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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="anproj_2.html#NM1">CollProjectiveSpace</a>;<br/>

<b>assume </b><a NAME="E1:21_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="anproj_2.html#V6">Pappian</a>
 ;<br/>

<div><i><font color="Green">E19</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>for <font color="Olive">b<sub>5</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>(  not <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a>  or  not <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 ;<br/><a NAME="E2:21_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> is  <a href="anproj_2.html#V7">up-3-dimensional</a> <a href="anproj_2.html#NM1">CollProjectiveSpace</a>
 <b>by </b><i><a class="ref" href="anproj_2.html#D14">ANPROJ_2:def 14</a></i>;<br/><b>hence </b><a NAME="E3:21_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="anproj_2.html#V5">Desarguesian</a>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>5</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 ;<br/><a NAME="E2:21_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> is  <a href="anproj_2.html#V6">Pappian</a> <a href="anproj_2.html#NM2">CollProjectivePlane</a>
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:21_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="anproj_2.html#D14">ANPROJ_2:def 14</a></i>;<br/><a NAME="E3:21_1_1_2"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font>, <font color="Olive">b<sub>6</sub></font>, <font color="Olive">b<sub>7</sub></font>, <font color="Olive">b<sub>8</sub></font>, <font color="Olive">b<sub>9</sub></font>, <font color="Olive">b<sub>10</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>5</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>6</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>6</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>7</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>7</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>6</sub></font>,<font color="Olive">b<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">b<sub>6</sub></font>,<font color="Olive">b<sub>7</sub></font>,<font color="Olive">b<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>7</sub></font>,<font color="Olive">b<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>6</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>7</sub></font> <a href="collsp.html#R1">is_collinear</a>  holds <br/><font color="Olive">b<sub>8</sub></font>,<font color="Olive">b<sub>9</sub></font>,<font color="Olive">b<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="hessenbe.html#T21" target="_self">Th21</a></i>;<br/><b>hence </b><a NAME="E4:21_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="anproj_2.html#V5">Desarguesian</a>
 <b>by </b><i><a class="ref" href="anproj_2.html#D12">ANPROJ_2:def 12</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:21_1_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="anproj_2.html#V5">Desarguesian</a>
 <b>by </b><i><a class="txt" href="hessenbe.html#E2:21_1_1"><i><font color="Green">E19</font></i></a></i>;<br/>


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