<?xml version="1.0"?>
<div class="add">

<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">u1</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">v1</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/>













<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:68_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">u1</font>,<font color="Maroon">v1</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="geomtrap.html#K5" target="_self">DTrapezium</a> <font color="Maroon">V</font>,<font color="Maroon">w</font>,<font color="Maroon">y</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">u1'</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>, <font color="Maroon">v1'</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:68_1"/><i><font color="Green">E45</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">u1</font>,<font color="Maroon">v1</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">u1'</font>,<font color="Maroon">v1'</font></span><a href="domain_1.html#K1">]</a></span> )
 <b>and </b><br/><a NAME="E4:68_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">u'</font>,<font color="Maroon">v'</font>,<font color="Maroon">u1'</font>,<font color="Maroon">v1'</font> <a href="geomtrap.html#R2" target="_self">are_DTr_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="geomtrap.html#T12" target="_self">Th12</a></i>;<br/><a NAME="E5:68_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">u1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u1'</font> &amp; <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1'</font> )
 <b>by </b><i><a class="ref" href="geomtrap.html#T14" target="_self">Th14</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E6:68_1"/>
<font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">u1</font>,<font color="Maroon">v1</font> <a href="geomtrap.html#R2" target="_self">are_DTr_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="geomtrap.html#E17"><i><font color="Green">Lemma65</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:68"/>
( <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">u1</font>,<font color="Maroon">v1</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="geomtrap.html#K5" target="_self">DTrapezium</a> <font color="Maroon">V</font>,<font color="Maroon">w</font>,<font color="Maroon">y</font> iff <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">u1</font>,<font color="Maroon">v1</font> <a href="geomtrap.html#R2" target="_self">are_DTr_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="geomtrap.html#T12" target="_self">Th12</a></i>;<br/>


</div>
