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

<b>consider </b><font color="Maroon" title="c1">V</font> being   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a><b> such that </b><br/><a NAME="E2:33_1_1"/><i><font color="Green" title="E27">A1</font></i>: 
 ex <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> st <br/>for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <span class="p1">(<span class="default"><font color="Olive" title="b3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> holds <br/>( <font color="Olive" title="b3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="ref" href="funcsdom.html#T37" title="FUNCSDOM:th.37">FUNCSDOM:37</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c2">X</font> =  <a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font> as   <a href="analoaf.html#NM1" title="ANALOAF:NM.1">OAffinSpace</a> <b>by </b><i><a class="txt" href="papdesaf.html#E2:33_1_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="analoaf.html#T38" title="ANALOAF:th.38">ANALOAF:38</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c2">X</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">X</font> is <a href="papdesaf.html#V1" title="PAPDESAF:attr.1">Pappian</a> &amp; <font color="Maroon" title="c2">X</font> is <a href="papdesaf.html#V2" title="PAPDESAF:attr.2">Desarguesian</a> &amp; <font color="Maroon" title="c2">X</font> is <a href="papdesaf.html#V3" title="PAPDESAF:attr.3">Moufangian</a> &amp; <font color="Maroon" title="c2">X</font> is <a href="papdesaf.html#V4" title="PAPDESAF:attr.4">translation</a> )</span><br/>

<b>set </b><font color="Maroon" title="c3">AS</font> =  <a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Maroon" title="c2">X</font>;<br/>
<a NAME="E4:33_1_1"/><i><font color="Green" title="E28">A2</font></i>: 
 <a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Maroon" title="c2">X</font> <a href="papdesaf.html#NR1" title="PAPDESAF:NR.1">satisfies_PAP'</a> 
 
<b>by </b><i><a class="ref" href="papdesaf.html#T19" target="_self" title="PAPDESAF:th.19">Th19</a></i>;<br/>
<a NAME="E5:33_1_1"/><i><font color="Green" title="E29">A3</font></i>: 
 <a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Maroon" title="c2">X</font> <a href="papdesaf.html#NR2" title="PAPDESAF:NR.2">satisfies_DES'</a> 
 
<b>by </b><i><a class="ref" href="papdesaf.html#T14" target="_self" title="PAPDESAF:th.14">Th14</a>, <a class="ref" href="papdesaf.html#T17" target="_self" title="PAPDESAF:th.17">Th17</a></i>;<br/>
<a NAME="E6:33_1_1"/><i><font color="Green" title="E30">A4</font></i>: 
 <a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Maroon" title="c2">X</font> <a href="papdesaf.html#NR3" title="PAPDESAF:NR.3">satisfies_TDES'</a> 
 
<b>by </b><i><a class="ref" href="papdesaf.html#T22" target="_self" title="PAPDESAF:th.22">Th22</a></i>;<br/>
<a NAME="E7:33_1_1"/>
 <a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Maroon" title="c2">X</font> <a href="papdesaf.html#NR4" title="PAPDESAF:NR.4">satisfies_des'</a> 
 
<b>by </b><i><a class="ref" href="papdesaf.html#T23" target="_self" title="PAPDESAF:th.23">Th23</a></i>;<br/>
<b>hence </b><a NAME="E8:33_1_1"/>
( <font color="Maroon" title="c2">X</font> is <a href="papdesaf.html#V1" title="PAPDESAF:attr.1">Pappian</a> &amp; <font color="Maroon" title="c2">X</font> is <a href="papdesaf.html#V2" title="PAPDESAF:attr.2">Desarguesian</a> &amp; <font color="Maroon" title="c2">X</font> is <a href="papdesaf.html#V3" title="PAPDESAF:attr.3">Moufangian</a> &amp; <font color="Maroon" title="c2">X</font> is <a href="papdesaf.html#V4" title="PAPDESAF:attr.4">translation</a> )
 <b>by </b><i><a class="txt" href="papdesaf.html#E4:33_1_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="papdesaf.html#E5:33_1_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="papdesaf.html#E6:33_1_1"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="papdesaf.html#D11" target="_self" title="PAPDESAF:def.11">Def11</a>, <a class="ref" href="papdesaf.html#D12" target="_self" title="PAPDESAF:def.12">Def12</a>, <a class="ref" href="papdesaf.html#D13" target="_self" title="PAPDESAF:def.13">Def13</a>, <a class="ref" href="papdesaf.html#D14" target="_self" title="PAPDESAF:def.14">Def14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
