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<span class="kw">let </span><font color="Maroon" title="c7">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c7">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> holds <br/> <a href="analmetr.html#K2" title="ANALMETR:func.2">AMSpace</a> (<font color="Maroon" title="c7">V</font>,<font color="Olive" title="b1">w</font>,<font color="Olive" title="b2">y</font>) is   <a href="analmetr.html#NM2" title="ANALMETR:NM.2">OrtAfPl</a></span><br/><span class="kw">let </span><font color="Maroon" title="c8">w</font>, <font color="Maroon" title="c9">y</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c7">V</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c8">w</font>,<font color="Maroon" title="c9">y</font> implies  <a href="analmetr.html#K2" title="ANALMETR:func.2">AMSpace</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c9">y</font>) is   <a href="analmetr.html#NM2" title="ANALMETR:NM.2">OrtAfPl</a> )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c10">POS</font> =  <a href="analmetr.html#K2" title="ANALMETR:func.2">AMSpace</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c9">y</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c11">X</font> =  <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K2" title="ANALMETR:func.2">AMSpace</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c9">y</font>)</span>)</span>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K2" title="ANALMETR:func.2">AMSpace</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c9">y</font>)</span>)</span> #);<br/>
<a NAME="E1:57"/><span class="lab"><font color="Green" title="E44">A1</font></span>: 
 <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K2" title="ANALMETR:func.2">AMSpace</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c9">y</font>)</span>)</span>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K2" title="ANALMETR:func.2">AMSpace</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c9">y</font>)</span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <span class="p1">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c7">V</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T20" target="_self" onmouseover="rs('analmetr/T20')" onmouseout="rh()">Th20</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:57"/><span class="lab"><font color="Green" title="E45">A2</font></span>: 
 <a href="analmetr.html#R1" title="ANALMETR:pred.1">Gen</a> <font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c9">y</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="analmetr.html#K2" title="ANALMETR:func.2">AMSpace</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c9">y</font>) is   <a href="analmetr.html#NM2" title="ANALMETR:NM.2">OrtAfPl</a></span><br/>

<a NAME="E3:57"/><span class="kw">then </span>
( (  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="xreal_0.html#NM1" title="XREAL_0: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="c8">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="c9">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="c7">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#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) ) &amp; (  for <font color="Olive" title="b1">w1</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c7">V</font>  ex <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <font color="Olive" title="b1">w1</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#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">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#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">y</font></span>)</span> ) )
 
;<br/>
<a NAME="E4:57"/><span class="kw">then </span>
 <a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c7">V</font> is   <a href="analoaf.html#NM2" title="ANALOAF:NM.2">OAffinPlane</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="analoaf.html#T28" onmouseover="rs('analoaf/T28')" onmouseout="rh()">ANALOAF:28</a></span>;<br/>
<a NAME="E5:57"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A3</font></span>: 
 <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K2" title="ANALMETR:func.2">AMSpace</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c9">y</font>)</span>)</span>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K2" title="ANALMETR:func.2">AMSpace</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c9">y</font>)</span>)</span> #) is   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="diraf.html#T45" onmouseover="rs('diraf/T45')" onmouseout="rh()">DIRAF:45</a></span>;<br/>
<a NAME="E6:57"/>
( (  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</font>, <font color="Olive" title="b5">p</font>, <font color="Olive" title="b6">q</font>, <font color="Olive" title="b7">r</font>, <font color="Olive" title="b8">s</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K2" title="ANALMETR:func.2">AMSpace</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c9">y</font>)</span>)</span> holds <br/> ( ( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> implies <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> ) &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b3">c</font> &amp; ( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font> implies ( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b4">d</font>,<font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> ) ) &amp; ( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b5">p</font>,<font color="Olive" title="b6">q</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">r</font>,<font color="Olive" title="b8">s</font> &amp;  not <font color="Olive" title="b5">p</font>,<font color="Olive" title="b6">q</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b7">r</font>,<font color="Olive" title="b8">s</font> implies <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> ) &amp; ( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b5">p</font>,<font color="Olive" title="b6">q</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b7">r</font>,<font color="Olive" title="b8">s</font> &amp;  not <font color="Olive" title="b5">p</font>,<font color="Olive" title="b6">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">r</font>,<font color="Olive" title="b8">s</font> implies <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> ) ) ) &amp; (  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K2" title="ANALMETR:func.2">AMSpace</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c9">y</font>)</span>)</span>  ex <font color="Olive" title="b4">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K2" title="ANALMETR:func.2">AMSpace</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c9">y</font>)</span>)</span> st <br/>( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">x</font> &amp; <font color="Olive" title="b3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">x</font> ) ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="analmetr.html#T23" target="_self" onmouseover="rs('analmetr/T23')" onmouseout="rh()">Th23</a>, <a class="ref" href="analmetr.html#T24" target="_self" onmouseover="rs('analmetr/T24')" onmouseout="rh()">Th24</a>, <a class="ref" href="analmetr.html#T25" target="_self" onmouseover="rs('analmetr/T25')" onmouseout="rh()">Th25</a>, <a class="ref" href="analmetr.html#T26" target="_self" onmouseover="rs('analmetr/T26')" onmouseout="rh()">Th26</a>, <a class="ref" href="analmetr.html#T27" target="_self" onmouseover="rs('analmetr/T27')" onmouseout="rh()">Th27</a>, <a class="ref" href="analmetr.html#T28" target="_self" onmouseover="rs('analmetr/T28')" onmouseout="rh()">Th28</a>, <a class="ref" href="analmetr.html#T30" target="_self" onmouseover="rs('analmetr/T30')" onmouseout="rh()">Th30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:57"/>
 <a href="analmetr.html#K2" title="ANALMETR:func.2">AMSpace</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c9">y</font>) is   <a href="analmetr.html#NM2" title="ANALMETR:NM.2">OrtAfPl</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:57"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="ref" href="analmetr.html#D9" target="_self" onmouseover="rs('analmetr/D9')" onmouseout="rh()">Def8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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