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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>given </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> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E1:99"/><i><font color="Green">E89</font></i>: 
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="real_1.html#NM1">Real</a> st <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>

<a NAME="E2:99"/><i><font color="Green">E90</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>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E1:99"><i><font color="Green">E89</font></i></a>, <a class="ref" href="anproj_2.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E3:99"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="anproj_2.html#V1" target="_self">up-3-dimensional</a>
 
<b>by </b><i><a class="txt" href="anproj_2.html#E1:99"><i><font color="Green">E89</font></i></a>, <a class="txt" href="anproj_2.html#E89"><i><font color="Green">Lemma82</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">c<sub>1</sub></font> as   <a href="anproj_2.html#NM1">CollProjectiveSpace</a> <b>by </b><i><a class="txt" href="anproj_2.html#E75"><i><font color="Green">Lemma71</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>thus </b><a NAME="E5:99"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>8</sub></font> =  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>9</sub></font> =  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>10</sub></font> =  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>6</sub></font> <b>by </b><i><a class="txt" href="anproj_2.html#E2:99"><i><font color="Green">E90</font></i></a>, <a class="ref" href="anproj_1.html#T42">ANPROJ_1:42</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="anproj_2.html#D14" target="_self">ANPROJ_2:def 14</a><br/>

<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>10</sub></font>
;<br/>

<b>thus </b><a NAME="E7:99"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>6</sub></font> holds <br/>(  not <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  or  not <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:99_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>6</sub></font> st <br/>( <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E3:99_1"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 ;<br/>
<a NAME="E4:99_1"/><i><font color="Green">E92</font></i>: 
( <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K4">]</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">ProjectiveSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K4">]</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">ProjectiveSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E3:99_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="collsp.html#D2">COLLSP:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</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="E6:99_1"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="ref" href="anproj_1.html#T42">ANPROJ_1:42</a></i>;<br/>
<a NAME="E7:99_1"/>
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="anproj_1.html#R2">are_LinDep</a>  &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="anproj_1.html#R2">are_LinDep</a>  )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E2:99"><i><font color="Green">E90</font></i></a>, <a class="txt" href="anproj_2.html#E4:99_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="anproj_2.html#E6:99_1"><i><font color="Green">E93</font></i></a>, <a class="ref" href="anproj_1.html#T41">ANPROJ_1:41</a></i>;<br/>
<b>hence </b><a NAME="E8:99_1"/>
contradiction
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:99"><i><font color="Green">E89</font></i></a>, <a class="txt" href="anproj_2.html#E6:99_1"><i><font color="Green">E93</font></i></a>, <a class="ref" href="anproj_2.html#T5" target="_self">Th5</a></i>;<br/>


</div><b>end;</b></div>


</div>
