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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="Red">::  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="Red">::  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="E1:39"/><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="E2:39"/><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="Red">::  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="E3:39"/><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="#E2:39"><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="E4:39"/><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="#E2:39"><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="E5:39"/>
<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#T17" title="ANALOAF:th.17">ANALOAF:17</a></span>;<br/>
<a NAME="E6:39"/><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="E7:39"/><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#T14" target="_self" title="GEOMTRAP:th.14">Th14</a></span>;<br/>
<a NAME="E8:39"/><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="E9:39"/><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="#E2:39"><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="E10:39"/>
<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="39_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_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="E1:39_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="39_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:39_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="Red">::  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="E2:39_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#T18" title="ANALOAF:th.18">ANALOAF:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="39_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:39_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="Red">::  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="E2:39_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="E3:39_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="#E9:39"><span class="lab"><font color="Green" title="E42">A8</font></span></a>, <a class="txt" href="#E7:39"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="ref" href="analoaf.html#T20" title="ANALOAF:th.20">ANALOAF:20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E11:39"/><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="E12:39"/>
<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#T18" title="ANALOAF:th.18">ANALOAF:18</a></span>;<br/>
<a NAME="E13:39"/><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="E14:39"/><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#T14" target="_self" title="GEOMTRAP:th.14">Th14</a></span>;<br/>
<a NAME="E15:39"/><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="E16:39"/><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="#E9:39"><span class="lab"><font color="Green" title="E42">A8</font></span></a>, <a class="ref" href="analoaf.html#T21" title="ANALOAF:th.21">ANALOAF:21</a></span>;<br/>
<a NAME="E17:39"/>
<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="39_2"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_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="E1:39_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="39_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:39_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="Red">::  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="E2:39_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#T18" title="ANALOAF:th.18">ANALOAF:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="39_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:39_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="Red">::  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="E2:39_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="E3:39_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="#E16:39"><span class="lab"><font color="Green" title="E47">A13</font></span></a>, <a class="txt" href="#E14:39"><span class="lab"><font color="Green" title="E45">A11</font></span></a>, <a class="ref" href="analoaf.html#T20" title="ANALOAF:th.20">ANALOAF:20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E18:39"/><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="E19:39"/>
<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#T17" title="ANALOAF:th.17">ANALOAF:17</a></span>;<br/>
<a NAME="E20:39"/><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="E21:39"/>
<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#T18" title="ANALOAF:th.18">ANALOAF:18</a></span>;<br/>
<a NAME="E22:39"/><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="E23:39"/><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="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_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="E1:39_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="#E23:39"><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="39_3_1"><span class="kw">suppose </span></a><a NAME="E1:39_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="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:39_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="#E1:39"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E8:39"><span class="lab"><font color="Green" title="E41">A7</font></span></a>, <a class="txt" href="#E6:39"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="txt" href="#E22:39"><span class="lab"><font color="Green" title="E50">A16</font></span></a>, <a class="txt" href="#E18:39"><span class="lab"><font color="Green" title="E48">A14</font></span></a>, <a class="txt" href="geomtrap.html#E38"><span class="lab"><font color="Green" title="E33">Lm11</font></span></a></span>;<br/><a NAME="E3:39_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="#E1:39"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E1:39_3_1"><span class="lab"><font color="Green" title="E52">A18</font></span></a>, <a class="txt" href="geomtrap.html#E28"><span class="lab"><font color="Green" title="E25">Lm10</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:39_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E23:39"><span class="lab"><font color="Green" title="E51">A17</font></span></a>, <a class="txt" href="geomtrap.html#E26"><span class="lab"><font color="Green" title="E23">Lm8</font></span></a>, <a class="ref" href="geomtrap.html#T9" target="_self" title="GEOMTRAP:th.9">Th9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="39_3_2"><span class="kw">suppose </span></a><a NAME="E1:39_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="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:39_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="#E1:39"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E15:39"><span class="lab"><font color="Green" title="E46">A12</font></span></a>, <a class="txt" href="#E20:39"><span class="lab"><font color="Green" title="E49">A15</font></span></a>, <a class="txt" href="#E13:39"><span class="lab"><font color="Green" title="E44">A10</font></span></a>, <a class="txt" href="#E11:39"><span class="lab"><font color="Green" title="E43">A9</font></span></a>, <a class="txt" href="geomtrap.html#E38"><span class="lab"><font color="Green" title="E33">Lm11</font></span></a></span>;<br/><a NAME="E3:39_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="#E1:39"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="txt" href="#E1:39_3_2"><span class="lab"><font color="Green" title="E52">A19</font></span></a>, <a class="txt" href="geomtrap.html#E28"><span class="lab"><font color="Green" title="E25">Lm10</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:39_3_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E23:39"><span class="lab"><font color="Green" title="E51">A17</font></span></a>, <a class="txt" href="geomtrap.html#E26"><span class="lab"><font color="Green" title="E23">Lm8</font></span></a>, <a class="ref" href="geomtrap.html#T9" target="_self" title="GEOMTRAP:th.9">Th9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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>
