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

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">w</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">v</font>, <font color="Olive" title="b4">u1</font>, <font color="Olive" title="b5">u2</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">v</font>,<font color="Olive" title="b4">u1</font>,<font color="Olive" title="b3">v</font>,<font color="Olive" title="b5">u2</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="b4">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">u2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">w</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c5">u1</font>, <font color="Maroon" title="c6">u2</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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">v</font>,<font color="Maroon" title="c5">u1</font>,<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c6">u2</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="c5">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u2</font> )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:37"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E2:37"/><span class="lab"><font color="Green" title="E36">A2</font></span>: 
<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u1</font>,<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c6">u2</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u2</font></span><br/>

<a NAME="E3:37"/><span class="lab"><font color="Green" title="E37">A3</font></span>: 
<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u1</font>,<font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u1</font>,<font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">u2</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="ref" href="geomtrap.html#D3" target="_self" title="GEOMTRAP:def.3">Def3</a></span>;<br/>
<a NAME="E4:37"/><span class="lab"><font color="Green" title="E38">A4</font></span>: 
<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c6">u2</font>,<font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u1</font>,<font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">u2</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="ref" href="geomtrap.html#D3" target="_self" title="GEOMTRAP:def.3">Def3</a></span>;<br/>
<a NAME="E5:37"/>
<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u1</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="analoaf.html#T8" title="ANALOAF:th.8">ANALOAF:8</a></span>;<br/>
<a NAME="E6:37"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A5</font></span>: 
<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u1</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="geomtrap.html#D1" target="_self" title="GEOMTRAP:def.1">Def1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">b</font> = <font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">c</font> = <font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">u2</font>;<br/>
<a NAME="E7:37"/><span class="lab"><font color="Green" title="E40">A6</font></span>: 
<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u1</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="geomtrap.html#T12" target="_self" title="GEOMTRAP:th.12">Th14</a></span>;<br/>
<a NAME="E8:37"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A7</font></span>: 
<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u1</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="geomtrap.html#D1" target="_self" title="GEOMTRAP:def.1">Def1</a></span>;<br/>
<a NAME="E9:37"/><span class="lab"><font color="Green" title="E42">A8</font></span>: 
<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u1</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c6">u2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="ref" href="geomtrap.html#D3" target="_self" title="GEOMTRAP:def.3">Def3</a></span>;<br/>
<a NAME="E10:37"/>
<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c6">u2</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:37_1_1"/>
( <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u1</font> or <font color="Maroon" title="c4">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:37_1_1_1"/>
<font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c6">u2</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u1</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:37_1_1_1"/>
<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c6">u2</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analoaf.html#T9" title="ANALOAF:th.9">ANALOAF:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:37_1_1_2"/>
<font color="Maroon" title="c4">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c6">u2</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u1</font></span><br/><div class="add"><a NAME="E2:37_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">u1</font>
 ;<br/><span class="kw">hence </span><a NAME="E3:37_1_1_2"/>
<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c6">u2</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:37"><span class="lab"><font color="Green" title="E42">A8</font></span></a>, <a class="txt" href="#E7:37"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="ref" href="analoaf.html#T11" title="ANALOAF:th.11">ANALOAF:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E11:37"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A9</font></span>: 
<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c6">u2</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="geomtrap.html#D1" target="_self" title="GEOMTRAP:def.1">Def1</a></span>;<br/>
<a NAME="E12:37"/>
<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c6">u2</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c4">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="analoaf.html#T9" title="ANALOAF:th.9">ANALOAF:9</a></span>;<br/>
<a NAME="E13:37"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A10</font></span>: 
<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c6">u2</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c4">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="geomtrap.html#D1" target="_self" title="GEOMTRAP:def.1">Def1</a></span>;<br/>
<a NAME="E14:37"/><span class="lab"><font color="Green" title="E45">A11</font></span>: 
<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c6">u2</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">u2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="geomtrap.html#T12" target="_self" title="GEOMTRAP:th.12">Th14</a></span>;<br/>
<a NAME="E15:37"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A12</font></span>: 
<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c6">u2</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">u2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="geomtrap.html#D1" target="_self" title="GEOMTRAP:def.1">Def1</a></span>;<br/>
<a NAME="E16:37"/><span class="lab"><font color="Green" title="E47">A13</font></span>: 
<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c6">u2</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:37"><span class="lab"><font color="Green" title="E42">A8</font></span></a>, <a class="ref" href="analoaf.html#T12" title="ANALOAF:th.12">ANALOAF:12</a></span>;<br/>
<a NAME="E17:37"/>
<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u1</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">u2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:37_2_1"/>
( <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">u2</font> or <font color="Maroon" title="c4">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">u2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:37_2_1_1"/>
<font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">u2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u1</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">u2</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:37_2_1_1"/>
<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u1</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">u2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analoaf.html#T9" title="ANALOAF:th.9">ANALOAF:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:37_2_1_2"/>
<font color="Maroon" title="c4">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">u2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u1</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">u2</font></span><br/><div class="add"><a NAME="E2:37_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">u2</font>
 ;<br/><span class="kw">hence </span><a NAME="E3:37_2_1_2"/>
<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u1</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">u2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:37"><span class="lab"><font color="Green" title="E47">A13</font></span></a>, <a class="txt" href="#E14:37"><span class="lab"><font color="Green" title="E45">A11</font></span></a>, <a class="ref" href="analoaf.html#T11" title="ANALOAF:th.11">ANALOAF:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E18:37"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A14</font></span>: 
<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u1</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">u2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="geomtrap.html#D1" target="_self" title="GEOMTRAP:def.1">Def1</a></span>;<br/>
<a NAME="E19:37"/>
<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c6">u2</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c6">u2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="analoaf.html#T8" title="ANALOAF:th.8">ANALOAF:8</a></span>;<br/>
<a NAME="E20:37"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A15</font></span>: 
<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c6">u2</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c6">u2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="geomtrap.html#D1" target="_self" title="GEOMTRAP:def.1">Def1</a></span>;<br/>
<a NAME="E21:37"/>
<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u1</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c4">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="analoaf.html#T9" title="ANALOAF:th.9">ANALOAF:9</a></span>;<br/>
<a NAME="E22:37"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A16</font></span>: 
<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u1</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c4">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="geomtrap.html#D1" target="_self" title="GEOMTRAP:def.1">Def1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E23:37"/><span class="lab"><font color="Green" title="E51">A17</font></span>: 
<font color="Maroon" title="c5">u1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">u2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_3"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:37_3"/>
( <font color="Maroon" title="c4">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">u1</font> or <font color="Maroon" title="c4">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">u2</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:37"><span class="lab"><font color="Green" title="E51">A17</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_3_1"><span class="kw">suppose </span></a><a NAME="E1:37_3_1"/><span class="lab"><font color="Green" title="E52">A18</font></span>: 
<font color="Maroon" title="c4">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">u1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:37_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u1</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u1</font>,<font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">u2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E8:37"><span class="lab"><font color="Green" title="E41">A7</font></span></a>, <a class="txt" href="#E6:37"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="txt" href="#E22:37"><span class="lab"><font color="Green" title="E50">A16</font></span></a>, <a class="txt" href="#E18:37"><span class="lab"><font color="Green" title="E48">A14</font></span></a>, <a class="txt" href="geomtrap.html#E36"><span class="lab"><font color="Green" title="E33">Lm11</font></span></a></span>;<br/><a NAME="E3:37_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u1</font>,<font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">u2</font>,<font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u1</font>,<font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">u2</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E3:37"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E1:37_3_1"><span class="lab"><font color="Green" title="E52">A18</font></span></a>, <a class="txt" href="geomtrap.html#E26"><span class="lab"><font color="Green" title="E25">Lm10</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:37_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E23:37"><span class="lab"><font color="Green" title="E51">A17</font></span></a>, <a class="txt" href="geomtrap.html#E24"><span class="lab"><font color="Green" title="E23">Lm8</font></span></a>, <a class="ref" href="geomtrap.html#T7" target="_self" title="GEOMTRAP:th.7">Th9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_3_2"><span class="kw">suppose </span></a><a NAME="E1:37_3_2"/><span class="lab"><font color="Green" title="E52">A19</font></span>: 
<font color="Maroon" title="c4">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">u2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:37_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c6">u2</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u1</font>,<font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">u2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E15:37"><span class="lab"><font color="Green" title="E46">A12</font></span></a>, <a class="txt" href="#E20:37"><span class="lab"><font color="Green" title="E49">A15</font></span></a>, <a class="txt" href="#E13:37"><span class="lab"><font color="Green" title="E44">A10</font></span></a>, <a class="txt" href="#E11:37"><span class="lab"><font color="Green" title="E43">A9</font></span></a>, <a class="txt" href="geomtrap.html#E36"><span class="lab"><font color="Green" title="E33">Lm11</font></span></a></span>;<br/><a NAME="E3:37_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u1</font>,<font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">u2</font>,<font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">u1</font>,<font color="Maroon" title="c4">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c6">u2</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E4:37"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="txt" href="#E1:37_3_2"><span class="lab"><font color="Green" title="E52">A19</font></span></a>, <a class="txt" href="geomtrap.html#E26"><span class="lab"><font color="Green" title="E25">Lm10</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:37_3_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E23:37"><span class="lab"><font color="Green" title="E51">A17</font></span></a>, <a class="txt" href="geomtrap.html#E24"><span class="lab"><font color="Green" title="E23">Lm8</font></span></a>, <a class="ref" href="geomtrap.html#T7" target="_self" title="GEOMTRAP:th.7">Th9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
