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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>assume </b><a NAME="E1:100"/><i><font color="Green">E58</font></i>: 
 ex <font color="Olive">u</font>, <font color="Olive">v</font>, <font color="Olive">u1</font>, <font color="Olive">v1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> st <br/>( ( for <font color="Olive">w</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">a1</font>, <font color="Olive">b1</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">w</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">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Olive">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">u1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">b1</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v1</font></span>)</span> ) &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">a1</font>, <font color="Olive">b1</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">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Olive">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">u1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">b1</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v1</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">a1</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b1</font> <a href="hidden.html#R1">=</a> 0 ) ) )
 ;<br/>

<b>then consider </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">u1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">v1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:100"/>
for <font color="Olive">w</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">a1</font>, <font color="Olive">b1</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">w</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">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="p3">(<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="p2">(<span class="default"><font color="Olive">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">b1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span>
 <b>and </b><br/><a NAME="E4:100"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">a1</font>, <font color="Olive">b1</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">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="p3">(<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="p2">(<span class="default"><font color="Olive">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">b1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</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">a1</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b1</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>
<b>consider </b><font color="Maroon">CS1</font> being   <a href="anproj_2.html#NM1">CollProjectiveSpace</a><b> such that </b><br/><a NAME="E6:100"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">CS1</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E7:100"/><i><font color="Green">E61</font></i>: 
not <font color="Maroon">CS1</font> is <a href="anproj_2.html#V7" target="_self">2-dimensional</a>
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">CS2</font> being   <a href="anproj_2.html#NM1">CollProjectiveSpace</a><b> such that </b><br/><a NAME="E9:100"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">CS2</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E10:100"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">CS2</font> is <a href="anproj_2.html#V8" target="_self">at_most-3-dimensional</a>
 <b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E11:100"/>
 ex <font color="Olive">CS</font> being  <a href="anproj_2.html#NM1">CollProjectiveSpace</a> st <br/>( <font color="Olive">CS</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font> &amp; not <font color="Olive">CS</font> is <a href="anproj_2.html#V7" target="_self">2-dimensional</a> &amp; <font color="Olive">CS</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#E7"><i><font color="Green">Lemma92</font></i></a>, <a class="txt" href="anproj_2.html#E8"><i><font color="Green">Lemma93</font></i></a>, <a class="txt" href="anproj_2.html#E9"><i><font color="Green">Lemma94</font></i></a>, <a class="txt" href="anproj_2.html#E10"><i><font color="Green">Lemma95</font></i></a></i>;<br/>


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