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<b>consider </b><font color="Maroon" title="c1">V</font> being  non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a><b> such that </b><br/><a NAME="E2:101_2_1"/><i><font color="Green" title="E90">A5</font></i>: 
 ex <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font>, <font color="Olive" title="b4">u1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( ( for <font color="Olive" title="b5">a</font>, <font color="Olive" title="b6">b</font>, <font color="Olive" title="b7">c</font>, <font color="Olive" title="b8">d</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b5">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Olive" title="b6">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b2">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b7">c</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b3">w</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b8">d</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b4">u1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> holds <br/>( <font color="Olive" title="b5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b8">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) &amp; ( for <font color="Olive" title="b5">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>  ex <font color="Olive" title="b6">a</font>, <font color="Olive" title="b7">b</font>, <font color="Olive" title="b8">c</font>, <font color="Olive" title="b9">d</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <font color="Olive" title="b5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b6">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Olive" title="b7">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b2">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b8">c</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b3">w</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b9">d</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b4">u1</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="anproj_2.html#T23" target="_self" title="ANPROJ_2:th.23">Th23</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c2">V</font> = <font color="Maroon" title="c1">V</font> as  <a href="anproj_2.html#V1" title="ANPROJ_2:attr.1">up-3-dimensional</a> <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a> <b>by </b><i><a class="txt" href="anproj_2.html#E2:101_2_1"><i><font color="Green" title="E90">A5</font></i></a>, <a class="txt" href="anproj_2.html#E89"><i><font color="Green" title="E82">Lm43</font></i></a></i>;<br/>
<b>take </b><font color="Maroon" title="c3">CS</font> =  <a href="anproj_1.html#K6" title="ANPROJ_1:func.6">ProjectiveSpace</a> <font color="Maroon" title="c2">V</font>;<br/>

<b>thus </b><a NAME="E4:101_2_1"/>
( <font color="Maroon" title="c3">CS</font> is <a href="collsp.html#V1" title="COLLSP:attr.1">strict</a> &amp; <font color="Maroon" title="c3">CS</font> is <a href="anproj_2.html#V4" title="ANPROJ_2:attr.4">Fanoian</a> &amp; <font color="Maroon" title="c3">CS</font> is <a href="anproj_2.html#V5" title="ANPROJ_2:attr.5">Desarguesian</a> &amp; <font color="Maroon" title="c3">CS</font> is <a href="anproj_2.html#V6" title="ANPROJ_2:attr.6">Pappian</a> )
 ;<br/>

<a NAME="E5:101_2_1"/>
 ex <font color="Olive" title="b1">CS</font> being  <a href="anproj_2.html#NM1" title="ANPROJ_2:NM.1">CollProjectiveSpace</a> st <br/>( <font color="Olive" title="b1">CS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K6" title="ANPROJ_1:func.6">ProjectiveSpace</a> <font color="Maroon" title="c2">V</font> &amp; not <font color="Olive" title="b1">CS</font> is <a href="anproj_2.html#V7" title="ANPROJ_2:attr.7">2-dimensional</a> &amp; <font color="Olive" title="b1">CS</font> is <a href="anproj_2.html#V8" title="ANPROJ_2:attr.8">at_most-3-dimensional</a> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E2:101_2_1"><i><font color="Green" title="E90">A5</font></i></a>, <a class="ref" href="anproj_2.html#T36" target="_self" title="ANPROJ_2:th.36">Th36</a></i>;<br/>
<b>hence </b><a NAME="E6:101_2_1"/>
( <font color="Maroon" title="c3">CS</font> is <a href="anproj_2.html#V8" title="ANPROJ_2:attr.8">at_most-3-dimensional</a> &amp; not <font color="Maroon" title="c3">CS</font> is <a href="anproj_2.html#V7" title="ANPROJ_2:attr.7">2-dimensional</a> )
 ;<br/>


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