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<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  holds <br/>(  not <font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Olive" title="b1">u</font>,<font color="Olive" title="b3">w</font> or <font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Olive" title="b2">v</font>,<font color="Olive" title="b3">w</font> or <font color="Olive" title="b1">u</font>,<font color="Olive" title="b3">w</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Olive" title="b3">w</font>,<font color="Olive" title="b2">v</font> )</span><br/><b>let </b><font color="Maroon" title="c2">u</font>, <font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c4">w</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</font> or <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font> or <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c3">v</font> )</span><br/>







<b>assume </b><a NAME="E1:14"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font> or <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c3">v</font> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_1"/>
( <font color="Maroon" title="c2">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">v</font> &amp; <font color="Maroon" title="c2">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">w</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font> or <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c3">v</font> )</span><br/><b>then consider </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:14_1"/><i><font color="Green" title="E16">A2</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">u</font></span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">a</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">b</font> )
 <b>by </b><i><a class="txt" href="analoaf.html#E1:14"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="analoaf.html#T16" target="_self" title="ANALOAF:th.16">Th16</a></i>;<br/><font color="Maroon" title="c3">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">w</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">u</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="c2">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">w</font></span>)</span>
<b>by </b><i><a class="ref" href="analoaf.html#T4" target="_self" title="ANALOAF:th.4">Th4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">u</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T47" title="RLVECT_1:th.47">RLVECT_1:47</a></i>
;<br/><b>then </b><i><font color="Green" title="E17">A3</font></i>: <font color="Maroon" title="c6">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">w</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">u</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="analoaf.html#E3:14_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="rlvect_1.html#T48" title="RLVECT_1:th.48">RLVECT_1:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">u</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T49" title="RLVECT_1:th.49">RLVECT_1:49</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">v</font></span>)</span>
<b>by </b><i><a class="ref" href="analoaf.html#T11" target="_self" title="ANALOAF:th.11">Th11</a></i>
;<br/><font color="Maroon" title="c4">w</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">u</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="c2">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">v</font></span>)</span>
<b>by </b><i><a class="ref" href="analoaf.html#T4" target="_self" title="ANALOAF:th.4">Th4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">u</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T47" title="RLVECT_1:th.47">RLVECT_1:47</a></i>
;<br/><b>then </b><i><font color="Green" title="E18">A4</font></i>: <font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">v</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">u</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="analoaf.html#E3:14_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="rlvect_1.html#T48" title="RLVECT_1:th.48">RLVECT_1:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">u</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T49" title="RLVECT_1:th.49">RLVECT_1:49</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">w</font></span>)</span>
<b>by </b><i><a class="ref" href="analoaf.html#T11" target="_self" title="ANALOAF:th.11">Th11</a></i>
;<br/><div><i><font color="Green" title="E19">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_1_5"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font> or <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c3">v</font> )</span><br/><a NAME="E2:14_1_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">w</font>
 <b>by </b><i><a class="txt" href="analoaf.html#E3:14_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="rlvect_1.html#T50" title="RLVECT_1:th.50">RLVECT_1:50</a></i>;<br/><a NAME="E3:14_1_5"/><b>then </b>
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T21" title="RLVECT_1:th.21">RLVECT_1:21</a></i>;<br/><b>hence </b><a NAME="E4:14_1_5"/>
( <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font> or <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c3">v</font> )
 <b>by </b><i><a class="ref" href="analoaf.html#D1" target="_self" title="ANALOAF:def.1">Def1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_1_6"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font> or <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c3">v</font> )</span><br/><a NAME="E2:14_1_6"/><b>then </b>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">a</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">b</font> or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">a</font> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T57" title="XREAL_1:th.57">XREAL_1:57</a></i>;<br/><a NAME="E3:14_1_6"/><b>then </b>
( <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">u</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c3">v</font> or <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c2">u</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font> )
 <b>by </b><i><a class="txt" href="analoaf.html#E3:14_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="analoaf.html#E5:14_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="analoaf.html#E7:14_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="analoaf.html#D1" target="_self" title="ANALOAF:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E4:14_1_6"/>
( <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font> or <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c3">v</font> )
 <b>by </b><i><a class="ref" href="analoaf.html#T21" target="_self" title="ANALOAF:th.21">Th21</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E10:14_1"/>
( <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font> or <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c3">v</font> )
 <b>by </b><i><a class="txt" href="analoaf.html#E8:14_1"><i><font color="Green" title="E19">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:14"/>
( <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font> or <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c3">v</font> )
 <b>by </b><i><a class="ref" href="analoaf.html#D1" target="_self" title="ANALOAF:def.1">Def1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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