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

<span class="kw">let </span><font color="Maroon" title="c7">POS</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"> <font color="Maroon" title="c7">POS</font> is   <a href="analmetr.html#NM1" title="ANALMETR:NM.1">OrtAfSp</a></span><br/>

<a NAME="E1:61"/>
 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 <font color="Maroon" title="c7">POS</font>  st <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="b5">p</font>,<font color="Olive" title="b8">s</font> 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="b6">q</font>,<font color="Olive" title="b8">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c9">b</font>, <font color="Maroon" title="c10">c</font>, <font color="Maroon" title="c11">d</font>, <font color="Maroon" title="c12">p</font>, <font color="Maroon" title="c13">q</font>, <font color="Maroon" title="c14">r</font>, <font color="Maroon" title="c15">s</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</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="c8">a</font>,<font color="Maroon" title="c9">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c13">q</font> &amp; <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c15">s</font> implies <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c15">s</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:61_1"/><span class="lab"><font color="Green" title="E49">A1</font></span>: 
<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c13">q</font>
 <span class="kw">and </span><br/><a NAME="E2:61_1"/><span class="lab"><font color="Green" title="E50">A2</font></span>: 
<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c15">s</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="c8">a</font>,<font color="Maroon" title="c9">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c15">s</font></span><br/>

<div><span class="lab"><font color="Green" title="E51">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c16">p9</font> = <font color="Maroon" title="c12">p</font>, <font color="Maroon" title="c17">q9</font> = <font color="Maroon" title="c13">q</font>, <font color="Maroon" title="c18">s9</font> = <font color="Maroon" title="c15">s</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c7">POS</font></span>)</span> ;<br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:61_1_1"/><span class="lab"><font color="Green" title="E52">A4</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b</font>
 <span class="kw">and </span><br/><a NAME="E3:61_1_1"/><span class="lab"><font color="Green" title="E53">A5</font></span>: 
<font color="Maroon" title="c12">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">q</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="c8">a</font>,<font color="Maroon" title="c9">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c15">s</font></span><br/><a NAME="E4:61_1_1"/>
<font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c13">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c15">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="txt" href="#E2:61_1"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="txt" href="#E2:61_1_1"><span class="lab"><font color="Green" title="E52">A4</font></span></a>, <a class="ref" href="analmetr.html#D9" target="_self" title="ANALMETR:def.9">Def10</a></span>;<br/><a NAME="E5:61_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">p9</font>,<font color="Maroon" title="c17">q9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c16">p9</font>,<font color="Maroon" title="c18">s9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T36" target="_self" title="ANALMETR:th.36">Th48</a></span>;<br/><a NAME="E6:61_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">q9</font>,<font color="Maroon" title="c16">p9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c17">q9</font>,<font color="Maroon" title="c18">s9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T40" title="DIRAF:th.40">DIRAF:40</a></span>;<br/><a NAME="E7:61_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">p9</font>,<font color="Maroon" title="c17">q9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c17">q9</font>,<font color="Maroon" title="c18">s9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T4" title="AFF_1:th.4">AFF_1:4</a></span>;<br/><a NAME="E8:61_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A6</font></span>: 
<font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c13">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c15">s</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T36" target="_self" title="ANALMETR:th.36">Th48</a></span>;<br/><a NAME="E9:61_1_1"/>
<font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c13">q</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="ref" href="analmetr.html#D9" target="_self" title="ANALMETR:def.9">Def10</a></span>;<br/><span class="kw">hence </span><a NAME="E10:61_1_1"/>
<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c15">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1"><span class="lab"><font color="Green" title="E53">A5</font></span></a>, <a class="txt" href="#E8:61_1_1"><span class="lab"><font color="Green" title="E54">A6</font></span></a>, <a class="ref" href="analmetr.html#D9" target="_self" title="ANALMETR:def.9">Def10</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/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:61_1_2"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</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="c8">a</font>,<font color="Maroon" title="c9">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c15">s</font></span><br/><a NAME="E2:61_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c15">s</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#D9" target="_self" title="ANALMETR:def.9">Def10</a></span>;<br/><span class="kw">hence </span><a NAME="E3:61_1_2"/>
<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c15">s</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#D9" target="_self" title="ANALMETR:def.9">Def10</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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:61_1"/>
<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c15">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_1"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="txt" href="#E3:61_1"><span class="lab"><font color="Green" title="E51">A3</font></span></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/>


</div><span class="kw">end;</span></div>
<a NAME="E2:61"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A7</font></span>: 
 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 <font color="Maroon" title="c7">POS</font> 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="b5">p</font>,<font color="Olive" title="b8">s</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="b6">q</font>,<font color="Olive" title="b8">s</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#D9" target="_self" title="ANALMETR:def.9">Def10</a></span>;<br/>
<a NAME="E3:61"/><span class="lab"><font color="Green" title="E50">A8</font></span>: 
 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 <font color="Maroon" title="c7">POS</font>  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> holds <br/> ex <font color="Olive" title="b4">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font> 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="b1">a</font>,<font color="Olive" title="b4">x</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="b4">x</font>,<font color="Olive" title="b3">c</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c9">b</font>, <font color="Maroon" title="c10">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</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="c8">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b</font> implies  ex <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font> st <br/>( <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a</font>,<font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c10">c</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:61_2"/><span class="lab"><font color="Green" title="E51">A9</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font> st <br/>( <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a</font>,<font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c10">c</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c11">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font><span class="kw"> such that </span><br/><a NAME="E3:61_2"/><span class="lab"><font color="Green" title="E52">A10</font></span>: 
<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">y</font>
 <span class="kw">and </span><br/><a NAME="E4:61_2"/><span class="lab"><font color="Green" title="E53">A11</font></span>: 
<font color="Maroon" title="c10">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#D9" target="_self" title="ANALMETR:def.9">Def10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">a9</font> = <font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c13">b9</font> = <font color="Maroon" title="c9">b</font>, <font color="Maroon" title="c14">c9</font> = <font color="Maroon" title="c10">c</font>, <font color="Maroon" title="c15">y9</font> = <font color="Maroon" title="c11">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c7">POS</font></span>)</span> ;<br/>
<a NAME="E6:61_2"/>
 not <font color="Maroon" title="c12">a9</font>,<font color="Maroon" title="c13">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c14">c9</font>,<font color="Maroon" title="c15">y9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:61_2_1"/>
<font color="Maroon" title="c12">a9</font>,<font color="Maroon" title="c13">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c14">c9</font>,<font color="Maroon" title="c15">y9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:61_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T36" target="_self" title="ANALMETR:th.36">Th48</a></span>;<br/>
<a NAME="E3:61_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">y</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_2"><span class="lab"><font color="Green" title="E51">A9</font></span></a>, <a class="txt" href="#E3:61_2"><span class="lab"><font color="Green" title="E52">A10</font></span></a>, <a class="ref" href="analmetr.html#D9" target="_self" title="ANALMETR:def.9">Def10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:61_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:61_2"><span class="lab"><font color="Green" title="E53">A11</font></span></a>, <a class="ref" href="analmetr.html#D9" target="_self" title="ANALMETR:def.9">Def10</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/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">x9</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#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c7">POS</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E8:61_2"/><span class="lab"><font color="Green" title="E54">A12</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">a9</font>,<font color="Maroon" title="c13">b9</font>,<font color="Maroon" title="c16">x9</font>
 <span class="kw">and </span><br/><a NAME="E9:61_2"/><span class="lab"><font color="Green" title="E55">A13</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c14">c9</font>,<font color="Maroon" title="c15">y9</font>,<font color="Maroon" title="c16">x9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T60" title="AFF_1:th.60">AFF_1:60</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">x</font> = <font color="Maroon" title="c16">x9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font> ;<br/>
<a NAME="E11:61_2"/>
<font color="Maroon" title="c14">c9</font>,<font color="Maroon" title="c15">y9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c14">c9</font>,<font color="Maroon" title="c16">x9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:61_2"><span class="lab"><font color="Green" title="E55">A13</font></span></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E12:61_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A14</font></span>: 
<font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c17">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T36" target="_self" title="ANALMETR:th.36">Th48</a></span>;<br/>
<a NAME="E13:61_2"/>
<font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">y</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_2"><span class="lab"><font color="Green" title="E52">A10</font></span></a>, <a class="ref" href="analmetr.html#D9" target="_self" title="ANALMETR:def.9">Def10</a></span>;<br/>
<a NAME="E14:61_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c17">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:61_2"><span class="lab"><font color="Green" title="E53">A11</font></span></a>, <a class="txt" href="#E12:61_2"><span class="lab"><font color="Green" title="E56">A14</font></span></a>, <a class="ref" href="analmetr.html#D9" target="_self" title="ANALMETR:def.9">Def10</a></span>;<br/>
<a NAME="E15:61_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A15</font></span>: 
<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c10">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#D9" target="_self" title="ANALMETR:def.9">Def10</a></span>;<br/>
<a NAME="E16:61_2"/>
<font color="Maroon" title="c12">a9</font>,<font color="Maroon" title="c13">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c12">a9</font>,<font color="Maroon" title="c16">x9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:61_2"><span class="lab"><font color="Green" title="E54">A12</font></span></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E17:61_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c17">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T36" target="_self" title="ANALMETR:th.36">Th48</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:61_2"/>
 ex <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font> st <br/>( <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a</font>,<font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c10">c</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:61_2"><span class="lab"><font color="Green" title="E57">A15</font></span></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/>


</div><span class="kw">end;</span></div>
<a NAME="E4:61"/>
(  <a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c7">POS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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 <font color="Maroon" title="c7">POS</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c7">POS</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 <font color="Maroon" title="c7">POS</font>  ex <font color="Olive" title="b4">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font> 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="ref" href="analmetr.html#D9" target="_self" title="ANALMETR:def.9">Def10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:61"/>
<font color="Maroon" title="c7">POS</font> is   <a href="analmetr.html#NM1" title="ANALMETR:NM.1">OrtAfSp</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61"><span class="lab"><font color="Green" title="E50">A8</font></span></a>, <a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="ref" href="analmetr.html#D8" target="_self" title="ANALMETR:def.8">Def9</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/>


</div>
