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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="incproj.html#V6">2-dimensional</a> <a href="incproj.html#V10">Desarguesian</a> <a href="incproj.html#NM2">IncProjSp</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><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="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>















<b>assume </b><a NAME="E1:25"/><i><font color="Green">E19</font></i>: 
( not <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>4</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>6</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>6</sub></font> &amp; not <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="projred2.html#R1" target="_self">are_concurrent</a>  &amp; <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="projred2.html#R1" target="_self">are_concurrent</a>  &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <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> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>

<a NAME="E2:25"/>
not <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="projred2.html#R1" target="_self">are_concurrent</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1"/>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="projred2.html#R1" target="_self">are_concurrent</a> 
 ;<br/>

<a NAME="E2:25_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="ref" href="projred2.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E3:25_1"/>
contradiction
 <b>by </b><i><a class="txt" href="projred2.html#E1:25"><i><font color="Green">E19</font></i></a>, <a class="ref" href="projred2.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:25"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>8</sub></font> &amp; not <font color="Maroon">c<sub>9</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Maroon">c<sub>9</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>and </b><br/><a NAME="E5:25"/><i><font color="Green">E21</font></i>: 
<span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="projred2.html#E1:25"><i><font color="Green">E19</font></i></a>, <a class="ref" href="projred2.html#T21" target="_self">Th21</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/>

<b>thus </b><a NAME="E6:25"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>8</sub></font> &amp; not <font color="Maroon">c<sub>9</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Maroon">c<sub>9</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="projred2.html#E4:25"><i><font color="Green">E20</font></i></a></i>;<br/>

<a NAME="E7:25"/><i><font color="Green">E22</font></i>: 
(  <a href="projred1.html#K1">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="projred1.html#K1">IncProj</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> is <a href="funct_1.html#V2">one-to-one</a> )
 
<b>by </b><i><a class="txt" href="projred2.html#E1:25"><i><font color="Green">E19</font></i></a>, <a class="txt" href="projred2.html#E4:25"><i><font color="Green">E20</font></i></a>, <a class="ref" href="projred2.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E8:25"/><i><font color="Green">E23</font></i>: 
(  <a href="projred1.html#K1">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="projred1.html#K1">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> is <a href="funct_1.html#V2">one-to-one</a> )
 
<b>by </b><i><a class="txt" href="projred2.html#E1:25"><i><font color="Green">E19</font></i></a>, <a class="ref" href="projred2.html#T8" target="_self">Th8</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="projred2.html#E1:25"><i><font color="Green">E19</font></i></a>, <a class="ref" href="projred2.html#T9" target="_self">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="projred2.html#E1:25"><i><font color="Green">E19</font></i></a>, <a class="ref" href="projred2.html#T9" target="_self">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> 
<b>by </b><i><a class="txt" href="projred2.html#E5:25"><i><font color="Green">E21</font></i></a>, <a class="txt" href="projred2.html#E8:25"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funct_1.html#T66">FUNCT_1:66</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="projred2.html#E7:25"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_1.html#T66">FUNCT_1:66</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="projred2.html#E4:25"><i><font color="Green">E20</font></i></a>, <a class="ref" href="projred2.html#T9" target="_self">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="projred2.html#E1:25"><i><font color="Green">E19</font></i></a>, <a class="ref" href="projred2.html#T9" target="_self">Th9</a></i>
;<br/>


</div>
