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<b>let </b><font color="Maroon">MS</font> be   <a href="analmetr.html#NM2">OrtAfPl</a>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">w</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/>







<b>assume </b><a NAME="E1:39"/><i><font color="Green">E34</font></i>: 
(  <a href="analmetr.html#R1">Gen</a> <font color="Maroon">w</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">MS</font> <a href="hidden.html#R1">=</a>  <a href="analmetr.html#K2">AMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">w</font>,<font color="Maroon">y</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">MS'</font> =  <a href="analmetr.html#K2">AMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">w</font>,<font color="Maroon">y</font> as   <a href="analmetr.html#NM2">OrtAfPl</a> ;<br/>
<a NAME="E3:39"/>
 <a href="conaffm.html#NR5" target="_self">LIN_holds_in</a> <font color="Maroon">MS'</font>
 
<b>by </b><i><a class="ref" href="euclmetr.html#T1" target="_self">Th1</a>, </i>;<br/>
<a NAME="E4:39"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="conaffm.html#NR3" target="_self">3H_holds_in</a> <font color="Maroon">MS'</font>
 
<b>by </b><i><a class="ref" href="conaffm.html#T6">CONAFFM:6</a></i>;<br/>
<a NAME="E5:39"/>
( ( 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="Maroon">w</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="Maroon">y</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 ) ) &amp; ( for <font color="Olive">u</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</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="Maroon">y</font></span>)</span> ) )
 
<b>by </b><i><a class="ref" href="euclmetr.html#T1" target="_self">Th1</a>, <a class="ref" href="analmetr.html#D1">ANALMETR:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">OAS</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="ref" href="analoaf.html#T38">ANALOAF:38</a></i>;<br/>
<a NAME="E7:39"/><i><font color="Green">E44</font></i>: 
 <a href="analmetr.html#K3">Af</a> <font color="Maroon">MS'</font> <a href="hidden.html#R1">=</a>  <a href="diraf.html#K2">Lambda</a> <font color="Maroon">OAS</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T30">ANALMETR:30</a></i>;<br/>
<a NAME="E8:39"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="analmetr.html#K3">Af</a> <font color="Maroon">MS'</font> is <a href="aff_2.html#V2">Pappian</a>
 
<b>by </b><i><a class="ref" href="papdesaf.html#T19">PAPDESAF:19</a></i>;<br/>
<a NAME="E9:39"/><i><font color="Green">E51</font></i>: 
 <a href="analmetr.html#K3">Af</a> <font color="Maroon">MS'</font> is <a href="aff_2.html#V4">Desarguesian</a>
 
<b>by </b><i><a class="txt" href="euclmetr.html#E4:39"><i><font color="Green">E35</font></i></a>, <a class="ref" href="papdesaf.html#T20">PAPDESAF:20</a></i>;<br/>
<a NAME="E10:39"/><i><font color="Green">E64</font></i>: 
 <a href="analmetr.html#K3">Af</a> <font color="Maroon">MS'</font> is <a href="aff_2.html#V7">Moufangian</a>
 
<b>by </b><i><a class="txt" href="euclmetr.html#E4:39"><i><font color="Green">E35</font></i></a>, <a class="ref" href="papdesaf.html#T22">PAPDESAF:22</a></i>;<br/>
<a NAME="E11:39"/>
 <a href="analmetr.html#K3">Af</a> <font color="Maroon">MS'</font> is <a href="aff_2.html#V11">translational</a>
 
<b>by </b><i><a class="txt" href="euclmetr.html#E4:39"><i><font color="Green">E35</font></i></a>, <a class="ref" href="papdesaf.html#T23">PAPDESAF:23</a></i>;<br/>
<b>hence </b><a NAME="E12:39"/>
<font color="Maroon">MS</font> is  <a href="euclmetr.html#V1" target="_self">Euclidean</a> <a href="euclmetr.html#V2" target="_self">Pappian</a> <a href="euclmetr.html#V3" target="_self">Desarguesian</a> <a href="euclmetr.html#V4" target="_self">Fanoian</a> <a href="euclmetr.html#V5" target="_self">Moufangian</a> <a href="euclmetr.html#V6" target="_self">translation</a> <a href="euclmetr.html#V7" target="_self">Homogeneous</a> <a href="analmetr.html#NM2">OrtAfPl</a>
 <b>by </b><i><a class="ref" href="euclmetr.html#T1" target="_self">Th1</a>, <a class="txt" href="euclmetr.html#E1:39"><i><font color="Green">E34</font></i></a>, <a class="txt" href="euclmetr.html#E4:39"><i><font color="Green">E35</font></i></a>, , , , , , , , , , </i>;<br/>


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