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

<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">OAS</font> be   <a href="analoaf.html#NM1" title="ANALOAF:NM.1">OAffinSpace</a>;<br/>



<b>assume </b><a NAME="E1:25"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c2">OAS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font>
 ;<br/>

<a NAME="E2:25"/>
for <font color="Olive" title="b1">o</font>, <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</font>, <font color="Olive" title="b5">a1</font>, <font color="Olive" title="b6">b1</font>, <font color="Olive" title="b7">c1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">OAS</font>  st <font color="Olive" title="b2">a</font>,<font color="Olive" title="b1">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b5">a1</font> &amp; <font color="Olive" title="b3">b</font>,<font color="Olive" title="b1">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b6">b1</font> &amp; <font color="Olive" title="b4">c</font>,<font color="Olive" title="b1">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b7">c1</font> &amp; not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font> &amp; not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">a</font>,<font color="Olive" title="b4">c</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">b1</font>,<font color="Olive" title="b5">a1</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">c1</font>,<font color="Olive" title="b5">a1</font> holds <br/><font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">c1</font>,<font color="Olive" title="b6">b1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">o</font>, <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font>, <font color="Maroon" title="c7">a1</font>, <font color="Maroon" title="c8">b1</font>, <font color="Maroon" title="c9">c1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">OAS</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:25_1"/><i><font color="Green" title="E20">A2</font></i>: 
( <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c7">a1</font> &amp; <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c3">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c8">b1</font> &amp; <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c3">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c9">c1</font> )
 <b>and </b><br/><a NAME="E2:25_1"/><i><font color="Green" title="E21">A3</font></i>: 
( not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> &amp; not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font> )
 <b>and </b><br/><a NAME="E3:25_1"/><i><font color="Green" title="E22">A4</font></i>: 
( <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b1</font>,<font color="Maroon" title="c7">a1</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c7">a1</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c3">o</font>, <font color="Maroon" title="c11">u</font> = <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c12">v</font> = <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c13">w</font> = <font color="Maroon" title="c6">c</font>, <font color="Maroon" title="c14">u1</font> = <font color="Maroon" title="c7">a1</font>, <font color="Maroon" title="c15">v1</font> = <font color="Maroon" title="c8">b1</font>, <font color="Maroon" title="c16">w1</font> = <font color="Maroon" title="c9">c1</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="papdesaf.html#E1:25"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="papdesaf.html#T4" target="_self" title="PAPDESAF:th.4">Th4</a></i>;<br/>
<a NAME="E5:25_1"/><i><font color="Green" title="E23">A5</font></i>: 
( <font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">c</font> &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">c</font> )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E2:25_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></i>;<br/>
<div><i><font color="Green" title="E24">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_1_1"/><i><font color="Green" title="E25">A7</font></i>: 
<font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a1</font>
 ;<br/><a NAME="E2:25_1_1"/><i><font color="Green" title="E26">A8</font></i>: 
<font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1_1_1"/><i><font color="Green" title="E27">A9</font></i>: 
<font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b1</font>
 ;<br/>

<a NAME="E2:25_1_1_1"/>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c8">b1</font>,<font color="Maroon" title="c3">o</font>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E3:25_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="papdesaf.html#E1:25_1_1"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="diraf.html#D4" title="DIRAF:def.4">DIRAF:def 4</a></i>;<br/>
<a NAME="E3:25_1_1_1"/><b>then </b><i><font color="Green" title="E28">A10</font></i>: 
<font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c8">b1</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font>
 
<b>by </b><i><a class="ref" href="diraf.html#T27" title="DIRAF:th.27">DIRAF:27</a></i>;<br/>
<a NAME="E4:25_1_1_1"/>
<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c3">o</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c8">b1</font>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E1:25_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="diraf.html#D4" title="DIRAF:def.4">DIRAF:def 4</a></i>;<br/>
<a NAME="E5:25_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c8">b1</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">b</font>
 
<b>by </b><i><a class="ref" href="diraf.html#T27" title="DIRAF:th.27">DIRAF:27</a></i>;<br/>
<a NAME="E6:25_1_1_1"/><b>then </b>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c8">b1</font>,<font color="Maroon" title="c5">b</font>
 
<b>by </b><i><a class="ref" href="diraf.html#D5" title="DIRAF:def.5">DIRAF:def 5</a></i>;<br/>
<a NAME="E7:25_1_1_1"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c8">b1</font>,<font color="Maroon" title="c4">a</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c8">b1</font>,<font color="Maroon" title="c5">b</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c8">b1</font>,<font color="Maroon" title="c3">o</font> )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E1:25_1_1_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="papdesaf.html#E3:25_1_1_1"><i><font color="Green" title="E28">A10</font></i></a>, <a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a>, <a class="ref" href="diraf.html#T39" title="DIRAF:th.39">DIRAF:39</a></i>;<br/>
<b>hence </b><a NAME="E8:25_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="papdesaf.html#E2:25_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="papdesaf.html#E1:25_1_1_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:25_1_1"/>
<font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">c1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1_1_2"/><i><font color="Green" title="E27">A11</font></i>: 
<font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c1</font>
 ;<br/>

<a NAME="E2:25_1_1_2"/>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c3">o</font>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E3:25_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="papdesaf.html#E1:25_1_1"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="diraf.html#D4" title="DIRAF:def.4">DIRAF:def 4</a></i>;<br/>
<a NAME="E3:25_1_1_2"/><b>then </b><i><font color="Green" title="E28">A12</font></i>: 
<font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c9">c1</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">a</font>
 
<b>by </b><i><a class="ref" href="diraf.html#T27" title="DIRAF:th.27">DIRAF:27</a></i>;<br/>
<a NAME="E4:25_1_1_2"/>
<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c3">o</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c9">c1</font>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E1:25_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="diraf.html#D4" title="DIRAF:def.4">DIRAF:def 4</a></i>;<br/>
<a NAME="E5:25_1_1_2"/><b>then </b>
<font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c9">c1</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c6">c</font>
 
<b>by </b><i><a class="ref" href="diraf.html#T27" title="DIRAF:th.27">DIRAF:27</a></i>;<br/>
<a NAME="E6:25_1_1_2"/><b>then </b>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c6">c</font>
 
<b>by </b><i><a class="ref" href="diraf.html#D5" title="DIRAF:def.5">DIRAF:def 5</a></i>;<br/>
<a NAME="E7:25_1_1_2"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c4">a</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c6">c</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c3">o</font> )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E1:25_1_1_2"><i><font color="Green" title="E27">A11</font></i></a>, <a class="txt" href="papdesaf.html#E3:25_1_1_2"><i><font color="Green" title="E28">A12</font></i></a>, <a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a>, <a class="ref" href="diraf.html#T39" title="DIRAF:th.39">DIRAF:39</a></i>;<br/>
<b>hence </b><a NAME="E8:25_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="papdesaf.html#E2:25_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="papdesaf.html#E1:25_1_1_2"><i><font color="Green" title="E27">A11</font></i></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:25_1_1"/>
<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c8">b1</font>
 <b>by </b><i><a class="txt" href="papdesaf.html#E2:25_1_1"><i><font color="Green" title="E26">A8</font></i></a>, <a class="ref" href="diraf.html#T7" title="DIRAF:th.7">DIRAF:7</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_1_2"/><i><font color="Green" title="E25">A13</font></i>: 
<font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a1</font>
 ;<br/><a NAME="E2:25_1_2"/>
<font color="Maroon" title="c11">u</font>,<font color="Maroon" title="c10">y</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c14">u1</font>
 <b>by </b><i><a class="txt" href="papdesaf.html#E1:25"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="papdesaf.html#E1:25_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="geomtrap.html#T2" title="GEOMTRAP:th.2">GEOMTRAP:2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c17">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:25_1_2"/><i><font color="Green" title="E26">A14</font></i>: 
<font color="Maroon" title="c10">y</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">r</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">u1</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c10">y</font></span>)</span>
 <b>and </b><br/><a NAME="E5:25_1_2"/><i><font color="Green" title="E27">A15</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c17">r</font>
 <b>by </b><i><a class="txt" href="papdesaf.html#E5:25_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="txt" href="papdesaf.html#E1:25_1_2"><i><font color="Green" title="E25">A13</font></i></a>, <a class="txt" href="papdesaf.html#E32"><i><font color="Green" title="E17">Lm1</font></i></a></i>;<br/><a NAME="E6:25_1_2"/>
( <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b1</font>,<font color="Maroon" title="c3">o</font> &amp; <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c3">o</font> )
 <b>by </b><i><a class="txt" href="papdesaf.html#E1:25_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/><a NAME="E7:25_1_2"/><b>then </b>
( <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c12">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c15">v1</font>,<font color="Maroon" title="c10">y</font> &amp; <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c13">w</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c16">w1</font>,<font color="Maroon" title="c10">y</font> )
 <b>by </b><i><a class="txt" href="papdesaf.html#E1:25"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="geomtrap.html#T2" title="GEOMTRAP:th.2">GEOMTRAP:2</a></i>;<br/><a NAME="E8:25_1_2"/><b>then </b><i><font color="Green" title="E28">A16</font></i>: 
( <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c12">v</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c15">v1</font> &amp; <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c13">w</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c16">w1</font> )
 <b>by </b><i><a class="ref" href="geomtrap.html#D1" title="GEOMTRAP:def.1">GEOMTRAP:def 1</a></i>;<br/><a NAME="E9:25_1_2"/>
( <font color="Maroon" title="c11">u</font>,<font color="Maroon" title="c12">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c15">v1</font>,<font color="Maroon" title="c14">u1</font> &amp; <font color="Maroon" title="c11">u</font>,<font color="Maroon" title="c13">w</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c16">w1</font>,<font color="Maroon" title="c14">u1</font> )
 <b>by </b><i><a class="txt" href="papdesaf.html#E1:25"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="papdesaf.html#E3:25_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="geomtrap.html#T2" title="GEOMTRAP:th.2">GEOMTRAP:2</a></i>;<br/><a NAME="E10:25_1_2"/><b>then </b><i><font color="Green" title="E29">A17</font></i>: 
( <font color="Maroon" title="c11">u</font>,<font color="Maroon" title="c12">v</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c14">u1</font>,<font color="Maroon" title="c15">v1</font> &amp; <font color="Maroon" title="c11">u</font>,<font color="Maroon" title="c13">w</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c14">u1</font>,<font color="Maroon" title="c16">w1</font> )
 <b>by </b><i><a class="ref" href="geomtrap.html#D1" title="GEOMTRAP:def.1">GEOMTRAP:def 1</a></i>;<br/><a NAME="E11:25_1_2"/>
( not <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c11">u</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c12">v</font> &amp; not <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c11">u</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c13">w</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1_2_1"/>
( <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c11">u</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c12">v</font> or <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c11">u</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c13">w</font> )
 ;<br/>

<a NAME="E2:25_1_2_1"/><b>then </b>
( <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c11">u</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c12">v</font> or <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c11">u</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c12">v</font>,<font color="Maroon" title="c10">y</font> or <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c11">u</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c13">w</font> or <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c11">u</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c13">w</font>,<font color="Maroon" title="c10">y</font> )
 
<b>by </b><i><a class="ref" href="geomtrap.html#D1" title="GEOMTRAP:def.1">GEOMTRAP:def 1</a></i>;<br/>
<a NAME="E3:25_1_2_1"/><b>then </b>
( <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">b</font> or <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c3">o</font> or <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c6">c</font> or <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c3">o</font> )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E1:25"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="geomtrap.html#T2" title="GEOMTRAP:th.2">GEOMTRAP:2</a></i>;<br/>
<a NAME="E4:25_1_2_1"/><b>then </b>
( <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">b</font> or <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c6">c</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#D4" title="DIRAF:def.4">DIRAF:def 4</a></i>;<br/>
<b>hence </b><a NAME="E5:25_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="papdesaf.html#E2:25_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="diraf.html#D5" title="DIRAF:def.5">DIRAF:def 5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:25_1_2"/><b>then </b>
( <font color="Maroon" title="c15">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c17">r</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">v</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c11">u</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c16">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c17">r</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">w</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c11">u</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="papdesaf.html#E4:25_1_2"><i><font color="Green" title="E26">A14</font></i></a>, <a class="txt" href="papdesaf.html#E5:25_1_2"><i><font color="Green" title="E27">A15</font></i></a>, <a class="txt" href="papdesaf.html#E8:25_1_2"><i><font color="Green" title="E28">A16</font></i></a>, <a class="txt" href="papdesaf.html#E10:25_1_2"><i><font color="Green" title="E29">A17</font></i></a>, <a class="ref" href="papdesaf.html#T15" target="_self" title="PAPDESAF:th.15">Th15</a></i>;<br/><b>then </b><i><font color="Green" title="E30">A18</font></i>: 1 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">w1</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c15">v1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c17">r</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">w</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c11">u</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c17">r</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">v</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c11">u</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c17">r</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">w</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c11">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c14">u1</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c14">u1</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c17">r</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">v</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c11">u</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T41" title="RLVECT_1:th.41">RLVECT_1:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c17">r</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">w</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c11">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c17">r</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">v</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c11">u</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlsub_2.html#T78" title="RLSUB_2:th.78">RLSUB_2:78</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c17">r</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">w</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">v</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c11">u</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T48" title="RLVECT_1:th.48">RLVECT_1:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c17">r</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">w</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">v</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">u</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T41" title="RLVECT_1:th.41">RLVECT_1:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c17">r</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">w</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c12">v</font></span>)</span>
<b>by </b><i><a class="ref" href="rlsub_2.html#T78" title="RLSUB_2:th.78">RLSUB_2:78</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">w</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c12">v</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T224" title="XCMPLX_1:th.224">XCMPLX_1:224</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">w</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c12">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T38" title="RLVECT_1:th.38">RLVECT_1:38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c13">w</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T47" title="RLVECT_1:th.47">RLVECT_1:47</a></i>
;<br/><a NAME="E14:25_1_2"/>
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c17">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a>  )
 <b>by </b><i><a class="txt" href="papdesaf.html#E5:25_1_2"><i><font color="Green" title="E27">A15</font></i></a>, <a class="ref" href="xreal_1.html#T124" title="XREAL_1:th.124">XREAL_1:124</a></i>;<br/><a NAME="E15:25_1_2"/><b>then </b>
<font color="Maroon" title="c13">w</font>,<font color="Maroon" title="c12">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c15">v1</font>,<font color="Maroon" title="c16">w1</font>
 <b>by </b><i><a class="txt" href="papdesaf.html#E13:25_1_2"><i><font color="Green" title="E30">A18</font></i></a>, <a class="ref" href="analoaf.html#D1" title="ANALOAF:def.1">ANALOAF:def 1</a></i>;<br/><a NAME="E16:25_1_2"/><b>then </b>
<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b1</font>,<font color="Maroon" title="c9">c1</font>
 <b>by </b><i><a class="txt" href="papdesaf.html#E1:25"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="geomtrap.html#T2" title="GEOMTRAP:th.2">GEOMTRAP:2</a></i>;<br/><b>hence </b><a NAME="E17:25_1_2"/>
<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c8">b1</font>
 <b>by </b><i><a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:25_1"/>
<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c8">b1</font>
 <b>by </b><i><a class="txt" href="papdesaf.html#E6:25_1"><i><font color="Green" title="E24">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:25"/>
<font color="Maroon" title="c2">OAS</font> <a href="papdesaf.html#NR7" title="PAPDESAF:NR.7">satisfies_DES_1</a> 
 <b>by </b><i><a class="ref" href="papdesaf.html#D16" target="_self" title="PAPDESAF:def.16">Def16</a></i>;<br/>


</div>
