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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">p</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<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>assume </b><b>that </b><br/><a NAME="E1:11"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font>
 <b>and </b><br/><a NAME="E2:11"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">u</font>,<font color="Maroon">v</font>
 <b>and </b><br/><a NAME="E3:11"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> )
 ;<br/><b>then consider </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:11_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> )
 <b>by </b><i>, , </i>;<br/><b>consider </b><font color="Maroon">c</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">d</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:11_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">w</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">d</font> )
 <b>by </b><i>, , , </i>;<br/><i><font color="Green">E46</font></i>: <font color="Maroon">q</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">p</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
;<br/><i><font color="Green">E47</font></i>: <font color="Maroon">q</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">p</font> = 
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">w</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">d</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
;<br/><a NAME="E8:11_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font> <a href="real_1.html#K2">"</a>  &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c</font> <a href="real_1.html#K2">"</a>  )
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/><a NAME="E9:11_1"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">d</font> )
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T131">XREAL_1:131</a></i>;<br/><b>hence </b><a NAME="E10:11_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>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:11"/>
<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/>;<br/>


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