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<b>let </b><font color="Maroon">V</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">u</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">w</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/>









<b>thus </b><a NAME="E1:27"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">w</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="analoaf.html#K1" target="_self">DirPar</a> <font color="Maroon">V</font> implies <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">w</font>,<font color="Maroon">y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:27_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">w</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="analoaf.html#K1" target="_self">DirPar</a> <font color="Maroon">V</font>
 ;<br/>

<b>then consider </b><font color="Maroon">u'</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>, <font color="Maroon">v'</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>, <font color="Maroon">w'</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>, <font color="Maroon">y'</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:27_1"/><i><font color="Green">E32</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">u'</font>,<font color="Maroon">v'</font></span><a href="domain_1.html#K1">]</a></span> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">w</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">w'</font>,<font color="Maroon">y'</font></span><a href="domain_1.html#K1">]</a></span> )
 <b>and </b><br/><a NAME="E4:27_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">u'</font>,<font color="Maroon">v'</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">w'</font>,<font color="Maroon">y'</font>
 <b>by </b><i><a class="txt" href="analoaf.html#E2"><i><font color="Green">Lemma27</font></i></a></i>;<br/>
<a NAME="E5:27_1"/>
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u'</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v'</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w'</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font> )
 
<b>by </b><i><a class="ref" href="analoaf.html#T4" target="_self">Th4</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E6:27_1"/>
<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="analoaf.html#T6" target="_self">Th6</a></i>;<br/>


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

<b>thus </b><a NAME="E2:27"/>
( <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">w</font>,<font color="Maroon">y</font> implies <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">w</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="analoaf.html#K1" target="_self">DirPar</a> <font color="Maroon">V</font> )
 <b>by </b><i><a class="txt" href="analoaf.html#E2"><i><font color="Green">Lemma27</font></i></a></i>;<br/>


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