7</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>8</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  )
 ;<br/>

<a NAME="E2:13"/><b>then </b><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>5</sub></font>,<font color="Maroon">c<sub>6</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>5</sub></font>,<font color="Maroon">c<sub>7</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>5</sub></font>,<font color="Maroon">c<sub>8</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>6</sub></font>,<font color="Maroon">c<sub>8</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>7</sub></font>,<font color="Maroon">c<sub>8</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>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  )
 
<b>by </b><i><a class="txt" href="projdes1.html#E16"><i><font color="Green">Lemma9</font></i></a></i>;<br/>
<a NAME="E3:13"/><i><font color="Green">E13</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>
 
<b>by </b><i><a class="txt" href="projdes1.html#E1:13"><i><font color="Green">E11</font></i></a>, <a class="ref" href="anproj_2.html#D7">ANPROJ_2:def 7</a></i>;<br/>
<div><i><font color="Green">E14</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_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> <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>6</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>7</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 ;<br/><a NAME="E2:13_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="txt" href="projdes1.html#E3:13"><i><font color="Green">E13</font></i></a>, <a class="ref" href="anproj_2.html#D8">ANPROJ_2:def 8</a></i>;<br/><b>hence </b><a NAME="E3:13_1"/>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</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><i><font color="Green">E15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2"/><i><font color="Green">E16</font></i>: 
not <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> 
 ;<br/><a NAME="E2:13_2"/><b>then </b><i><font color="Green">E17</font></i>: 
not <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</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/><a NAME="E3:13_2"/>
( <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<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="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<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="projdes1.html#R1" target="_self">are_coplanar</a>  )
 <b>by </b><i><a class="txt" href="projdes1.html#E2:13"><i><font color="Green">E12</font></i></a>, <a class="ref" href="projdes1.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E4:13_2"/><b>then </b><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<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="projdes1.html#R1" target="_self">are_coplanar</a>  )
 <b>by </b><i><a class="txt" href="projdes1.html#E2:13_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="projdes1.html#E16"><i><font color="Green">Lemma9</font></i></a></i>;<br/><a NAME="E5:13_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="projdes1.html#E1:13_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="anproj_2.html#D7">ANPROJ_2:def 7</a></i>;<br/><div><i><font color="Green">E20</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2_1"/>
( <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>5</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>  )
 ;<br/><a NAME="E2:13_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="txt" href="projdes1.html#E5:13_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="collsp.html#T11">COLLSP:11</a></i>;<br/><b>hence </b><a NAME="E3:13_2_1"/>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</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><i><font color="Green">E21</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2_2"/>
not <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/><a NAME="E2:13_2_2"/><b>then </b><i><font color="Green">E22</font></i>: 
not <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</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/><a NAME="E3:13_2_2"/>
( <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  )
 <b>by </b><i><a class="txt" href="projdes1.html#E4:13_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="projdes1.html#T11" target="_self">Th11</a></i>;<br/><b>hence </b><a NAME="E4:13_2_2"/>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a> 
 <b>by </b><i><a class="txt" href="projdes1.html#E2:13_2_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="projdes1.html#E16"><i><font color="Green">Lemma9</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2_3"/>
not <font color="Maroon">c<sub>5</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> 
 ;<br/><a NAME="E2:13_2_3"/><b>then </b><i><font color="Green">E22</font></i>: 
not <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</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/><a NAME="E3:13_2_3"/>
( <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  )
 <b>by </b><i><a class="txt" href="projdes1.html#E4:13_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="projdes1.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E4:13_2_3"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a> 
 <b>by </b><i><a class="txt" href="projdes1.html#E2:13_2_3"><i><font color="Green">E22</font></i></a>, <a class="txt" href="projdes1.html#E16"><i><font color="Green">Lemma9</font></i></a></i>;<br/><b>hence </b><a NAME="E5:13_2_3"/>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a> 
 <b>by </b><i><a class="ref" href="projdes1.html#T11" target="_self">Th11</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:13_2"/>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a> 
 <b>by </b><i><a class="txt" href="projdes1.html#E6:13_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projdes1.html#E7:13_2"><i><font color="Green">E21</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_3"/><i><font color="Green">E17</font></i>: 
not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/><a NAME="E2:13_3"/><b>then </b><i><font color="Green">E18</font></i>: 
not <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</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/><a NAME="E3:13_3"/>
( <font color="Maroon">c<sub>6</sub></font>,<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="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; <font color="Maroon">c<sub>6</sub></font>,<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="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; <font color="Maroon">c<sub>6</sub></font>,<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="projdes1.html#R1" target="_self">are_coplanar</a>  )
 <b>by </b><i><a class="txt" href="projdes1.html#E2:13"><i><font color="Green">E12</font></i></a>, <a class="ref" href="projdes1.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E4:13_3"/><b>then </b><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; <font color="Maroon">c<sub>6</sub></font>,<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="projdes1.html#R1" target="_self">are_coplanar</a>  )
 <b>by </b><i><a class="txt" href="projdes1.html#E2:13_3"><i><font color="Green">E18</font></i></a>, <a class="txt" href="projdes1.html#E16"><i><font color="Green">Lemma9</font></i></a></i>;<br/><a NAME="E5:13_3"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="projdes1.html#E1:13_3"><i><font color="Green">E17</font></i></a>, <a class="ref" href="anproj_2.html#D7">ANPROJ_2:def 7</a></i>;<br/><div><i><font color="Green">E21</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_3_1"/>
( <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>6</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>  )
 ;<br/><a NAME="E2:13_3_1"/><b>then </b>
<font color="Maroon">c<sub>6</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="txt" href="projdes1.html#E5:13_3"><i><font color="Green">E20</font></i></a>, <a class="ref" href="collsp.html#T11">COLLSP:11</a></i>;<br/><a NAME="E3:13_3_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="E4:13_3_1"/>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</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><i><font color="Green">E22</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_3_2"/>
not <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/><a NAME="E2:13_3_2"/><b>then </b><i><font color="Green">E23</font></i>: 
not <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</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/><a NAME="E3:13_3_2"/>
( <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  )
 <b>by </b><i><a class="txt" href="projdes1.html#E4:13_3"><i><font color="Green">E19</font></i></a>, <a class="ref" href="projdes1.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E4:13_3_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a> 
 <b>by </b><i><a class="txt" href="projdes1.html#E2:13_3_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="projdes1.html#E16"><i><font color="Green">Lemma9</font></i></a></i>;<br/><b>hence </b><a NAME="E5:13_3_2"/>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a> 
 <b>by </b><i><a class="ref" href="projdes1.html#T11" target="_self">Th11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_3_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_3_3"/>
not <font color="Maroon">c<sub>6</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> 
 ;<br/><a NAME="E2:13_3_3"/><b>then </b><i><font color="Green">E23</font></i>: 
not <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</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/><a NAME="E3:13_3_3"/>
( <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</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>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  )
 <b>by </b><i><a class="txt" href="projdes1.html#E4:13_3"><i><font color="Green">E19</font></i></a>, <a class="ref" href="projdes1.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E4:13_3_3"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a> 
 <b>by </b><i><a class="txt" href="projdes1.html#E2:13_3_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="projdes1.html#E16"><i><font color="Green">Lemma9</font></i></a></i>;<br/><b>hence </b><a NAME="E5:13_3_3"/>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a> 
 <b>by </b><i><a class="ref" href="projdes1.html#T11" target="_self">Th11</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:13_3"/>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a> 
 <b>by </b><i><a class="txt" href="projdes1.html#E6:13_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="projdes1.html#E7:13_3"><i><font color="Green">E22</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_4"/><i><font color="Green">E17</font></i>: 
not <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> 
 ;<br/><a NAME="E2:13_4"/><b>then </b><i><font color="Green">E18</font></i>: 
not <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</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/><a NAME="E3:13_4"/>
( <font color="Maroon">c<sub>7</sub></font>,<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="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; <font color="Maroon">c<sub>7</sub></font>,<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="projdes1.html#R1" target="_self">are_coplanar</a>  )
 <b>by </b><i><a class="txt" href="projdes1.html#E2:13"><i><font color="Green">E12</font></i></a>, <a class="ref" href="projdes1.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E4:13_4"/><b>then </b><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</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>  &amp; <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  )
 <b>by </b><i><a class="txt" href="projdes1.html#E2:13_4"><i><font color="Green">E18</font></i></a>, <a class="txt" href="projdes1.html#E16"><i><font color="Green">Lemma9</font></i></a></i>;<br/><a NAME="E5:13_4"/><i><font color="Green">E20</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>2</sub></font>
 <b>by </b><i><a class="txt" href="projdes1.html#E1:13_4"><i><font color="Green">E17</font></i></a>, <a class="ref" href="anproj_2.html#D7">ANPROJ_2:def 7</a></i>;<br/><div><i><font color="Green">E21</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_4_1"/>
( <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</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>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 ;<br/><a NAME="E2:13_4_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#E5:13_4"><i><font color="Green">E20</font></i></a>, <a class="ref" href="collsp.html#T11">COLLSP:11</a></i>;<br/><a NAME="E3:13_4_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="E4:13_4_1"/>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</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><i><font color="Green">E22</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_4_2"/>
not <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/><a NAME="E2:13_4_2"/><b>then </b><i><font color="Green">E23</font></i>: 
not <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</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/><a NAME="E3:13_4_2"/>
( <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  )
 <b>by </b><i><a class="txt" href="projdes1.html#E4:13_4"><i><font color="Green">E19</font></i></a>, <a class="ref" href="projdes1.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E4:13_4_2"/><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>,<font color="Maroon">c<sub>8</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a> 
 <b>by </b><i><a class="txt" href="projdes1.html#E2:13_4_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="projdes1.html#E16"><i><font color="Green">Lemma9</font></i></a></i>;<br/><b>hence </b><a NAME="E5:13_4_2"/>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a> 
 <b>by </b><i><a class="ref" href="projdes1.html#T11" target="_self">Th11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_4_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_4_3"/>
not <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/><a NAME="E2:13_4_3"/><b>then </b><i><font color="Green">E23</font></i>: 
not <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</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/><a NAME="E3:13_4_3"/>
( <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</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>7</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  )
 <b>by </b><i><a class="txt" href="projdes1.html#E4:13_4"><i><font color="Green">E19</font></i></a>, <a class="ref" href="projdes1.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E4:13_4_3"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a> 
 <b>by </b><i><a class="txt" href="projdes1.html#E2:13_4_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="projdes1.html#E16"><i><font color="Green">Lemma9</font></i></a></i>;<br/><b>hence </b><a NAME="E5:13_4_3"/>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a> 
 <b>by </b><i><a class="ref" href="projdes1.html#T11" target="_self">Th11</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:13_4"/>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a> 
 <b>by </b><i><a class="txt" href="projdes1.html#E6:13_4"><i><font color="Green">E21</font></i></a>, <a class="txt" href="projdes1.html#E7:13_4"><i><font color="Green">E22</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:13"/>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a> 
 <b>by </b><i><a class="txt" href="projdes1.html#E4:13"><i><font color="Green">E14</font></i></a>, <a class="txt" href="projdes1.html#E5:13"><i><font color="Green">E15</font></i></a>, <a class="txt" href="projdes1.html#E6:13"><i><font color="Green">E16</font></i></a></i>;<br/>


</div>
