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<b>consider </b><font color="Maroon">V</font> being   <a href="rlvect_1.html#NM2">RealLinearSpace</a><b> such that </b><br/><a NAME="E2:35_1_1"/><i><font color="Green">E28</font></i>: 
 ex <font color="Olive">u</font>, <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <br/>for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a>  st <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> holds <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="funcsdom.html#T37">FUNCSDOM:37</a></i>;<br/>
<b>reconsider </b><font color="Maroon">X</font> =  <a href="analoaf.html#K2">OASpace</a> <font color="Maroon">V</font> as   <a href="analoaf.html#NM1">OAffinSpace</a> <b>by </b><i><a class="txt" href="papdesaf.html#E2:35_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="analoaf.html#T38">ANALOAF:38</a></i>;<br/>
<b>take </b>
<font color="Maroon">X</font>
;<br/>

<b>set </b><font color="Maroon">AS</font> =  <a href="diraf.html#K2">Lambda</a> <font color="Maroon">X</font>;<br/>
<a NAME="E4:35_1_1"/><i><font color="Green">E29</font></i>: 
 <a href="diraf.html#K2">Lambda</a> <font color="Maroon">X</font> <a href="papdesaf.html#NR1" target="_self">satisfies_PAP'</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E5:35_1_1"/><i><font color="Green">E38</font></i>: 
 <a href="diraf.html#K2">Lambda</a> <font color="Maroon">X</font> <a href="papdesaf.html#NR2" target="_self">satisfies_DES'</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E6:35_1_1"/><i><font color="Green">E43</font></i>: 
 <a href="diraf.html#K2">Lambda</a> <font color="Maroon">X</font> <a href="papdesaf.html#NR3" target="_self">satisfies_TDES'</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E7:35_1_1"/>
 <a href="diraf.html#K2">Lambda</a> <font color="Maroon">X</font> <a href="papdesaf.html#NR4" target="_self">satisfies_des'</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:35_1_1"/>
( <font color="Maroon">X</font> is <a href="papdesaf.html#V1" target="_self">Pappian</a> &amp; <font color="Maroon">X</font> is <a href="papdesaf.html#V2" target="_self">Desarguesian</a> &amp; <font color="Maroon">X</font> is <a href="papdesaf.html#V3" target="_self">Moufangian</a> &amp; <font color="Maroon">X</font> is <a href="papdesaf.html#V4" target="_self">translation</a> )
 <b>by </b><i><a class="txt" href="papdesaf.html#E4:35_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="papdesaf.html#T4" target="_self">Th4</a>, , , , , </i>;<br/>


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