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<b>let </b><font color="Maroon">V</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">u</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">v</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">w</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><a NAME="E1:92"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="real_1.html#NM1">Real</a>  st <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">c</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> holds <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> 0 )
 <b>and </b><br/><a NAME="E2:92"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>  ex <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">c</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">V'</font> = <font color="Maroon">V</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>, </i>;<br/>
<b>take </b>
 <a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V'</font>
;<br/>

<a NAME="E4:92"/>
 ex <font color="Olive">x1</font>, <font color="Olive">x2</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> st <br/>( <font color="Olive">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">x2</font> &amp; ( for <font color="Olive">r1</font>, <font color="Olive">r2</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>  ex <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> st <br/>( <font color="Olive">x1</font>,<font color="Olive">x2</font>,<font color="Olive">q</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">r1</font>,<font color="Olive">r2</font>,<font color="Olive">q</font> <a href="collsp.html#R1">is_collinear</a>  ) ) )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E5:92"/><b>then </b>
for <font color="Olive">p</font>, <font color="Olive">p1</font>, <font color="Olive">q</font>, <font color="Olive">q1</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>  ex <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> st <br/>( <font color="Olive">p</font>,<font color="Olive">p1</font>,<font color="Olive">r</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">q</font>,<font color="Olive">q1</font>,<font color="Olive">r</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:92"/>
(  <a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V'</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font> &amp;  <a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V'</font> is <a href="anproj_2.html#V7" target="_self">2-dimensional</a> )
 <b>by </b><i/>;<br/>


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