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

<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font>, <font color="Maroon" title="c4">u</font>, <font color="Maroon" title="c5">v</font>, <font color="Maroon" title="c6">w</font>, <font color="Maroon" title="c7">y</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>;<br/>













<b>assume </b><b>that </b><br/><a NAME="E1:11"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">q</font>
 <b>and </b><br/><a NAME="E2:11"/><i><font color="Green" title="E16">A2</font></i>: 
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font>
 <b>and </b><br/><a NAME="E3:11"/><i><font color="Green" title="E17">A3</font></i>: 
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c7">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" title="E18">A4</font></i>: 
( <font color="Maroon" title="c4">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">v</font> &amp; <font color="Maroon" title="c6">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">y</font> )
 ;<br/><b>then consider </b><font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c9">b</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:11_1"/><i><font color="Green" title="E19">A5</font></i>: 
( <font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c4">u</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">a</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c9">b</font> )
 <b>by </b><i><a class="txt" href="analoaf.html#E1:11"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="analoaf.html#E2:11"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="analoaf.html#T16" target="_self" title="ANALOAF:th.16">Th16</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">c</font>, <font color="Maroon" title="c11">d</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:11_1"/><i><font color="Green" title="E20">A6</font></i>: 
( <font color="Maroon" title="c10">c</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">d</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c6">w</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">c</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c11">d</font> )
 <b>by </b><i><a class="txt" href="analoaf.html#E1:11"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="analoaf.html#E3:11"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="analoaf.html#E1:11_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="analoaf.html#T16" target="_self" title="ANALOAF:th.16">Th16</a></i>;<br/><i><font color="Green" title="E21">A7</font></i>: <font color="Maroon" title="c3">q</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">p</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c4">u</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="analoaf.html#E3:11_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="analoaf.html#T13" target="_self" title="ANALOAF:th.13">Th13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c4">u</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
;<br/><i><font color="Green" title="E22">A8</font></i>: <font color="Maroon" title="c3">q</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">p</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">c</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">d</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c6">w</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="analoaf.html#E5:11_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="analoaf.html#T13" target="_self" title="ANALOAF:th.13">Th13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">c</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c11">d</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c6">w</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
;<br/><a NAME="E8:11_1"/>
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">a</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a>  &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">c</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a>  )
 <b>by </b><i><a class="txt" href="analoaf.html#E3:11_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="analoaf.html#E5:11_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="xreal_1.html#T124" title="XREAL_1:th.124">XREAL_1:124</a></i>;<br/><a NAME="E9:11_1"/><b>then </b>
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c9">b</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">c</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c11">d</font> )
 <b>by </b><i><a class="txt" href="analoaf.html#E3:11_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="analoaf.html#E5:11_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="xreal_1.html#T131" title="XREAL_1:th.131">XREAL_1:131</a></i>;<br/><b>hence </b><a NAME="E10:11_1"/>
<font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c7">y</font>
 <b>by </b><i><a class="txt" href="analoaf.html#E6:11_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="analoaf.html#E7:11_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="ref" href="analoaf.html#D1" target="_self" title="ANALOAF:def.1">Def1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:11"/>
<font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c7">y</font>
 <b>by </b><i><a class="ref" href="analoaf.html#D1" target="_self" title="ANALOAF:def.1">Def1</a></i>;<br/>


</div>
