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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">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>assume </b><a NAME="E1:17"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u1</font>,<font color="Maroon">v1</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_1"/>
( <font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font> &amp; <font color="Maroon">u1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v1</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="E3:17_1"/><i><font color="Green">E46</font></i>: 
( 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> &amp; <font color="Maroon">a</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> <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">v1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="analoaf.html#D1">ANALOAF:def 1</a></i>;<br/><b>set </b><font color="Maroon">p</font> = <font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font>;<br/><b>set </b><font color="Maroon">q</font> = <font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</font>;<br/><a NAME="E4:17_1"/><i><font color="Green">E47</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T36">XREAL_1:36</a></i>;<br/><a NAME="E5:17_1"/><i><font color="Green">E67</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> <a href="real_1.html#K4">*</a> 2
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T131">XREAL_1:131</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K4">*</a> 2</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font></span>)</span></span>)</span> = 
<font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default">2 <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default">2 <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T48">RLVECT_1:48</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default">1 <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default">2 <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default">1 <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default">2 <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default">2 <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default">2 <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<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#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<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#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="rlvect_1.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default">1 <a href="rlvect_1.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<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#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default">1 <a href="rlvect_1.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<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#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<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#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u1</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<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#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T41">RLVECT_1:41</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font></span>)</span></span>)</span> <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#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u1</font></span>)</span> <a href="rlvect_1.html#K4">+</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><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<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">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</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>, <a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K3">+</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/><b>hence </b><a NAME="E7:17_1"/>
<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font>,<font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</font>
 <b>by </b><i>, , <a class="ref" href="analoaf.html#D1">ANALOAF:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:17"/>
<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">u1</font>,<font color="Maroon">v</font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">v1</font>
 <b>by </b><i>, <a class="ref" href="analoaf.html#T18">ANALOAF:18</a></i>;<br/>


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