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<b>consider </b><font color="Maroon">c<sub>1</sub></font> being  non <a href="realset2.html#V3">trivial</a> <a href="rlvect_1.html#NM2">RealLinearSpace</a><b> such that </b><br/><a NAME="E2:101_2_1"/><i><font color="Green">E90</font></i>: 
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>, <font color="Olive">b<sub>6</sub></font>, <font color="Olive">b<sub>7</sub></font>, <font color="Olive">b<sub>8</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>5</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>6</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>7</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>8</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>4</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>5</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>8</sub></font> <a href="hidden.html#R1">=</a> 0 ) ) &amp; ( 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>ex <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> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">b<sub>6</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>7</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>8</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>9</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>4</sub></font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="anproj_2.html#T23" target="_self">Th23</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  <a href="anproj_2.html#V1" target="_self">up-3-dimensional</a> <a href="rlvect_1.html#NM2">RealLinearSpace</a> <b>by </b><i><a class="txt" href="anproj_2.html#E2:101_2_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="anproj_2.html#E89"><i><font color="Green">Lemma82</font></i></a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>3</sub></font> =  <a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>thus </b><a NAME="E4:101_2_1"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="collsp.html#V1">strict</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="anproj_2.html#V4" target="_self">Fanoian</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="anproj_2.html#V5" target="_self">Desarguesian</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="anproj_2.html#V6" target="_self">Pappian</a> )
 ;<br/>

<a NAME="E5:101_2_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="anproj_2.html#NM1">CollProjectiveSpace</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">c<sub>2</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> is <a href="anproj_2.html#V7" target="_self">2-dimensional</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="anproj_2.html#V8" target="_self">at_most-3-dimensional</a> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E2:101_2_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="anproj_2.html#T36" target="_self">Th36</a></i>;<br/>
<b>hence </b><a NAME="E6:101_2_1"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="anproj_2.html#V8" target="_self">at_most-3-dimensional</a> &amp; not <font color="Maroon">c<sub>3</sub></font> is <a href="anproj_2.html#V7" target="_self">2-dimensional</a> )
 ;<br/>


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