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<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">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/><span class="kw">let </span><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:()"><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">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/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:31"/><span class="lab"><font color="Green" title="E29">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="E2:31"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">set </span><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/>
<span class="kw">set </span><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"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="geomtrap.html#D3" target="_self" title="GEOMTRAP:def.3">Def3</a></span>;<br/>
<a NAME="E4:31"/><span class="lab"><font color="Green" title="E32">A4</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="geomtrap.html#D3" target="_self" title="GEOMTRAP:def.3">Def3</a></span>;<br/>
<a NAME="E5:31"/><span class="lab"><font color="Green" title="E33">A5</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="geomtrap.html#D3" target="_self" title="GEOMTRAP:def.3">Def3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_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="E1:31_1"/>
( <font color="Maroon" title="c4">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v2</font> or <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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1"><span class="kw">suppose </span></a><a NAME="E1:31_1_1"/>
<font color="Maroon" title="c4">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v2</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">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/><div class="add"><span class="kw">hence </span><a NAME="E2:31_1_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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="geomtrap.html#T17" target="_self" title="GEOMTRAP:th.17">Th19</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="31_1_2"><span class="kw">suppose </span></a><a NAME="E1:31_1_2"/>
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><div class="add"><a NAME="E2:31_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A6</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="geomtrap.html#T7" target="_self" title="GEOMTRAP:th.7">Th9</a></span>;<br/><a NAME="E3:31_1_2"/>
<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>
 <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="E4:31_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A7</font></span>: 
<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>
 <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="E5:31_1_2"/><span class="lab"><font color="Green" title="E36">A8</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="ref" href="geomtrap.html#T13" target="_self" title="GEOMTRAP:th.13">Th15</a></span>;<br/><a NAME="E6:31_1_2"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="geomtrap.html#E15"><span class="lab"><font color="Green" title="E14">Lm1</font></span></a></span>;<br/><a NAME="E7:31_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A9</font></span>: 
<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>
 <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="E8:31_1_2"/><span class="lab"><font color="Green" title="E38">A10</font></span>: 
<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_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:31_1_2_1"/><span class="lab"><font color="Green" title="E39">A11</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:31_1_2_1"/><span class="lab"><font color="Green" title="E40">A12</font></span>: 
<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_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:31_1_2_1_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:31_1_2_1_1"/><span class="kw">then </span>
<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/>
<span class="kw">hence </span><a NAME="E3:31_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1_2_1"><span class="lab"><font color="Green" title="E39">A11</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>
<a NAME="E3:31_1_2_1"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="txt" href="#E4:31_1_2"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="txt" href="#E1:31_1_2_1"><span class="lab"><font color="Green" title="E39">A11</font></span></a>, <a class="txt" href="geomtrap.html#E26"><span class="lab"><font color="Green" title="E25">Lm10</font></span></a></span>;<br/>
<a NAME="E4:31_1_2_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E2:31_1_2"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="txt" href="#E7:31_1_2"><span class="lab"><font color="Green" title="E37">A9</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="E5:31_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E2:31_1_2_1"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="txt" href="geomtrap.html#E24"><span class="lab"><font color="Green" title="E23">Lm8</font></span></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><a NAME="E9:31_1_2"/>
<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>
 <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="E10:31_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A13</font></span>: 
<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>
 <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="E11:31_1_2"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31_1_2"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="txt" href="geomtrap.html#E15"><span class="lab"><font color="Green" title="E14">Lm1</font></span></a></span>;<br/><a NAME="E12:31_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A14</font></span>: 
<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>
 <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="E13:31_1_2"/>
<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_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:31_1_2_2"/><span class="lab"><font color="Green" title="E41">A15</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:31_1_2_2"/><span class="lab"><font color="Green" title="E42">A16</font></span>: 
<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_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:31_1_2_2_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:31_1_2_2_1"/><span class="kw">then </span>
<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/>
<span class="kw">hence </span><a NAME="E3:31_1_2_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1_2_2"><span class="lab"><font color="Green" title="E41">A15</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>
<a NAME="E3:31_1_2_2"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E10:31_1_2"><span class="lab"><font color="Green" title="E39">A13</font></span></a>, <a class="txt" href="#E1:31_1_2_2"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="txt" href="geomtrap.html#E26"><span class="lab"><font color="Green" title="E25">Lm10</font></span></a></span>;<br/>
<a NAME="E4:31_1_2_2"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E2:31_1_2"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="txt" href="#E12:31_1_2"><span class="lab"><font color="Green" title="E40">A14</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="E5:31_1_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E2:31_1_2_2"><span class="lab"><font color="Green" title="E42">A16</font></span></a>, <a class="txt" href="geomtrap.html#E24"><span class="lab"><font color="Green" title="E23">Lm8</font></span></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><span class="kw">hence </span><a NAME="E14:31_1_2"/>
( <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E8:31_1_2"><span class="lab"><font color="Green" title="E38">A10</font></span></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>
