<?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>; <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">w</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">v1</font>, <font color="Olive" title="b4">u</font>, <font color="Olive" title="b5">v2</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st  <a href="analmetr.html#R1" title="ANALMETR:pred.1">Gen</a> <font color="Olive" title="b1">w</font>,<font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b3">v1</font>,<font color="Olive" title="b4">u</font>,<font color="Olive" title="b4">u</font>,<font color="Olive" title="b5">v2</font> <a href="geomtrap.html#R2" title="GEOMTRAP:pred.2">are_DTr_wrt</a> <font color="Olive" title="b1">w</font>,<font color="Olive" title="b2">y</font> holds <br/>( <font color="Olive" title="b3">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">u</font> &amp; <font color="Olive" title="b4">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">v2</font> )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <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> &amp; <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> 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> ) )</span><br/>











<b>assume </b><b>that </b><br/><a NAME="E1:31"/><i><font color="Green" title="E29">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="E2:31"/><i><font color="Green" title="E30">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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> )</span><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="E3:31"/><i><font color="Green" title="E31">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="#E2:31"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="geomtrap.html#D3" target="_self" title="GEOMTRAP:def.3">Def3</a></i>;<br/>
<a NAME="E4:31"/>
( <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="31_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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> )</span><br/>

<a NAME="E2:31_1"/><b>then </b><i><font color="Green" title="E32">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="E3:31_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="#E3:31"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="geomtrap.html#T15" target="_self" title="GEOMTRAP:th.15">Th15</a></i>;<br/>
<a NAME="E4:31_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="E5:31_1"/><b>then </b><i><font color="Green" title="E33">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="E6:31_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="E7:31_1"/><b>then </b><i><font color="Green" title="E34">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="E8:31_1"/><i><font color="Green" title="E35">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="31_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1_1"/><i><font color="Green" title="E36">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:31_1_1"/><i><font color="Green" title="E37">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="31_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:31_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="E3:31_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:31_1_1"><i><font color="Green" title="E36">A8</font></i></a>, <a class="ref" href="geomtrap.html#T9" target="_self" title="GEOMTRAP:th.9">Th9</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>
<a NAME="E3:31_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="#E1:31"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="#E3:31"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="#E7:31_1"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="#E1:31_1_1"><i><font color="Green" title="E36">A8</font></i></a>, <a class="txt" href="geomtrap.html#E28"><i><font color="Green" title="E25">Lm9</font></i></a></i>;<br/>
<a NAME="E4:31_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="#E1:31"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="#E2:31_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="#E5:31_1"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="geomtrap.html#E28"><i><font color="Green" title="E25">Lm9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:31_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:31"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="#E2:31_1_1"><i><font color="Green" title="E37">A9</font></i></a>, <a class="txt" href="geomtrap.html#E26"><i><font color="Green" title="E23">Lm7</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>
<a NAME="E9:31_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="31_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1_2"/><i><font color="Green" title="E36">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:31_1_2"/><i><font color="Green" title="E37">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="31_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:31_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="E3:31_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:31_1_2"><i><font color="Green" title="E36">A10</font></i></a>, <a class="ref" href="geomtrap.html#T9" target="_self" title="GEOMTRAP:th.9">Th9</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>
<a NAME="E3:31_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="#E1:31"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="#E3:31"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="#E7:31_1"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="#E1:31_1_2"><i><font color="Green" title="E36">A10</font></i></a>, <a class="txt" href="geomtrap.html#E28"><i><font color="Green" title="E25">Lm9</font></i></a></i>;<br/>
<a NAME="E4:31_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="#E1:31"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="#E2:31_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="#E5:31_1"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="geomtrap.html#E28"><i><font color="Green" title="E25">Lm9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:31_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:31"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="#E2:31_1_2"><i><font color="Green" title="E37">A11</font></i></a>, <a class="txt" href="geomtrap.html#E26"><i><font color="Green" title="E23">Lm7</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="E10:31_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="#E8:31_1"><i><font color="Green" title="E35">A7</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="E5:31"/>
( <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="#E2:31"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="geomtrap.html#T19" target="_self" title="GEOMTRAP:th.19">Th19</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>
