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<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="E2:37_1_1"/><i><font color="Green" title="E33">A1</font></i>: 
 ex <font color="Olive" title="b1">w</font>, <font color="Olive" title="b2">y</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>
 <b>by </b><i><a class="ref" href="analmetr.html#T7" title="ANALMETR:th.7">ANALMETR:7</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c2">w</font>, <font color="Maroon" title="c3">y</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E4:37_1_1"/><i><font color="Green" title="E34">A2</font></i>: 
 <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>
 <b>by </b><i><a class="txt" href="euclmetr.html#E2:37_1_1"><i><font color="Green" title="E33">A1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">MS</font> =  <a href="analmetr.html#K2" title="ANALMETR:func.2">AMSpace</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font> as   <a href="analmetr.html#NM2" title="ANALMETR:NM.2">OrtAfPl</a> <b>by </b><i><a class="txt" href="euclmetr.html#E4:37_1_1"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="analmetr.html#T46" title="ANALMETR:th.46">ANALMETR:46</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">MS</font>
;<br/>

<a NAME="E6:37_1_1"/>
 <a href="conaffm.html#NR5" title="CONAFFM:NR.5">LIN_holds_in</a> <font color="Maroon" title="c4">MS</font>
 
<b>by </b><i><a class="txt" href="euclmetr.html#E4:37_1_1"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="euclmetr.html#T21" target="_self" title="EUCLMETR:th.21">Th21</a></i>;<br/>
<a NAME="E7:37_1_1"/><b>then </b><i><font color="Green" title="E35">A3</font></i>: 
 <a href="conaffm.html#NR3" title="CONAFFM:NR.3">3H_holds_in</a> <font color="Maroon" title="c4">MS</font>
 
<b>by </b><i><a class="ref" href="conaffm.html#T6" title="CONAFFM:th.6">CONAFFM:6</a></i>;<br/>
<a NAME="E8:37_1_1"/>
( ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">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="b1">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c2">w</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="b2">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c3">y</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="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) ) &amp; ( for <font color="Olive" title="b1">u</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  ex <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c2">w</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="b3">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c3">y</font></span>)</span> ) )
 
<b>by </b><i><a class="txt" href="euclmetr.html#E4:37_1_1"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="analmetr.html#D1" title="ANALMETR:def.1">ANALMETR:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">OAS</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="ref" href="analoaf.html#T38" title="ANALOAF:th.38">ANALOAF:38</a></i>;<br/>
<a NAME="E10:37_1_1"/><i><font color="Green" title="E36">A4</font></i>: 
 <a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c4">MS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Maroon" title="c5">OAS</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T30" title="ANALMETR:th.30">ANALMETR:30</a></i>;<br/>
<a NAME="E11:37_1_1"/><b>then </b><i><font color="Green" title="E37">A5</font></i>: 
 <a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c4">MS</font> is <a href="aff_2.html#V2" title="AFF_2:attr.2">Pappian</a>
 
<b>by </b><i><a class="ref" href="papdesaf.html#T19" title="PAPDESAF:th.19">PAPDESAF:19</a></i>;<br/>
<a NAME="E12:37_1_1"/><i><font color="Green" title="E38">A6</font></i>: 
 <a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c4">MS</font> is <a href="aff_2.html#V4" title="AFF_2:attr.4">Desarguesian</a>
 
<b>by </b><i><a class="txt" href="euclmetr.html#E10:37_1_1"><i><font color="Green" title="E36">A4</font></i></a>, <a class="ref" href="papdesaf.html#T20" title="PAPDESAF:th.20">PAPDESAF:20</a></i>;<br/>
<a NAME="E13:37_1_1"/><i><font color="Green" title="E39">A7</font></i>: 
 <a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c4">MS</font> is <a href="aff_2.html#V7" title="AFF_2:attr.7">Moufangian</a>
 
<b>by </b><i><a class="txt" href="euclmetr.html#E10:37_1_1"><i><font color="Green" title="E36">A4</font></i></a>, <a class="ref" href="papdesaf.html#T22" title="PAPDESAF:th.22">PAPDESAF:22</a></i>;<br/>
<a NAME="E14:37_1_1"/>
 <a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c4">MS</font> is <a href="aff_2.html#V11" title="AFF_2:attr.11">translational</a>
 
<b>by </b><i><a class="txt" href="euclmetr.html#E10:37_1_1"><i><font color="Green" title="E36">A4</font></i></a>, <a class="ref" href="papdesaf.html#T23" title="PAPDESAF:th.23">PAPDESAF:23</a></i>;<br/>
<b>hence </b><a NAME="E15:37_1_1"/>
( <font color="Maroon" title="c4">MS</font> is <a href="euclmetr.html#V1" title="EUCLMETR:attr.1">Euclidean</a> &amp; <font color="Maroon" title="c4">MS</font> is <a href="euclmetr.html#V2" title="EUCLMETR:attr.2">Pappian</a> &amp; <font color="Maroon" title="c4">MS</font> is <a href="euclmetr.html#V3" title="EUCLMETR:attr.3">Desarguesian</a> &amp; <font color="Maroon" title="c4">MS</font> is <a href="euclmetr.html#V7" title="EUCLMETR:attr.7">Homogeneous</a> &amp; <font color="Maroon" title="c4">MS</font> is <a href="euclmetr.html#V6" title="EUCLMETR:attr.6">translation</a> &amp; <font color="Maroon" title="c4">MS</font> is <a href="euclmetr.html#V4" title="EUCLMETR:attr.4">Fanoian</a> &amp; <font color="Maroon" title="c4">MS</font> is <a href="euclmetr.html#V5" title="EUCLMETR:attr.5">Moufangian</a> )
 <b>by </b><i><a class="txt" href="euclmetr.html#E4:37_1_1"><i><font color="Green" title="E34">A2</font></i></a>, <a class="txt" href="euclmetr.html#E7:37_1_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="txt" href="euclmetr.html#E10:37_1_1"><i><font color="Green" title="E36">A4</font></i></a>, <a class="txt" href="euclmetr.html#E11:37_1_1"><i><font color="Green" title="E37">A5</font></i></a>, <a class="txt" href="euclmetr.html#E12:37_1_1"><i><font color="Green" title="E38">A6</font></i></a>, <a class="txt" href="euclmetr.html#E13:37_1_1"><i><font color="Green" title="E39">A7</font></i></a>, <a class="ref" href="euclmetr.html#D2" target="_self" title="EUCLMETR:def.2">Def2</a>, <a class="ref" href="euclmetr.html#D3" target="_self" title="EUCLMETR:def.3">Def3</a>, <a class="ref" href="euclmetr.html#D4" target="_self" title="EUCLMETR:def.4">Def4</a>, <a class="ref" href="euclmetr.html#D5" target="_self" title="EUCLMETR:def.5">Def5</a>, <a class="ref" href="euclmetr.html#D6" target="_self" title="EUCLMETR:def.6">Def6</a>, <a class="ref" href="euclmetr.html#T8" target="_self" title="EUCLMETR:th.8">Th8</a>, <a class="ref" href="euclmetr.html#T23" target="_self" title="EUCLMETR:th.23">Th23</a></i>;<br/>


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