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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>;<br/><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">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><a NAME="E1:52"/><i><font color="Green" title="E32">A1</font></i>: 
 <a href="analmetr.html#R1" title="ANALMETR:pred.1">Gen</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>
 ;<br/>

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





<div><i><font color="Green" title="E33">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_1"/><i><font color="Green" title="E34">A3</font></i>: 
<font color="Maroon" title="c4">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font>
 ;<br/><b>take </b><font color="Maroon" title="c7">u1</font> = <font color="Maroon" title="c6">w</font> <a href="analort.html#K1" title="ANALORT:func.1">+</a> <font color="Maroon" title="c2">x</font>;<br/><a NAME="E2:52_1"/>
 <a href="analort.html#K2" title="ANALORT:func.2">Ortm</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c6">w</font>, <a href="analort.html#K2" title="ANALORT:func.2">Ortm</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c7">u1</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>by </b><i><a class="txt" href="analort.html#E1:52_1"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="analoaf.html#T18" title="ANALOAF:th.18">ANALOAF:18</a></i>;<br/><a NAME="E3:52_1"/><b>then </b><i><font color="Green" title="E35">A4</font></i>: 
<font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c7">u1</font>,<font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="ref" href="analort.html#D4" target="_self" title="ANALORT:def.4">Def4</a></i>;<br/><a NAME="E4:52_1"/>
<font color="Maroon" title="c6">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">u1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:52_1_1"/>
<font color="Maroon" title="c6">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">u1</font>
 ;<br/>

<a NAME="E2:52_1_1"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T22" title="RLVECT_1:th.22">RLVECT_1:22</a></i>;<br/>
<b>hence </b><a NAME="E3:52_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="analort.html#E1:52"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="analort.html#E4"><i><font color="Green" title="E4">Lm4</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:52_1"/>
 ex <font color="Olive" title="b1">u1</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c6">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">u1</font> &amp; <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">w</font>,<font color="Olive" title="b1">u1</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> )
 <b>by </b><i><a class="txt" href="analort.html#E1:52"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="analort.html#E3:52_1"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="analort.html#T19" target="_self" title="ANALORT:th.19">Th19</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_2"/><i><font color="Green" title="E34">A5</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>
 ;<br/><b>consider </b><font color="Maroon" title="c7">u2</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E3:52_2"/><i><font color="Green" title="E35">A6</font></i>: 
 <a href="analort.html#K2" title="ANALORT:func.2">Ortm</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c7">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">u</font>
 <b>by </b><i><a class="txt" href="analort.html#E1:52"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="analort.html#T8" target="_self" title="ANALORT:th.8">Th8</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">v2</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E5:52_2"/><i><font color="Green" title="E36">A7</font></i>: 
 <a href="analort.html#K2" title="ANALORT:func.2">Ortm</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c8">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font>
 <b>by </b><i><a class="txt" href="analort.html#E1:52"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="analort.html#T8" target="_self" title="ANALORT:th.8">Th8</a></i>;<br/><b>take </b><font color="Maroon" title="c9">u1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c8">v2</font> <a href="analort.html#K1" title="ANALORT:func.1">+</a> <font color="Maroon" title="c6">w</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c7">u2</font>;<br/><a NAME="E6:52_2"/>
<font color="Maroon" title="c7">u2</font>,<font color="Maroon" title="c8">v2</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c9">u1</font>
 <b>by </b><i><a class="ref" href="analoaf.html#T25" title="ANALOAF:th.25">ANALOAF:25</a></i>;<br/><a NAME="E7:52_2"/><b>then </b>
<font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c9">u1</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c7">u2</font>,<font color="Maroon" title="c8">v2</font>
 <b>by </b><i><a class="ref" href="analoaf.html#T21" title="ANALOAF:th.21">ANALOAF:21</a></i>;<br/><a NAME="E8:52_2"/><b>then </b>
 <a href="analort.html#K2" title="ANALORT:func.2">Ortm</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c6">w</font>, <a href="analort.html#K2" title="ANALORT:func.2">Ortm</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c9">u1</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a>  <a href="analort.html#K2" title="ANALORT:func.2">Ortm</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c7">u2</font>, <a href="analort.html#K2" title="ANALORT:func.2">Ortm</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c8">v2</font>
 <b>by </b><i><a class="txt" href="analort.html#E1:52"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="analort.html#T17" target="_self" title="ANALORT:th.17">Th17</a></i>;<br/><a NAME="E9:52_2"/><b>then </b><i><font color="Green" title="E37">A8</font></i>: 
<font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c9">u1</font>,<font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="txt" href="analort.html#E3:52_2"><i><font color="Green" title="E35">A6</font></i></a>, <a class="txt" href="analort.html#E5:52_2"><i><font color="Green" title="E36">A7</font></i></a>, <a class="ref" href="analort.html#D4" target="_self" title="ANALORT:def.4">Def4</a></i>;<br/><a NAME="E10:52_2"/>
<font color="Maroon" title="c6">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">u1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:52_2_1"/>
<font color="Maroon" title="c6">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">u1</font>
 ;<br/>

<a NAME="E2:52_2_1"/><b>then </b>
<font color="Maroon" title="c6">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">w</font> <a href="analort.html#K1" title="ANALORT:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">v2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c7">u2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>;<br/>
<a NAME="E3:52_2_1"/><b>then </b>
<font color="Maroon" title="c8">v2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c7">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T22" title="RLVECT_1:th.22">RLVECT_1:22</a></i>;<br/>
<b>hence </b><a NAME="E4:52_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="analort.html#E1:52_2"><i><font color="Green" title="E34">A5</font></i></a>, <a class="txt" href="analort.html#E3:52_2"><i><font color="Green" title="E35">A6</font></i></a>, <a class="txt" href="analort.html#E5:52_2"><i><font color="Green" title="E36">A7</font></i></a>, <a class="ref" href="rlvect_1.html#T35" title="RLVECT_1:th.35">RLVECT_1:35</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E11:52_2"/>
 ex <font color="Olive" title="b1">u1</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c6">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">u1</font> &amp; <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">w</font>,<font color="Olive" title="b1">u1</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> )
 <b>by </b><i><a class="txt" href="analort.html#E1:52"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="analort.html#E9:52_2"><i><font color="Green" title="E37">A8</font></i></a>, <a class="ref" href="analort.html#T19" target="_self" title="ANALORT:th.19">Th19</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:52"/>
 ex <font color="Olive" title="b1">u1</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c6">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">u1</font> &amp; <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">w</font>,<font color="Olive" title="b1">u1</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> )
 <b>by </b><i><a class="txt" href="analort.html#E2:52"><i><font color="Green" title="E33">A2</font></i></a></i>;<br/>


</div>
