<?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">w</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">v1</font>, <font color="Maroon" title="c5">u</font>, <font color="Maroon" title="c6">v2</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="E1:30"/><i><font color="Green" title="E28">A1</font></i>: 
 <a href="analmetr.html#R1" title="ANALMETR:pred.1">Gen</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>
 <b>and </b><br/><a NAME="E2:30"/><i><font color="Green" title="E29">A2</font></i>: 
<font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">v2</font> <a href="geomtrap.html#R2" title="GEOMTRAP:pred.2">are_DTr_wrt</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c4">v1</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u</font>;<br/>
<b>set </b><font color="Maroon" title="c8">b</font> = <font color="Maroon" title="c5">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">v2</font>;<br/>
<a NAME="E3:30"/><i><font color="Green" title="E30">A3</font></i>: 
( <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">u</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">v2</font> &amp; <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c4">v1</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c5">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">v2</font> <a href="analmetr.html#R3" title="ANALMETR:pred.3">are_Ort_wrt</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font> &amp; <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">v2</font>,<font color="Maroon" title="c4">v1</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c5">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">v2</font> <a href="analmetr.html#R3" title="ANALMETR:pred.3">are_Ort_wrt</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font> )
 
<b>by </b><i><a class="txt" href="geomtrap.html#E2:30"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="geomtrap.html#D3" target="_self" title="GEOMTRAP:def.3">Def3</a></i>;<br/>
<a NAME="E4:30"/>
( <font color="Maroon" title="c4">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">v2</font> implies ( <font color="Maroon" title="c4">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">u</font> &amp; <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v2</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1"/>
<font color="Maroon" title="c4">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">v2</font>
 ;<br/>

<a NAME="E2:30_1"/><b>then </b><i><font color="Green" title="E31">A4</font></i>: 
<font color="Maroon" title="c4">v1</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">v2</font>
 
<b>by </b><i><a class="ref" href="geomtrap.html#T9" target="_self" title="GEOMTRAP:th.9">Th9</a></i>;<br/>
<a NAME="E3:30_1"/>
<font color="Maroon" title="c4">v1</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c5">u</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c5">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">v2</font>
 
<b>by </b><i><a class="txt" href="geomtrap.html#E3:30"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="geomtrap.html#T15" target="_self" title="GEOMTRAP:th.15">Th15</a></i>;<br/>
<a NAME="E4:30_1"/><b>then </b>
( <font color="Maroon" title="c4">v1</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c5">u</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">v1</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c5">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">v2</font> &amp; <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c5">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">v2</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">v1</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c5">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">v2</font> )
 
<b>by </b><i><a class="txt" href="geomtrap.html#E17"><i><font color="Green" title="E14">Lm1</font></i></a></i>;<br/>
<a NAME="E5:30_1"/><b>then </b><i><font color="Green" title="E32">A5</font></i>: 
( <font color="Maroon" title="c4">v1</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c5">u</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c4">v1</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c5">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">v2</font> &amp; <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c5">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">v2</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c4">v1</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c5">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">v2</font> )
 
<b>by </b><i><a class="ref" href="geomtrap.html#D1" target="_self" title="GEOMTRAP:def.1">Def1</a></i>;<br/>
<a NAME="E6:30_1"/>
( <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">v2</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c5">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">v2</font> &amp; <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">u</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">v1</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c5">u</font> )
 
<b>by </b><i><a class="ref" href="geomtrap.html#T14" target="_self" title="GEOMTRAP:th.14">Th14</a></i>;<br/>
<a NAME="E7:30_1"/><b>then </b><i><font color="Green" title="E33">A6</font></i>: 
( <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">u</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c4">v1</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c5">u</font> &amp; <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">v2</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c5">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">v2</font> )
 
<b>by </b><i><a class="ref" href="geomtrap.html#D1" target="_self" title="GEOMTRAP:def.1">Def1</a></i>;<br/>
<a NAME="E8:30_1"/><i><font color="Green" title="E34">A7</font></i>: 
<font color="Maroon" title="c4">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">u</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1_1"/><i><font color="Green" title="E35">A8</font></i>: 
<font color="Maroon" title="c4">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">u</font>
 ;<br/>

<a NAME="E2:30_1_1"/><i><font color="Green" title="E36">A9</font></i>: 
<font color="Maroon" title="c5">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">v1</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1_1_1"/>
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v1</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u</font>
 ;<br/>

<a NAME="E2:30_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">v1</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u</font>
 
;<br/>
<b>hence </b><a NAME="E3:30_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="geomtrap.html#E1:30_1_1"><i><font color="Green" title="E35">A8</font></i></a>, <a class="ref" href="geomtrap.html#T9" target="_self" title="GEOMTRAP:th.9">Th9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:30_1_1"/>
<font color="Maroon" title="c4">v1</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c4">v1</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c5">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">v2</font> <a href="analmetr.html#R3" title="ANALMETR:pred.3">are_Ort_wrt</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>
 
<b>by </b><i><a class="txt" href="geomtrap.html#E1:30"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="geomtrap.html#E3:30"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="geomtrap.html#E7:30_1"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="geomtrap.html#E1:30_1_1"><i><font color="Green" title="E35">A8</font></i></a>, <a class="txt" href="geomtrap.html#E27"><i><font color="Green" title="E24">Lm9</font></i></a></i>;<br/>
<a NAME="E4:30_1_1"/><b>then </b>
<font color="Maroon" title="c4">v1</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c4">v1</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c5">u</font> <a href="analmetr.html#R3" title="ANALMETR:pred.3">are_Ort_wrt</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>
 
<b>by </b><i><a class="txt" href="geomtrap.html#E1:30"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="geomtrap.html#E2:30_1"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="geomtrap.html#E5:30_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="geomtrap.html#E27"><i><font color="Green" title="E24">Lm9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:30_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="geomtrap.html#E1:30"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="geomtrap.html#E2:30_1_1"><i><font color="Green" title="E36">A9</font></i></a>, <a class="txt" href="geomtrap.html#E25"><i><font color="Green" title="E22">Lm7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:30_1"/>
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1_2"/><i><font color="Green" title="E35">A10</font></i>: 
<font color="Maroon" title="c5">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">v2</font>
 ;<br/>

<a NAME="E2:30_1_2"/><i><font color="Green" title="E36">A11</font></i>: 
<font color="Maroon" title="c5">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">v2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1_2_1"/>
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">v2</font>
 ;<br/>

<a NAME="E2:30_1_2_1"/><b>then </b>
<font color="Maroon" title="c5">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u</font>
 
;<br/>
<b>hence </b><a NAME="E3:30_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="geomtrap.html#E1:30_1_2"><i><font color="Green" title="E35">A10</font></i></a>, <a class="ref" href="geomtrap.html#T9" target="_self" title="GEOMTRAP:th.9">Th9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:30_1_2"/>
<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c5">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">v2</font>,<font color="Maroon" title="c4">v1</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c5">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">v2</font> <a href="analmetr.html#R3" title="ANALMETR:pred.3">are_Ort_wrt</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>
 
<b>by </b><i><a class="txt" href="geomtrap.html#E1:30"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="geomtrap.html#E3:30"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="geomtrap.html#E7:30_1"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="geomtrap.html#E1:30_1_2"><i><font color="Green" title="E35">A10</font></i></a>, <a class="txt" href="geomtrap.html#E27"><i><font color="Green" title="E24">Lm9</font></i></a></i>;<br/>
<a NAME="E4:30_1_2"/><b>then </b>
<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c5">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">v2</font>,<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c5">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">v2</font> <a href="analmetr.html#R3" title="ANALMETR:pred.3">are_Ort_wrt</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>
 
<b>by </b><i><a class="txt" href="geomtrap.html#E1:30"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="geomtrap.html#E2:30_1"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="geomtrap.html#E5:30_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="geomtrap.html#E27"><i><font color="Green" title="E24">Lm9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:30_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="geomtrap.html#E1:30"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="geomtrap.html#E2:30_1_2"><i><font color="Green" title="E36">A11</font></i></a>, <a class="txt" href="geomtrap.html#E25"><i><font color="Green" title="E22">Lm7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:30_1"/>
( <font color="Maroon" title="c4">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">u</font> &amp; <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v2</font> )
 <b>by </b><i><a class="txt" href="geomtrap.html#E8:30_1"><i><font color="Green" title="E34">A7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:30"/>
( <font color="Maroon" title="c4">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">u</font> &amp; <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v2</font> )
 <b>by </b><i><a class="txt" href="geomtrap.html#E2:30"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="geomtrap.html#T19" target="_self" title="GEOMTRAP:th.19">Th19</a></i>;<br/>


</div>
