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<span class="kw">let </span><font color="Maroon" title="c1">MS</font> be   <a href="analmetr.html#NM2" title="ANALMETR:NM.2">OrtAfPl</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</font> being   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a><br/>  for <font color="Olive" title="b2">w</font>, <font color="Olive" title="b3">y</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Olive" title="b1">V</font>  st  <a href="analmetr.html#R1" title="ANALMETR:pred.1">Gen</a> <font color="Olive" title="b2">w</font>,<font color="Olive" title="b3">y</font> &amp; <font color="Maroon" title="c1">MS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="analmetr.html#K2" title="ANALMETR:func.2">AMSpace</a> (<font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">w</font>,<font color="Olive" title="b3">y</font>) holds <br/><font color="Maroon" title="c1">MS</font> is   <a href="euclmetr.html#V1" title="EUCLMETR:attr.1">Euclidean</a>   <a href="euclmetr.html#V2" title="EUCLMETR:attr.2">Pappian</a>   <a href="euclmetr.html#V3" title="EUCLMETR:attr.3">Desarguesian</a>   <a href="euclmetr.html#V4" title="EUCLMETR:attr.4">Fanoian</a>   <a href="euclmetr.html#V5" title="EUCLMETR:attr.5">Moufangian</a>   <a href="euclmetr.html#V6" title="EUCLMETR:attr.6">translation</a>   <a href="euclmetr.html#V7" title="EUCLMETR:attr.7">Homogeneous</a>  <a href="analmetr.html#NM2" title="ANALMETR:NM.2">OrtAfPl</a></span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <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="c2">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> &amp; <font color="Maroon" title="c1">MS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="analmetr.html#K2" title="ANALMETR:func.2">AMSpace</a> (<font color="Maroon" title="c2">V</font>,<font color="Olive" title="b1">w</font>,<font color="Olive" title="b2">y</font>) holds <br/><font color="Maroon" title="c1">MS</font> is   <a href="euclmetr.html#V1" title="EUCLMETR:attr.1">Euclidean</a>   <a href="euclmetr.html#V2" title="EUCLMETR:attr.2">Pappian</a>   <a href="euclmetr.html#V3" title="EUCLMETR:attr.3">Desarguesian</a>   <a href="euclmetr.html#V4" title="EUCLMETR:attr.4">Fanoian</a>   <a href="euclmetr.html#V5" title="EUCLMETR:attr.5">Moufangian</a>   <a href="euclmetr.html#V6" title="EUCLMETR:attr.6">translation</a>   <a href="euclmetr.html#V7" title="EUCLMETR:attr.7">Homogeneous</a>  <a href="analmetr.html#NM2" title="ANALMETR:NM.2">OrtAfPl</a></span><br/><span class="kw">let </span><font color="Maroon" title="c3">w</font>, <font color="Maroon" title="c4">y</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="analmetr.html#R1" title="ANALMETR:pred.1">Gen</a> <font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c1">MS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="analmetr.html#K2" title="ANALMETR:func.2">AMSpace</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c4">y</font>) implies <font color="Maroon" title="c1">MS</font> is   <a href="euclmetr.html#V1" title="EUCLMETR:attr.1">Euclidean</a>   <a href="euclmetr.html#V2" title="EUCLMETR:attr.2">Pappian</a>   <a href="euclmetr.html#V3" title="EUCLMETR:attr.3">Desarguesian</a>   <a href="euclmetr.html#V4" title="EUCLMETR:attr.4">Fanoian</a>   <a href="euclmetr.html#V5" title="EUCLMETR:attr.5">Moufangian</a>   <a href="euclmetr.html#V6" title="EUCLMETR:attr.6">translation</a>   <a href="euclmetr.html#V7" title="EUCLMETR:attr.7">Homogeneous</a>  <a href="analmetr.html#NM2" title="ANALMETR:NM.2">OrtAfPl</a> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:39"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
 <a href="analmetr.html#R1" title="ANALMETR:pred.1">Gen</a> <font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c4">y</font>
 <span class="kw">and </span><br/><a NAME="E2:39"/><span class="lab"><font color="Green" title="E34">A2</font></span>: 
<font color="Maroon" title="c1">MS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="analmetr.html#K2" title="ANALMETR:func.2">AMSpace</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c4">y</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">MS</font> is   <a href="euclmetr.html#V1" title="EUCLMETR:attr.1">Euclidean</a>   <a href="euclmetr.html#V2" title="EUCLMETR:attr.2">Pappian</a>   <a href="euclmetr.html#V3" title="EUCLMETR:attr.3">Desarguesian</a>   <a href="euclmetr.html#V4" title="EUCLMETR:attr.4">Fanoian</a>   <a href="euclmetr.html#V5" title="EUCLMETR:attr.5">Moufangian</a>   <a href="euclmetr.html#V6" title="EUCLMETR:attr.6">translation</a>   <a href="euclmetr.html#V7" title="EUCLMETR:attr.7">Homogeneous</a>  <a href="analmetr.html#NM2" title="ANALMETR:NM.2">OrtAfPl</a></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">MS9</font> =  <a href="analmetr.html#K2" title="ANALMETR:func.2">AMSpace</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c4">y</font>) as   <a href="analmetr.html#NM2" title="ANALMETR:NM.2">OrtAfPl</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39"><span class="lab"><font color="Green" title="E34">A2</font></span></a></span>;<br/>
<a NAME="E4:39"/><span class="lab"><font color="Green" title="E35">A3</font></span>: 
<font color="Maroon" title="c5">MS9</font> is  <a href="conaffm.html#V3" title="CONAFFM:attr.3">satisfying_3H</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="euclmetr.html#T20" target="_self" title="EUCLMETR:th.20">Th21</a>, <a class="ref" href="conaffm.html#T6" title="CONAFFM:th.6">CONAFFM:6</a></span>;<br/>
<a NAME="E5:39"/>
 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#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">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#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">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="c2">V</font> holds <br/>( <font color="Olive" title="b1">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="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="analmetr.html#D1" title="ANALMETR:def.1">ANALMETR:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">OAS</font> =  <a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c2">V</font> as   <a href="analoaf.html#NM1" title="ANALOAF:NM.1">OAffinSpace</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="analoaf.html#T26" title="ANALOAF:th.26">ANALOAF:26</a></span>;<br/>
<a NAME="E7:39"/><span class="lab"><font color="Green" title="E36">A4</font></span>: 
 <a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c5">MS9</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="c6">OAS</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T20" title="ANALMETR:th.20">ANALMETR:20</a></span>;<br/>
<a NAME="E8:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A5</font></span>: 
(  <a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c5">MS9</font> is  <a href="aff_2.html#V7" title="AFF_2:attr.7">Moufangian</a>  &amp;  <a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c5">MS9</font> is  <a href="aff_2.html#V11" title="AFF_2:attr.11">translational</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="papdesaf.html#T16" title="PAPDESAF:th.16">PAPDESAF:16</a>, <a class="ref" href="papdesaf.html#T17" title="PAPDESAF:th.17">PAPDESAF:17</a></span>;<br/>
<a NAME="E9:39"/>
(  <a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c5">MS9</font> is  <a href="aff_2.html#V2" title="AFF_2:attr.2">Pappian</a>  &amp;  <a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c5">MS9</font> is  <a href="aff_2.html#V4" title="AFF_2:attr.4">Desarguesian</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:39"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="ref" href="papdesaf.html#T13" title="PAPDESAF:th.13">PAPDESAF:13</a>, <a class="ref" href="papdesaf.html#T14" title="PAPDESAF:th.14">PAPDESAF:14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:39"/>
<font color="Maroon" title="c1">MS</font> is   <a href="euclmetr.html#V1" title="EUCLMETR:attr.1">Euclidean</a>   <a href="euclmetr.html#V2" title="EUCLMETR:attr.2">Pappian</a>   <a href="euclmetr.html#V3" title="EUCLMETR:attr.3">Desarguesian</a>   <a href="euclmetr.html#V4" title="EUCLMETR:attr.4">Fanoian</a>   <a href="euclmetr.html#V5" title="EUCLMETR:attr.5">Moufangian</a>   <a href="euclmetr.html#V6" title="EUCLMETR:attr.6">translation</a>   <a href="euclmetr.html#V7" title="EUCLMETR:attr.7">Homogeneous</a>  <a href="analmetr.html#NM2" title="ANALMETR:NM.2">OrtAfPl</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E2:39"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="txt" href="#E7:39"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="txt" href="#E8:39"><span class="lab"><font color="Green" title="E37">A5</font></span></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#T22" target="_self" title="EUCLMETR:th.22">Th23</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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