<?xml version="1.0"?>
<div class="add">

<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="E1:52"/><i><font color="Green">E38</font></i>: 
 <a href="analmetr.html#R1" target="_self">Gen</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>
 ;<br/>

<b>set </b><font color="Maroon">POS</font> =  <a href="analmetr.html#K2" target="_self">AMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">w</font>,<font color="Maroon">y</font>;<br/>
<a NAME="E2:52"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">d</font>, <font color="Olive">p</font>, <font color="Olive">q</font>, <font color="Olive">r</font>, <font color="Olive">s</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K2" target="_self">AMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">w</font>,<font color="Maroon">y</font></span>)</span> holds <br/> ( ( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">a</font>,<font color="Olive">b</font> implies <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> ) &amp; <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">c</font>,<font color="Olive">c</font> &amp; ( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">c</font>,<font color="Olive">d</font> implies ( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">d</font>,<font color="Olive">c</font> &amp; <font color="Olive">c</font>,<font color="Olive">d</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">a</font>,<font color="Olive">b</font> ) ) &amp; ( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">p</font>,<font color="Olive">q</font> &amp; <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">r</font>,<font color="Olive">s</font> &amp; not <font color="Olive">p</font>,<font color="Olive">q</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">r</font>,<font color="Olive">s</font> implies <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> ) &amp; ( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">p</font>,<font color="Olive">q</font> &amp; <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">p</font>,<font color="Olive">s</font> implies <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">q</font>,<font color="Olive">s</font> ) )
 
<b>by </b><i><a class="ref" href="analmetr.html#D2" target="_self">Def2</a>, , , , , , <a class="txt" href="analmetr.html#E1"><i><font color="Green">Lemma37</font></i></a></i>;<br/>
<a NAME="E3:52"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K2" target="_self">AMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">w</font>,<font color="Maroon">y</font></span>)</span>  st <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> holds <br/> ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K2" target="_self">AMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">w</font>,<font color="Maroon">y</font></span>)</span> st <br/>( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">a</font>,<font color="Olive">x</font> &amp; <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">x</font>,<font color="Olive">c</font> )
 
<b>by </b><i><a class="ref" href="analmetr.html#D2" target="_self">Def2</a>, <a class="txt" href="analmetr.html#E1:52"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E4:52"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K2" target="_self">AMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">w</font>,<font color="Maroon">y</font></span>)</span>  ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K2" target="_self">AMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">w</font>,<font color="Maroon">y</font></span>)</span> st <br/>( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">c</font>,<font color="Olive">x</font> &amp; <font color="Olive">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">x</font> )
 
<b>by </b><i><a class="ref" href="analmetr.html#D2" target="_self">Def2</a>, </i>;<br/>
<b>set </b><font color="Maroon">X</font> =  <a href="analoaf.html#G1">AffinStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K2" target="_self">AMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">w</font>,<font color="Maroon">y</font></span>)</span>,the <a href="analoaf.html#U1">CONGR</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K2" target="_self">AMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">w</font>,<font color="Maroon">y</font></span>)</span> #);<br/>
<a NAME="E5:52"/>
 <a href="analoaf.html#G1">AffinStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K2" target="_self">AMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">w</font>,<font color="Maroon">y</font></span>)</span>,the <a href="analoaf.html#U1">CONGR</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K2" target="_self">AMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">w</font>,<font color="Maroon">y</font></span>)</span> #) <a href="hidden.html#R1">=</a>  <a href="analmetr.html#K3" target="_self">Af</a> <span class="p1">(<span class="default"><a href="analmetr.html#K2" target="_self">AMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">w</font>,<font color="Maroon">y</font></span>)</span>
 
;<br/>
<a NAME="E6:52"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="analoaf.html#G1">AffinStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K2" target="_self">AMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">w</font>,<font color="Maroon">y</font></span>)</span>,the <a href="analoaf.html#U1">CONGR</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K2" target="_self">AMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">w</font>,<font color="Maroon">y</font></span>)</span> #) <a href="hidden.html#R1">=</a>  <a href="diraf.html#K2">Lambda</a> <span class="p1">(<span class="default"><a href="analoaf.html#K2">OASpace</a> <font color="Maroon">V</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E7:52"/>
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 )
 
<b>by </b><i><a class="ref" href="analmetr.html#D2" target="_self">Def2</a>, </i>;<br/>
<a NAME="E8:52"/><b>then </b>
 <a href="analoaf.html#K2">OASpace</a> <font color="Maroon">V</font> is   <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="E9:52"/><b>then </b>
 <a href="analoaf.html#G1">AffinStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K2" target="_self">AMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">w</font>,<font color="Maroon">y</font></span>)</span>,the <a href="analoaf.html#U1">CONGR</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K2" target="_self">AMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">w</font>,<font color="Maroon">y</font></span>)</span> #) is   <a href="diraf.html#NM1">AffinSpace</a>
 
<b>by </b><i>, <a class="ref" href="diraf.html#T48">DIRAF:48</a></i>;<br/>
<b>hence </b><a NAME="E10:52"/>
 <a href="analmetr.html#K2" target="_self">AMSpace</a> <font color="Maroon">V</font>,<font color="Maroon">w</font>,<font color="Maroon">y</font> is   <a href="analmetr.html#NM1">OrtAfSp</a>
 <b>by </b><i><a class="txt" href="analmetr.html#E6"><i><font color="Green">Lemma44</font></i></a>, <a class="ref" href="analmetr.html#T5" target="_self">Th5</a>, <a class="txt" href="analmetr.html#E3:52"><i><font color="Green">E46</font></i></a>, <a class="ref" href="analmetr.html#D1" target="_self">Def1</a></i>;<br/>


</div>
