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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">y</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>, <font color="Maroon">u</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>, <font color="Maroon">v</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="E1:27"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">y</font>,<font color="Maroon">u</font> <a href="geomtrap.html#R1">'||'</a> <font color="Maroon">y</font>,<font color="Maroon">v</font>
 <b>and </b><br/><a NAME="E2:27"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">u</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font> )
 ;<br/>

<a NAME="E3:27"/>
( <font color="Maroon">y</font>,<font color="Maroon">u</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">y</font>,<font color="Maroon">v</font> or <font color="Maroon">y</font>,<font color="Maroon">u</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">v</font>,<font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="papdesaf.html#T3" target="_self">Th3</a>, <a class="ref" href="geomtrap.html#D1">GEOMTRAP:def 1</a></i>;<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="E5:27"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</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">y</font></span>)</span>
 <b>and </b><br/><a NAME="E6:27"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i><a class="ref" href="analmetr.html#T18">ANALMETR:18</a></i>;<br/>
<b>take </b><font color="Maroon">r</font> = <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>;<br/>

<div><i><font color="Green">E59</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:27_1"/><b>then </b>
(  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> <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">y</font></span>)</span> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i>, , <a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>;<br/><a NAME="E3:27_1"/><b>then </b>
<font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T24">RLVECT_1:24</a></i>;<br/><b>hence </b><a NAME="E4:27_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#T35">RLVECT_1:35</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E60</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:27_2"/><b>then </b>
(  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i>, , <a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>;<br/><a NAME="E3:27_2"/><b>then </b>
<font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T24">RLVECT_1:24</a></i>;<br/><b>hence </b><a NAME="E4:27_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#T35">RLVECT_1:35</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:27"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">a</font> <a href="real_1.html#K2">"</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T203">XCMPLX_1:203</a></i>;<br/>
<a NAME="E10:27"/>
<font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</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">y</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3"><b>proof </b></a><div class="add">

<b>thus </b><font color="Maroon">r</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">y</font></span>)</span> = 
<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">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></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">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
1 <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>
<br/>.= 
<font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:27"/>
( <font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</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">y</font></span>)</span> &amp; <font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i>, , <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/>


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