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<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">V</font> is  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> iff  ex <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="anproj_1.html#K1" title="ANPROJ_1:func.1">Proper_Vectors_of</a> <font color="Maroon" title="c1">V</font> )</span><br/>

<a NAME="E1:39"/><i><font color="Green" title="E30">A1</font></i>: 
( <font color="Maroon" title="c1">V</font> is  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> implies  ex <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="anproj_1.html#K1" title="ANPROJ_1:func.1">Proper_Vectors_of</a> <font color="Maroon" title="c1">V</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1"/>
<font color="Maroon" title="c1">V</font> is  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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="anproj_1.html#K1" title="ANPROJ_1:func.1">Proper_Vectors_of</a> <font color="Maroon" title="c1">V</font></span><br/>

<b>then consider </b><font color="Maroon" title="c2">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E3:39_1"/><i><font color="Green" title="E31">A2</font></i>: 
<font color="Maroon" title="c2">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="ref" href="anproj_1.html#D8" target="_self" title="ANPROJ_1:def.8">Def8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">u</font> = <font color="Maroon" title="c2">u</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<a NAME="E5:39_1"/><i><font color="Green" title="E32">A3</font></i>: 
( <font color="Maroon" title="c3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="anproj_1.html#K1" title="ANPROJ_1:func.1">Proper_Vectors_of</a> <font color="Maroon" title="c1">V</font> iff not <font color="Maroon" title="c3">u</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> )
 
<b>by </b><i><a class="ref" href="anproj_1.html#T26" target="_self" title="ANPROJ_1:th.26">Th26</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">u</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="anproj_1.html#K1" title="ANPROJ_1:func.1">Proper_Vectors_of</a> <font color="Maroon" title="c1">V</font></span><br/>

<b>thus </b><a NAME="E6:39_1"/>
<font color="Maroon" title="c3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="anproj_1.html#K1" title="ANPROJ_1:func.1">Proper_Vectors_of</a> <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="txt" href="#E3:39_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="#E5:39_1"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="struct_0.html#D15" title="STRUCT_0:def.15">STRUCT_0:def 15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:39"/>
(  ex <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="anproj_1.html#K1" title="ANPROJ_1:func.1">Proper_Vectors_of</a> <font color="Maroon" title="c1">V</font> implies <font color="Maroon" title="c1">V</font> is  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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c2">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><a NAME="E2:39_2"/><i><font color="Green" title="E31">A4</font></i>: 
<font color="Maroon" title="c2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="anproj_1.html#K1" title="ANPROJ_1:func.1">Proper_Vectors_of</a> <font color="Maroon" title="c1">V</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">V</font> is  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></span><br/>

<a NAME="E3:39_2"/>
<font color="Maroon" title="c2">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font>
 
<b>by </b><i><a class="txt" href="#E2:39_2"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="anproj_1.html#D4" target="_self" title="ANPROJ_1:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E4:39_2"/>
<font color="Maroon" title="c1">V</font> is  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>by </b><i><a class="ref" href="anproj_1.html#D8" target="_self" title="ANPROJ_1:def.8">Def8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:39"/>
( <font color="Maroon" title="c1">V</font> is  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> iff  ex <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="anproj_1.html#K1" title="ANPROJ_1:func.1">Proper_Vectors_of</a> <font color="Maroon" title="c1">V</font> )
 <b>by </b><i><a class="txt" href="#E1:39"><i><font color="Green" title="E30">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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