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

<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>;<br/>

<b>given </b><font color="Maroon" title="c2">u</font>, <font color="Maroon" title="c3">v</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><a NAME="E1:30"/><i><font color="Green" title="E30">A1</font></i>: 
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#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">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 )
 ;<br/>

<a NAME="E2:30"/><i><font color="Green" title="E31">A2</font></i>: 
<font color="Maroon" title="c2">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">v</font>
 
<b>by </b><i><a class="txt" href="analoaf.html#E1:30"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="analoaf.html#T28" target="_self" title="ANALOAF:th.28">Th28</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">S</font> =  <a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font>;<br/>
<b>thus </b><a NAME="E3:30"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span> st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font>
 <b>by </b><i><a class="txt" href="analoaf.html#E2:30"><i><font color="Green" title="E31">A2</font></i></a></i>;<br/>

<b>thus </b><a NAME="E4:30"/>
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="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span> holds <br/> ( <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="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="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</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> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="analoaf.html#R2" title="ANALOAF:pred.2">//</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> implies <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> ) &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="b3">c</font>,<font color="Olive" title="b4">d</font> implies <font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">d</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="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> implies <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="b1">a</font>,<font color="Olive" title="b3">c</font> ) &amp; (  not <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="b1">a</font>,<font color="Olive" title="b3">c</font> or <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="b2">b</font>,<font color="Olive" title="b3">c</font> or <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b2">b</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">c</font>, <font color="Maroon" title="c8">d</font>, <font color="Maroon" title="c9">p</font>, <font color="Maroon" title="c10">q</font>, <font color="Maroon" title="c11">r</font>, <font color="Maroon" title="c12">s</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c13">a'</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c14">b'</font> = <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c15">c'</font> = <font color="Maroon" title="c7">c</font>, <font color="Maroon" title="c16">d'</font> = <font color="Maroon" title="c8">d</font>, <font color="Maroon" title="c17">p'</font> = <font color="Maroon" title="c9">p</font>, <font color="Maroon" title="c18">q'</font> = <font color="Maroon" title="c10">q</font>, <font color="Maroon" title="c19">r'</font> = <font color="Maroon" title="c11">r</font>, <font color="Maroon" title="c20">s'</font> = <font color="Maroon" title="c12">s</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<b>thus </b><a NAME="E2:30_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c7">c</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="analoaf.html#D2" target="_self" title="ANALOAF:def.2">ANALOAF:def 2</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:30_1_1"/>
<font color="Maroon" title="c13">a'</font>,<font color="Maroon" title="c14">b'</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c15">c'</font>,<font color="Maroon" title="c15">c'</font>
 
<b>by </b><i><a class="ref" href="analoaf.html#D1" target="_self" title="ANALOAF:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E2:30_1_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c7">c</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span>
 <b>by </b><i><a class="ref" href="analoaf.html#T33" target="_self" title="ANALOAF:th.33">Th33</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:30_1"/>
( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font> implies <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1_2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="analoaf.html#D2" target="_self" title="ANALOAF:def.2">ANALOAF:def 2</a><br/>

<a NAME="E2:30_1_2"/><b>then </b>
<font color="Maroon" title="c13">a'</font>,<font color="Maroon" title="c14">b'</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c14">b'</font>,<font color="Maroon" title="c13">a'</font>
 
<b>by </b><i><a class="ref" href="analoaf.html#T33" target="_self" title="ANALOAF:th.33">Th33</a></i>;<br/>
<b>hence </b><a NAME="E3:30_1_2"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="ref" href="analoaf.html#T19" target="_self" title="ANALOAF:th.19">Th19</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:30_1"/>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c10">q</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">r</font>,<font color="Maroon" title="c12">s</font> implies <font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c10">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">r</font>,<font color="Maroon" title="c12">s</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1_3"/>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c10">q</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">r</font>,<font color="Maroon" title="c12">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="analoaf.html#D2" target="_self" title="ANALOAF:def.2">ANALOAF:def 2</a><br/>

<a NAME="E2:30_1_3"/><b>then </b>
( <font color="Maroon" title="c13">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">b'</font> &amp; <font color="Maroon" title="c13">a'</font>,<font color="Maroon" title="c14">b'</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c17">p'</font>,<font color="Maroon" title="c18">q'</font> &amp; <font color="Maroon" title="c13">a'</font>,<font color="Maroon" title="c14">b'</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c19">r'</font>,<font color="Maroon" title="c20">s'</font> )
 
<b>by </b><i><a class="ref" href="analoaf.html#T33" target="_self" title="ANALOAF:th.33">Th33</a></i>;<br/>
<a NAME="E3:30_1_3"/><b>then </b>
<font color="Maroon" title="c17">p'</font>,<font color="Maroon" title="c18">q'</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c19">r'</font>,<font color="Maroon" title="c20">s'</font>
 
<b>by </b><i><a class="ref" href="analoaf.html#T20" target="_self" title="ANALOAF:th.20">Th20</a></i>;<br/>
<a NAME="E4:30_1_3"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c10">q</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">r</font>,<font color="Maroon" title="c12">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span>
 
<b>by </b><i><a class="ref" href="analoaf.html#T33" target="_self" title="ANALOAF:th.33">Th33</a></i>;<br/>
<b>hence </b><a NAME="E5:30_1_3"/>
<font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c10">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">r</font>,<font color="Maroon" title="c12">s</font>
 <b>by </b><i><a class="ref" href="analoaf.html#D2" target="_self" title="ANALOAF:def.2">Def2</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E5:30_1"/>
( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> implies <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c7">c</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1_4"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="analoaf.html#D2" target="_self" title="ANALOAF:def.2">ANALOAF:def 2</a><br/>

<a NAME="E2:30_1_4"/><b>then </b>
<font color="Maroon" title="c13">a'</font>,<font color="Maroon" title="c14">b'</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c15">c'</font>,<font color="Maroon" title="c16">d'</font>
 
<b>by </b><i><a class="ref" href="analoaf.html#T33" target="_self" title="ANALOAF:th.33">Th33</a></i>;<br/>
<a NAME="E3:30_1_4"/><b>then </b>
<font color="Maroon" title="c14">b'</font>,<font color="Maroon" title="c13">a'</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c16">d'</font>,<font color="Maroon" title="c15">c'</font>
 
<b>by </b><i><a class="ref" href="analoaf.html#T21" target="_self" title="ANALOAF:th.21">Th21</a></i>;<br/>
<a NAME="E4:30_1_4"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c7">c</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span>
 
<b>by </b><i><a class="ref" href="analoaf.html#T33" target="_self" title="ANALOAF:th.33">Th33</a></i>;<br/>
<b>hence </b><a NAME="E5:30_1_4"/>
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c7">c</font>
 <b>by </b><i><a class="ref" href="analoaf.html#D2" target="_self" title="ANALOAF:def.2">Def2</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E6:30_1"/>
( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> implies <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1_5"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="analoaf.html#D2" target="_self" title="ANALOAF:def.2">ANALOAF:def 2</a><br/>

<a NAME="E2:30_1_5"/><b>then </b>
<font color="Maroon" title="c13">a'</font>,<font color="Maroon" title="c14">b'</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c14">b'</font>,<font color="Maroon" title="c15">c'</font>
 
<b>by </b><i><a class="ref" href="analoaf.html#T33" target="_self" title="ANALOAF:th.33">Th33</a></i>;<br/>
<a NAME="E3:30_1_5"/><b>then </b>
<font color="Maroon" title="c13">a'</font>,<font color="Maroon" title="c14">b'</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c13">a'</font>,<font color="Maroon" title="c15">c'</font>
 
<b>by </b><i><a class="ref" href="analoaf.html#T22" target="_self" title="ANALOAF:th.22">Th22</a></i>;<br/>
<a NAME="E4:30_1_5"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span>
 
<b>by </b><i><a class="ref" href="analoaf.html#T33" target="_self" title="ANALOAF:th.33">Th33</a></i>;<br/>
<b>hence </b><a NAME="E5:30_1_5"/>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font>
 <b>by </b><i><a class="ref" href="analoaf.html#D2" target="_self" title="ANALOAF:def.2">Def2</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E7:30_1"/>
(  not <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> or <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> or <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1_6"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="analoaf.html#D2" target="_self" title="ANALOAF:def.2">ANALOAF:def 2</a><br/>

<a NAME="E2:30_1_6"/><b>then </b>
<font color="Maroon" title="c13">a'</font>,<font color="Maroon" title="c14">b'</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c13">a'</font>,<font color="Maroon" title="c15">c'</font>
 
<b>by </b><i><a class="ref" href="analoaf.html#T33" target="_self" title="ANALOAF:th.33">Th33</a></i>;<br/>
<a NAME="E3:30_1_6"/><b>then </b>
( <font color="Maroon" title="c13">a'</font>,<font color="Maroon" title="c14">b'</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c14">b'</font>,<font color="Maroon" title="c15">c'</font> or <font color="Maroon" title="c13">a'</font>,<font color="Maroon" title="c15">c'</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c15">c'</font>,<font color="Maroon" title="c14">b'</font> )
 
<b>by </b><i><a class="ref" href="analoaf.html#T23" target="_self" title="ANALOAF:th.23">Th23</a></i>;<br/>
<a NAME="E4:30_1_6"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="analoaf.html#T33" target="_self" title="ANALOAF:th.33">Th33</a></i>;<br/>
<b>hence </b><a NAME="E5:30_1_6"/>
( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> or <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font> )
 <b>by </b><i><a class="ref" href="analoaf.html#D2" target="_self" title="ANALOAF:def.2">Def2</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>

<b>thus </b><a NAME="E5:30"/>
 ex <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> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span> st <br/>( not <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="b3">c</font>,<font color="Olive" title="b4">d</font> &amp; not <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="b4">d</font>,<font color="Olive" title="b3">c</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c5">a'</font>, <font color="Maroon" title="c6">b'</font>, <font color="Maroon" title="c7">c'</font>, <font color="Maroon" title="c8">d'</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="E2:30_2"/><i><font color="Green" title="E32">A3</font></i>: 
( not <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c6">b'</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c7">c'</font>,<font color="Maroon" title="c8">d'</font> &amp; not <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c6">b'</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c8">d'</font>,<font color="Maroon" title="c7">c'</font> )
 <b>by </b><i><a class="txt" href="analoaf.html#E1:30"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="analoaf.html#T29" target="_self" title="ANALOAF:th.29">Th29</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c5">a'</font>, <font color="Maroon" title="c10">b</font> = <font color="Maroon" title="c6">b'</font>, <font color="Maroon" title="c11">c</font> = <font color="Maroon" title="c7">c'</font>, <font color="Maroon" title="c12">d</font> = <font color="Maroon" title="c8">d'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span> ;<br/>
<a NAME="E4:30_2"/>
( not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c12">d</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">d</font>,<font color="Maroon" title="c11">c</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="analoaf.html#E2:30_2"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="analoaf.html#T33" target="_self" title="ANALOAF:th.33">Th33</a></i>;<br/>
<a NAME="E5:30_2"/><b>then </b>
( not <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c12">d</font> &amp; not <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c12">d</font>,<font color="Maroon" title="c11">c</font> )
 
<b>by </b><i><a class="ref" href="analoaf.html#D2" target="_self" title="ANALOAF:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E6:30_2"/>
 ex <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> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span> st <br/>( not <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="b3">c</font>,<font color="Olive" title="b4">d</font> &amp; not <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="b4">d</font>,<font color="Olive" title="b3">c</font> )
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E6:30"/>
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="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span>  ex <font color="Olive" title="b4">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span> st <br/>( <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="b3">c</font>,<font color="Olive" title="b4">d</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b4">d</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">d</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c8">a'</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c9">b'</font> = <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c10">c'</font> = <font color="Maroon" title="c7">c</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<b>consider </b><font color="Maroon" title="c11">d'</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="E3:30_3"/><i><font color="Green" title="E32">A4</font></i>: 
( <font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c9">b'</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c10">c'</font>,<font color="Maroon" title="c11">d'</font> &amp; <font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c10">c'</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c11">d'</font> &amp; <font color="Maroon" title="c9">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">d'</font> )
 <b>by </b><i><a class="txt" href="analoaf.html#E2:30"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="analoaf.html#T26" target="_self" title="ANALOAF:th.26">Th26</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">d</font> = <font color="Maroon" title="c11">d'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span> ;<br/>
<a NAME="E5:30_3"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c12">d</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c12">d</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="analoaf.html#E3:30_3"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="analoaf.html#T33" target="_self" title="ANALOAF:th.33">Th33</a></i>;<br/>
<a NAME="E6:30_3"/><b>then </b>
( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c12">d</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c12">d</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">d</font> )
 
<b>by </b><i><a class="txt" href="analoaf.html#E3:30_3"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="analoaf.html#D2" target="_self" title="ANALOAF:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E7:30_3"/>
 ex <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span> st <br/>( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">d</font> )
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E7:30"/>
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">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="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">b</font> &amp; <font color="Olive" title="b3">b</font>,<font color="Olive" title="b1">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b4">c</font> holds <br/> ex <font color="Olive" title="b5">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span> st <br/>( <font color="Olive" title="b2">a</font>,<font color="Olive" title="b1">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b5">d</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">c</font>,<font color="Olive" title="b5">d</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">p</font>, <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c8">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:30_4"/><i><font color="Green" title="E32">A5</font></i>: 
( <font color="Maroon" title="c5">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c8">c</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="analoaf.html#D2" target="_self" title="ANALOAF:def.2">ANALOAF:def 2</a><br/>

<b>reconsider </b><font color="Maroon" title="c9">p'</font> = <font color="Maroon" title="c5">p</font>, <font color="Maroon" title="c10">a'</font> = <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c11">b'</font> = <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c12">c'</font> = <font color="Maroon" title="c8">c</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<a NAME="E3:30_4"/>
( <font color="Maroon" title="c9">p'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">b'</font> &amp; <font color="Maroon" title="c11">b'</font>,<font color="Maroon" title="c9">p'</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c9">p'</font>,<font color="Maroon" title="c12">c'</font> )
 
<b>by </b><i><a class="txt" href="analoaf.html#E1:30_4"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="analoaf.html#T33" target="_self" title="ANALOAF:th.33">Th33</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">d'</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="E5:30_4"/><i><font color="Green" title="E33">A6</font></i>: 
( <font color="Maroon" title="c10">a'</font>,<font color="Maroon" title="c9">p'</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c9">p'</font>,<font color="Maroon" title="c13">d'</font> &amp; <font color="Maroon" title="c10">a'</font>,<font color="Maroon" title="c11">b'</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c12">c'</font>,<font color="Maroon" title="c13">d'</font> )
 <b>by </b><i><a class="ref" href="analoaf.html#T27" target="_self" title="ANALOAF:th.27">Th27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">d</font> = <font color="Maroon" title="c13">d'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span> ;<br/>
<a NAME="E7:30_4"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c14">d</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c14">d</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="analoaf.html#E5:30_4"><i><font color="Green" title="E33">A6</font></i></a>, <a class="ref" href="analoaf.html#T33" target="_self" title="ANALOAF:th.33">Th33</a></i>;<br/>
<a NAME="E8:30_4"/><b>then </b>
( <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c14">d</font> &amp; <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c14">d</font> )
 
<b>by </b><i><a class="ref" href="analoaf.html#D2" target="_self" title="ANALOAF:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E9:30_4"/>
 ex <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span> st <br/>( <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">p</font>,<font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">c</font>,<font color="Olive" title="b1">d</font> )
 ;<br/>


</div><b>end;</b></div>


</div>
