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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>;<br/><b>let </b><font color="Maroon" title="c2">u</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>;<br/>



<b>thus </b><a NAME="E1:32"/>
( <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> implies <font color="Maroon" title="c2">u</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_1"/>
<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>
 ;<br/>

<a NAME="E2:32_1"/><b>then </b>
<font color="Maroon" title="c2">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">V</font>
 
<b>by </b><i><a class="ref" href="anproj_1.html#D4" target="_self" title="ANPROJ_1:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E3:32_1"/>
<font color="Maroon" title="c2">u</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a> 
 <b>by </b><i><a class="ref" href="rlvect_1.html#D13" title="RLVECT_1:def.13">RLVECT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:32"/>
( <font color="Maroon" title="c2">u</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  implies <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> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_2"/>
<font color="Maroon" title="c2">u</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a> 
 ;<br/>

<a NAME="E2:32_2"/><b>then </b>
<font color="Maroon" title="c2">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">V</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D13" title="RLVECT_1:def.13">RLVECT_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E3:32_2"/>
<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>
 <b>by </b><i><a class="ref" href="anproj_1.html#D4" target="_self" title="ANPROJ_1:def.4">Def4</a></i>;<br/>


</div><b>end;</b></div>


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