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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="realset2.html#V3">trivial</a> <a href="rlvect_1.html#NM2">RealLinearSpace</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> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>







<b>assume </b><a NAME="E1:47"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  )
 ;<br/>

<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_1"/><i><font color="Green">E35</font></i>: 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2">(<span class="default"><a href="anproj_1.html#K3" target="_self">Dir</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="anproj_1.html#K3" target="_self">Dir</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="anproj_1.html#K3" target="_self">Dir</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="mcart_1.html#K3">]</a></span> <a href="hidden.html#R2">in</a> the <a href="collsp.html#U1">Collinearity</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6" target="_self">ProjectiveSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="anproj_1.html#K3" target="_self">Dir</a> <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>6</sub></font> =  <a href="anproj_1.html#K3" target="_self">Dir</a> <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>7</sub></font> =  <a href="anproj_1.html#K3" target="_self">Dir</a> <font color="Maroon">c<sub>4</sub></font> as    <a href="hidden.html#M1">set</a>  ;<br/><a NAME="E3:47_1"/>
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="mcart_1.html#K3">]</a></span> <a href="hidden.html#R2">in</a>  <a href="anproj_1.html#K5" target="_self">ProjectiveCollinearity</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="anproj_1.html#E1:47_1"><i><font color="Green">E35</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</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="E5:47_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3" target="_self">Dir</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3" target="_self">Dir</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3" target="_self">Dir</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">c<sub>9</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a>  )
 <b>by </b><i><a class="ref" href="anproj_1.html#D9" target="_self">Def9</a></i>;<br/><a NAME="E6:47_1"/>
(  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="anproj_1.html#E1:47"><i><font color="Green">E33</font></i></a>, <a class="txt" href="anproj_1.html#E5:47_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="anproj_1.html#T35" target="_self">Th35</a></i>;<br/><b>hence </b><a NAME="E7:47_1"/>
<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="anproj_1.html#R2" target="_self">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="anproj_1.html#E5:47_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="anproj_1.html#T9" target="_self">Th9</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:47"/>
( <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="anproj_1.html#R2" target="_self">are_LinDep</a>  implies <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2">(<span class="default"><a href="anproj_1.html#K3" target="_self">Dir</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="anproj_1.html#K3" target="_self">Dir</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="anproj_1.html#K3" target="_self">Dir</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="mcart_1.html#K3">]</a></span> <a href="hidden.html#R2">in</a>  <a href="anproj_1.html#K5" target="_self">ProjectiveCollinearity</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="anproj_1.html#E1:47"><i><font color="Green">E33</font></i></a>, <a class="ref" href="anproj_1.html#D9" target="_self">Def9</a></i>;<br/>
<b>hence </b><a NAME="E4:47"/>
( <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2">(<span class="default"><a href="anproj_1.html#K3" target="_self">Dir</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="anproj_1.html#K3" target="_self">Dir</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="anproj_1.html#K3" target="_self">Dir</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="mcart_1.html#K3">]</a></span> <a href="hidden.html#R2">in</a> the <a href="collsp.html#U1">Collinearity</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6" target="_self">ProjectiveSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> iff <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="anproj_1.html#R2" target="_self">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="anproj_1.html#E2:47"><i><font color="Green">E34</font></i></a></i>;<br/>


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