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<span class="kw">let </span><font color="Maroon" title="c7">POS</font> be   <a href="analmetr.html#NM1" title="ANALMETR:NM.1">OrtAfSp</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <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> #) is   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</a> implies <font color="Maroon" title="c7">POS</font> is   <a href="analmetr.html#NM2" title="ANALMETR:NM.2">OrtAfPl</a> )</span><br/>

<span class="kw">assume </span><a NAME="E1:63"/><span class="lab"><font color="Green" title="E46">A1</font></span>: 
 <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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> #) is   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</a>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">POS</font> is   <a href="analmetr.html#NM2" title="ANALMETR:NM.2">OrtAfPl</a></span><br/>

<div><span class="lab"><font color="Green" title="E47">A2</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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="b7">r</font>,<font color="Olive" title="b8">s</font> &amp;  not <font color="Olive" title="b5">p</font>,<font color="Olive" title="b6">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">r</font>,<font color="Olive" title="b8">s</font> implies <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> ) )</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <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="c8">a</font>,<font color="Maroon" title="c9">b</font> implies <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</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="c10">c</font>,<font color="Maroon" title="c10">c</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="c10">c</font>,<font color="Maroon" title="c11">d</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="c11">d</font>,<font color="Maroon" title="c10">c</font> &amp; <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">d</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> ) ) &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="c13">q</font> &amp; <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="c14">r</font>,<font color="Maroon" title="c15">s</font> &amp;  not <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="c14">r</font>,<font color="Maroon" title="c15">s</font> implies <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</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="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="c14">r</font>,<font color="Maroon" title="c15">s</font> &amp;  not <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="c14">r</font>,<font color="Maroon" title="c15">s</font> implies <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> ) )</span><br/><span class="kw">thus </span><a NAME="E1:63_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="c8">a</font>,<font color="Maroon" title="c9">b</font> implies <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</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="c10">c</font>,<font color="Maroon" title="c10">c</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="c10">c</font>,<font color="Maroon" title="c11">d</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="c11">d</font>,<font color="Maroon" title="c10">c</font> &amp; <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">d</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> ) ) &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="c13">q</font> &amp; <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="c14">r</font>,<font color="Maroon" title="c15">s</font> &amp;  not <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="c14">r</font>,<font color="Maroon" title="c15">s</font> implies <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#D8" target="_self" onmouseover="rs('analmetr/D8')" onmouseout="rh()">Def7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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="c14">r</font>,<font color="Maroon" title="c15">s</font> &amp;  not <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="c14">r</font>,<font color="Maroon" title="c15">s</font> implies <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> )</span><br/><span class="kw">thus </span><a NAME="E2:63_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="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="c14">r</font>,<font color="Maroon" title="c15">s</font> &amp;  not <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="c14">r</font>,<font color="Maroon" title="c15">s</font> implies <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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c16">a9</font> = <font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c17">b9</font> = <font color="Maroon" title="c9">b</font>, <font color="Maroon" title="c18">p9</font> = <font color="Maroon" title="c12">p</font>, <font color="Maroon" title="c19">q9</font> = <font color="Maroon" title="c13">q</font>, <font color="Maroon" title="c20">r9</font> = <font color="Maroon" title="c14">r</font>, <font color="Maroon" title="c21">s9</font> = <font color="Maroon" title="c15">s</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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> #) ;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:63_1_1"/><span class="lab"><font color="Green" title="E48">A3</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="E3:63_1_1"/><span class="lab"><font color="Green" title="E49">A4</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="c14">r</font>,<font color="Maroon" title="c15">s</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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="c14">r</font>,<font color="Maroon" title="c15">s</font> or <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> )</span><br/>

<a NAME="E4:63_1_1"/><span class="lab"><font color="Green" title="E50">A5</font></span>: 
<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="#E2:63_1_1"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="ref" href="analmetr.html#D8" target="_self" onmouseover="rs('analmetr/D8')" onmouseout="rh()">Def7</a></span>;<br/>
<a NAME="E5:63_1_1"/><span class="lab"><font color="Green" title="E51">A6</font></span>: 
<font color="Maroon" title="c14">r</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="txt" href="#E3:63_1_1"><span class="lab"><font color="Green" title="E49">A4</font></span></a>, <a class="ref" href="analmetr.html#D8" target="_self" onmouseover="rs('analmetr/D8')" onmouseout="rh()">Def7</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:63_1_1"/><span class="lab"><font color="Green" title="E52">A7</font></span>: 
(  not <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="c14">r</font>,<font color="Maroon" title="c15">s</font> &amp;  not <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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E7:63_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A8</font></span>: 
 not <font color="Maroon" title="c18">p9</font>,<font color="Maroon" title="c19">q9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">r9</font>,<font color="Maroon" title="c21">s9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T36" target="_self" onmouseover="rs('analmetr/T36')" onmouseout="rh()">Th36</a></span>;<br/>
<a NAME="E8:63_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A9</font></span>: 
<font color="Maroon" title="c18">p9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">q9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T3" onmouseover="rs('aff_1/T3')" onmouseout="rh()">AFF_1:3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c22">x9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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> #)<span class="kw"> such that </span><br/><a NAME="E10:63_1_1"/><span class="lab"><font color="Green" title="E55">A10</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c18">p9</font>,<font color="Maroon" title="c19">q9</font>,<font color="Maroon" title="c22">x9</font>
 <span class="kw">and </span><br/><a NAME="E11:63_1_1"/><span class="lab"><font color="Green" title="E56">A11</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c20">r9</font>,<font color="Maroon" title="c21">s9</font>,<font color="Maroon" title="c22">x9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E7:63_1_1"><span class="lab"><font color="Green" title="E53">A8</font></span></a>, <a class="ref" href="aff_1.html#T60" onmouseover="rs('aff_1/T60')" onmouseout="rh()">AFF_1:60</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">x</font> = <font color="Maroon" title="c22">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="E13:63_1_1"/><span class="lab"><font color="Green" title="E57">A12</font></span>: 
<font color="Maroon" title="c20">r9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c21">s9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:63_1_1"><span class="lab"><font color="Green" title="E53">A8</font></span></a>, <a class="ref" href="aff_1.html#T3" onmouseover="rs('aff_1/T3')" onmouseout="rh()">AFF_1:3</a></span>;<br/>
<a NAME="E14:63_1_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c21">s9</font>,<font color="Maroon" title="c20">r9</font>,<font color="Maroon" title="c22">x9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:63_1_1"><span class="lab"><font color="Green" title="E56">A11</font></span></a>, <a class="ref" href="aff_1.html#T6" onmouseover="rs('aff_1/T6')" onmouseout="rh()">AFF_1:6</a></span>;<br/>
<a NAME="E15:63_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">s9</font>,<font color="Maroon" title="c20">r9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c21">s9</font>,<font color="Maroon" title="c22">x9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/>
<a NAME="E16:63_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A13</font></span>: 
<font color="Maroon" title="c20">r9</font>,<font color="Maroon" title="c21">s9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c21">s9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T4" onmouseover="rs('aff_1/T4')" onmouseout="rh()">AFF_1:4</a></span>;<br/>
<a NAME="E17:63_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">r</font>,<font color="Maroon" title="c15">s</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c23">x</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" onmouseover="rs('analmetr/T36')" onmouseout="rh()">Th36</a></span>;<br/>
<a NAME="E18:63_1_1"/><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="c23">x</font>,<font color="Maroon" title="c15">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:63_1_1"><span class="lab"><font color="Green" title="E57">A12</font></span></a>, <a class="txt" href="#E5:63_1_1"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="ref" href="analmetr.html#D8" target="_self" onmouseover="rs('analmetr/D8')" onmouseout="rh()">Def7</a></span>;<br/>
<a NAME="E19:63_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A14</font></span>: 
<font color="Maroon" title="c23">x</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#D8" target="_self" onmouseover="rs('analmetr/D8')" onmouseout="rh()">Def7</a></span>;<br/>
<a NAME="E20:63_1_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c19">q9</font>,<font color="Maroon" title="c18">p9</font>,<font color="Maroon" title="c22">x9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:63_1_1"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="ref" href="aff_1.html#T6" onmouseover="rs('aff_1/T6')" onmouseout="rh()">AFF_1:6</a></span>;<br/>
<a NAME="E21:63_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">q9</font>,<font color="Maroon" title="c18">p9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c19">q9</font>,<font color="Maroon" title="c22">x9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/>
<a NAME="E22:63_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">p9</font>,<font color="Maroon" title="c19">q9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c19">q9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T4" onmouseover="rs('aff_1/T4')" onmouseout="rh()">AFF_1:4</a></span>;<br/>
<a NAME="E23:63_1_1"/><span class="kw">then </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="c23">x</font>,<font color="Maroon" title="c13">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T36" target="_self" onmouseover="rs('analmetr/T36')" onmouseout="rh()">Th36</a></span>;<br/>
<a NAME="E24:63_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">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="c23">x</font>,<font color="Maroon" title="c13">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:63_1_1"><span class="lab"><font color="Green" title="E54">A9</font></span></a>, <a class="txt" href="#E4:63_1_1"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="ref" href="analmetr.html#D8" target="_self" onmouseover="rs('analmetr/D8')" onmouseout="rh()">Def7</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E61">A16</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">q</font> implies  not <font color="Maroon" title="c23">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">s</font> )</span></a><div class="add"><span class="kw">consider </span><font color="Maroon" title="c24">y9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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> #)<span class="kw"> such that </span><br/><a NAME="E2:63_1_1_1"/><span class="lab"><font color="Green" title="E62">A17</font></span>: 
( <font color="Maroon" title="c16">a9</font>,<font color="Maroon" title="c17">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c19">q9</font>,<font color="Maroon" title="c24">y9</font> &amp; <font color="Maroon" title="c19">q9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">y9</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T40" onmouseover="rs('diraf/T40')" onmouseout="rh()">DIRAF:40</a></span>;<br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:63_1_1_1"/><span class="lab"><font color="Green" title="E63">A18</font></span>: 
<font color="Maroon" title="c23">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">q</font>
 <span class="kw">and </span><br/><a NAME="E4:63_1_1_1"/><span class="lab"><font color="Green" title="E64">A19</font></span>: 
<font color="Maroon" title="c23">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">s</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E5:63_1_1_1"/>
 not <font color="Maroon" title="c19">q9</font>,<font color="Maroon" title="c24">y9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c21">s9</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:63_1_1_1_1"/>
<font color="Maroon" title="c19">q9</font>,<font color="Maroon" title="c24">y9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c21">s9</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:63_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">q9</font>,<font color="Maroon" title="c24">y9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">r9</font>,<font color="Maroon" title="c21">s9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:63_1_1"><span class="lab"><font color="Green" title="E58">A13</font></span></a>, <a class="txt" href="#E4:63_1_1_1"><span class="lab"><font color="Green" title="E64">A19</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/>
<a NAME="E3:63_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">r9</font>,<font color="Maroon" title="c21">s9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c16">a9</font>,<font color="Maroon" title="c17">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63_1_1_1"><span class="lab"><font color="Green" title="E62">A17</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/>
<a NAME="E4:63_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">r</font>,<font color="Maroon" title="c15">s</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</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#T36" target="_self" onmouseover="rs('analmetr/T36')" onmouseout="rh()">Th36</a></span>;<br/>
<a NAME="E5:63_1_1_1_1"/><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="c8">a</font>,<font color="Maroon" title="c9">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:63_1_1"><span class="lab"><font color="Green" title="E57">A12</font></span></a>, <a class="txt" href="#E5:63_1_1"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="ref" href="analmetr.html#D8" target="_self" onmouseover="rs('analmetr/D8')" onmouseout="rh()">Def7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:63_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:63_1_1"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="ref" href="analmetr.html#D8" target="_self" onmouseover="rs('analmetr/D8')" onmouseout="rh()">Def7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">z9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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> #)<span class="kw"> such that </span><br/><a NAME="E7:63_1_1_1"/><span class="lab"><font color="Green" title="E65">A20</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c19">q9</font>,<font color="Maroon" title="c24">y9</font>,<font color="Maroon" title="c25">z9</font>
 <span class="kw">and </span><br/><a NAME="E8:63_1_1_1"/><span class="lab"><font color="Green" title="E66">A21</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c21">s9</font>,<font color="Maroon" title="c25">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="aff_1.html#T60" onmouseover="rs('aff_1/T60')" onmouseout="rh()">AFF_1:60</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c26">z</font> = <font color="Maroon" title="c25">z9</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="E10:63_1_1_1"/>
<font color="Maroon" title="c19">q9</font>,<font color="Maroon" title="c24">y9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c19">q9</font>,<font color="Maroon" title="c25">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:63_1_1_1"><span class="lab"><font color="Green" title="E65">A20</font></span></a>, <a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/><a NAME="E11:63_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">a9</font>,<font color="Maroon" title="c17">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c19">q9</font>,<font color="Maroon" title="c25">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63_1_1_1"><span class="lab"><font color="Green" title="E62">A17</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/><a NAME="E12:63_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A22</font></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="c13">q</font>,<font color="Maroon" title="c26">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T36" target="_self" onmouseover="rs('analmetr/T36')" onmouseout="rh()">Th36</a></span>;<br/><a NAME="E13:63_1_1_1"/><span class="lab"><font color="Green" title="E68">A23</font></span>: 
<font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c21">s9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c25">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:63_1_1_1"><span class="lab"><font color="Green" title="E66">A21</font></span></a>, <a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/><a NAME="E14:63_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c23">x</font>,<font color="Maroon" title="c15">s</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c23">x</font>,<font color="Maroon" title="c26">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T36" target="_self" onmouseover="rs('analmetr/T36')" onmouseout="rh()">Th36</a></span>;<br/><a NAME="E15:63_1_1_1"/><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="c23">x</font>,<font color="Maroon" title="c26">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:63_1_1"><span class="lab"><font color="Green" title="E59">A14</font></span></a>, <a class="txt" href="#E4:63_1_1_1"><span class="lab"><font color="Green" title="E64">A19</font></span></a>, <a class="ref" href="analmetr.html#D8" target="_self" onmouseover="rs('analmetr/D8')" onmouseout="rh()">Def7</a></span>;<br/><a NAME="E16:63_1_1_1"/><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="c13">q</font>,<font color="Maroon" title="c26">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:63_1_1"><span class="lab"><font color="Green" title="E60">A15</font></span></a>, <a class="ref" href="analmetr.html#D8" target="_self" onmouseover="rs('analmetr/D8')" onmouseout="rh()">Def7</a></span>;<br/><a NAME="E17:63_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c26">z</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c26">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:63_1_1"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="txt" href="#E12:63_1_1_1"><span class="lab"><font color="Green" title="E67">A22</font></span></a>, <a class="ref" href="analmetr.html#D8" target="_self" onmouseover="rs('analmetr/D8')" onmouseout="rh()">Def7</a></span>;<br/><a NAME="E18:63_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c21">s9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c19">q9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:63_1_1_1"><span class="lab"><font color="Green" title="E68">A23</font></span></a>, <a class="ref" href="analmetr.html#D8" target="_self" onmouseover="rs('analmetr/D8')" onmouseout="rh()">Def7</a></span>;<br/><a NAME="E19:63_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A24</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c21">s9</font>,<font color="Maroon" title="c19">q9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/><a NAME="E20:63_1_1_1"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c21">s9</font>,<font color="Maroon" title="c22">x9</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c19">q9</font>,<font color="Maroon" title="c18">p9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:63_1_1"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="ref" href="aff_1.html#T6" onmouseover="rs('aff_1/T6')" onmouseout="rh()">AFF_1:6</a>, <a class="ref" href="aff_1.html#T7" onmouseover="rs('aff_1/T7')" onmouseout="rh()">AFF_1:7</a></span>;<br/><a NAME="E21:63_1_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c21">s9</font>,<font color="Maroon" title="c18">p9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_1_1_1"><span class="lab"><font color="Green" title="E63">A18</font></span></a>, <a class="txt" href="#E19:63_1_1_1"><span class="lab"><font color="Green" title="E69">A24</font></span></a>, <a class="ref" href="aff_1.html#T11" onmouseover="rs('aff_1/T11')" onmouseout="rh()">AFF_1:11</a></span>;<br/><a NAME="E22:63_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c21">s9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c18">p9</font>,<font color="Maroon" title="c19">q9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:63_1_1_1"><span class="lab"><font color="Green" title="E69">A24</font></span></a>, <a class="ref" href="aff_1.html#T10" onmouseover="rs('aff_1/T10')" onmouseout="rh()">AFF_1:10</a></span>;<br/><a NAME="E23:63_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">p9</font>,<font color="Maroon" title="c19">q9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">r9</font>,<font color="Maroon" title="c21">s9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:63_1_1"><span class="lab"><font color="Green" title="E58">A13</font></span></a>, <a class="txt" href="#E4:63_1_1_1"><span class="lab"><font color="Green" title="E64">A19</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E24:63_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:63_1_1"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="ref" href="analmetr.html#T36" target="_self" onmouseover="rs('analmetr/T36')" onmouseout="rh()">Th36</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E26:63_1_1"/>
<font color="Maroon" title="c20">r9</font>,<font color="Maroon" title="c21">s9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">r9</font>,<font color="Maroon" title="c22">x9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:63_1_1"><span class="lab"><font color="Green" title="E56">A11</font></span></a>, <a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/>
<a NAME="E27:63_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A25</font></span>: 
<font color="Maroon" title="c20">r9</font>,<font color="Maroon" title="c21">s9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c20">r9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T4" onmouseover="rs('aff_1/T4')" onmouseout="rh()">AFF_1:4</a></span>;<br/>
<a NAME="E28:63_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">r</font>,<font color="Maroon" title="c15">s</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c23">x</font>,<font color="Maroon" title="c14">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T36" target="_self" onmouseover="rs('analmetr/T36')" onmouseout="rh()">Th36</a></span>;<br/>
<a NAME="E29:63_1_1"/><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="c23">x</font>,<font color="Maroon" title="c14">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:63_1_1"><span class="lab"><font color="Green" title="E57">A12</font></span></a>, <a class="txt" href="#E5:63_1_1"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="ref" href="analmetr.html#D8" target="_self" onmouseover="rs('analmetr/D8')" onmouseout="rh()">Def7</a></span>;<br/>
<a NAME="E30:63_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A26</font></span>: 
<font color="Maroon" title="c23">x</font>,<font color="Maroon" title="c14">r</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#D8" target="_self" onmouseover="rs('analmetr/D8')" onmouseout="rh()">Def7</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E64">A27</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_1_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">q</font> implies  not <font color="Maroon" title="c23">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">r</font> )</span></a><div class="add"><span class="kw">consider </span><font color="Maroon" title="c24">y9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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> #)<span class="kw"> such that </span><br/><a NAME="E2:63_1_1_2"/><span class="lab"><font color="Green" title="E65">A28</font></span>: 
( <font color="Maroon" title="c16">a9</font>,<font color="Maroon" title="c17">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c19">q9</font>,<font color="Maroon" title="c24">y9</font> &amp; <font color="Maroon" title="c19">q9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">y9</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T40" onmouseover="rs('diraf/T40')" onmouseout="rh()">DIRAF:40</a></span>;<br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:63_1_1_2"/><span class="lab"><font color="Green" title="E66">A29</font></span>: 
<font color="Maroon" title="c23">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">q</font>
 <span class="kw">and </span><br/><a NAME="E4:63_1_1_2"/><span class="lab"><font color="Green" title="E67">A30</font></span>: 
<font color="Maroon" title="c23">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">r</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E5:63_1_1_2"/>
 not <font color="Maroon" title="c19">q9</font>,<font color="Maroon" title="c24">y9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c20">r9</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:63_1_1_2_1"/>
<font color="Maroon" title="c19">q9</font>,<font color="Maroon" title="c24">y9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c20">r9</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:63_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">q9</font>,<font color="Maroon" title="c24">y9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">r9</font>,<font color="Maroon" title="c21">s9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:63_1_1"><span class="lab"><font color="Green" title="E62">A25</font></span></a>, <a class="txt" href="#E4:63_1_1_2"><span class="lab"><font color="Green" title="E67">A30</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/>
<a NAME="E3:63_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">r9</font>,<font color="Maroon" title="c21">s9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c16">a9</font>,<font color="Maroon" title="c17">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63_1_1_2"><span class="lab"><font color="Green" title="E65">A28</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/>
<a NAME="E4:63_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">r</font>,<font color="Maroon" title="c15">s</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</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#T36" target="_self" onmouseover="rs('analmetr/T36')" onmouseout="rh()">Th36</a></span>;<br/>
<a NAME="E5:63_1_1_2_1"/><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="c8">a</font>,<font color="Maroon" title="c9">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:63_1_1"><span class="lab"><font color="Green" title="E57">A12</font></span></a>, <a class="txt" href="#E5:63_1_1"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="ref" href="analmetr.html#D8" target="_self" onmouseover="rs('analmetr/D8')" onmouseout="rh()">Def7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:63_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:63_1_1"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="ref" href="analmetr.html#D8" target="_self" onmouseover="rs('analmetr/D8')" onmouseout="rh()">Def7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">z9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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> #)<span class="kw"> such that </span><br/><a NAME="E7:63_1_1_2"/><span class="lab"><font color="Green" title="E68">A31</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c19">q9</font>,<font color="Maroon" title="c24">y9</font>,<font color="Maroon" title="c25">z9</font>
 <span class="kw">and </span><br/><a NAME="E8:63_1_1_2"/><span class="lab"><font color="Green" title="E69">A32</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c20">r9</font>,<font color="Maroon" title="c25">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="aff_1.html#T60" onmouseover="rs('aff_1/T60')" onmouseout="rh()">AFF_1:60</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c26">z</font> = <font color="Maroon" title="c25">z9</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="E10:63_1_1_2"/>
<font color="Maroon" title="c19">q9</font>,<font color="Maroon" title="c24">y9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c19">q9</font>,<font color="Maroon" title="c25">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:63_1_1_2"><span class="lab"><font color="Green" title="E68">A31</font></span></a>, <a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/><a NAME="E11:63_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">a9</font>,<font color="Maroon" title="c17">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c19">q9</font>,<font color="Maroon" title="c25">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63_1_1_2"><span class="lab"><font color="Green" title="E65">A28</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/><a NAME="E12:63_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A33</font></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="c13">q</font>,<font color="Maroon" title="c26">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T36" target="_self" onmouseover="rs('analmetr/T36')" onmouseout="rh()">Th36</a></span>;<br/><a NAME="E13:63_1_1_2"/><span class="lab"><font color="Green" title="E71">A34</font></span>: 
<font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c20">r9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c25">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:63_1_1_2"><span class="lab"><font color="Green" title="E69">A32</font></span></a>, <a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/><a NAME="E14:63_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c23">x</font>,<font color="Maroon" title="c14">r</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c23">x</font>,<font color="Maroon" title="c26">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T36" target="_self" onmouseover="rs('analmetr/T36')" onmouseout="rh()">Th36</a></span>;<br/><a NAME="E15:63_1_1_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="c23">x</font>,<font color="Maroon" title="c26">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:63_1_1"><span class="lab"><font color="Green" title="E63">A26</font></span></a>, <a class="txt" href="#E4:63_1_1_2"><span class="lab"><font color="Green" title="E67">A30</font></span></a>, <a class="ref" href="analmetr.html#D8" target="_self" onmouseover="rs('analmetr/D8')" onmouseout="rh()">Def7</a></span>;<br/><a NAME="E16:63_1_1_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="c13">q</font>,<font color="Maroon" title="c26">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:63_1_1"><span class="lab"><font color="Green" title="E60">A15</font></span></a>, <a class="ref" href="analmetr.html#D8" target="_self" onmouseover="rs('analmetr/D8')" onmouseout="rh()">Def7</a></span>;<br/><a NAME="E17:63_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c26">z</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c26">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:63_1_1"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="txt" href="#E12:63_1_1_2"><span class="lab"><font color="Green" title="E70">A33</font></span></a>, <a class="ref" href="analmetr.html#D8" target="_self" onmouseover="rs('analmetr/D8')" onmouseout="rh()">Def7</a></span>;<br/><a NAME="E18:63_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c20">r9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c19">q9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:63_1_1_2"><span class="lab"><font color="Green" title="E71">A34</font></span></a>, <a class="ref" href="analmetr.html#D8" target="_self" onmouseover="rs('analmetr/D8')" onmouseout="rh()">Def7</a></span>;<br/><a NAME="E19:63_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A35</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c20">r9</font>,<font color="Maroon" title="c19">q9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/><a NAME="E20:63_1_1_2"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c20">r9</font>,<font color="Maroon" title="c22">x9</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c19">q9</font>,<font color="Maroon" title="c18">p9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:63_1_1"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="ref" href="aff_1.html#T6" onmouseover="rs('aff_1/T6')" onmouseout="rh()">AFF_1:6</a>, <a class="ref" href="aff_1.html#T7" onmouseover="rs('aff_1/T7')" onmouseout="rh()">AFF_1:7</a></span>;<br/><a NAME="E21:63_1_1_2"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c20">r9</font>,<font color="Maroon" title="c18">p9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_1_1_2"><span class="lab"><font color="Green" title="E66">A29</font></span></a>, <a class="txt" href="#E19:63_1_1_2"><span class="lab"><font color="Green" title="E72">A35</font></span></a>, <a class="ref" href="aff_1.html#T11" onmouseover="rs('aff_1/T11')" onmouseout="rh()">AFF_1:11</a></span>;<br/><a NAME="E22:63_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c20">r9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c18">p9</font>,<font color="Maroon" title="c19">q9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:63_1_1_2"><span class="lab"><font color="Green" title="E72">A35</font></span></a>, <a class="ref" href="aff_1.html#T10" onmouseover="rs('aff_1/T10')" onmouseout="rh()">AFF_1:10</a></span>;<br/><a NAME="E23:63_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">p9</font>,<font color="Maroon" title="c19">q9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">r9</font>,<font color="Maroon" title="c21">s9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:63_1_1"><span class="lab"><font color="Green" title="E62">A25</font></span></a>, <a class="txt" href="#E4:63_1_1_2"><span class="lab"><font color="Green" title="E67">A30</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E24:63_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:63_1_1"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="ref" href="analmetr.html#T36" target="_self" onmouseover="rs('analmetr/T36')" onmouseout="rh()">Th36</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E32:63_1_1"/>
<font color="Maroon" title="c18">p9</font>,<font color="Maroon" title="c19">q9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c18">p9</font>,<font color="Maroon" title="c22">x9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:63_1_1"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/>
<a NAME="E33:63_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">p9</font>,<font color="Maroon" title="c19">q9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c18">p9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T4" onmouseover="rs('aff_1/T4')" onmouseout="rh()">AFF_1:4</a></span>;<br/>
<a NAME="E34:63_1_1"/><span class="kw">then </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="c23">x</font>,<font color="Maroon" title="c12">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T36" target="_self" onmouseover="rs('analmetr/T36')" onmouseout="rh()">Th36</a></span>;<br/>
<a NAME="E35:63_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A36</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="c23">x</font>,<font color="Maroon" title="c12">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:63_1_1"><span class="lab"><font color="Green" title="E54">A9</font></span></a>, <a class="txt" href="#E4:63_1_1"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="ref" href="analmetr.html#D8" target="_self" onmouseover="rs('analmetr/D8')" onmouseout="rh()">Def7</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E66">A37</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_1_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">p</font> implies  not <font color="Maroon" title="c23">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">s</font> )</span></a><div class="add"><span class="kw">consider </span><font color="Maroon" title="c24">y9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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> #)<span class="kw"> such that </span><br/><a NAME="E2:63_1_1_3"/><span class="lab"><font color="Green" title="E67">A38</font></span>: 
( <font color="Maroon" title="c16">a9</font>,<font color="Maroon" title="c17">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c18">p9</font>,<font color="Maroon" title="c24">y9</font> &amp; <font color="Maroon" title="c18">p9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">y9</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T40" onmouseover="rs('diraf/T40')" onmouseout="rh()">DIRAF:40</a></span>;<br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:63_1_1_3"/><span class="lab"><font color="Green" title="E68">A39</font></span>: 
<font color="Maroon" title="c23">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">p</font>
 <span class="kw">and </span><br/><a NAME="E4:63_1_1_3"/><span class="lab"><font color="Green" title="E69">A40</font></span>: 
<font color="Maroon" title="c23">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">s</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E5:63_1_1_3"/>
 not <font color="Maroon" title="c18">p9</font>,<font color="Maroon" title="c24">y9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c21">s9</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:63_1_1_3_1"/>
<font color="Maroon" title="c18">p9</font>,<font color="Maroon" title="c24">y9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c21">s9</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:63_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">p9</font>,<font color="Maroon" title="c24">y9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">r9</font>,<font color="Maroon" title="c21">s9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:63_1_1"><span class="lab"><font color="Green" title="E58">A13</font></span></a>, <a class="txt" href="#E4:63_1_1_3"><span class="lab"><font color="Green" title="E69">A40</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/>
<a NAME="E3:63_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">r9</font>,<font color="Maroon" title="c21">s9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c16">a9</font>,<font color="Maroon" title="c17">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63_1_1_3"><span class="lab"><font color="Green" title="E67">A38</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/>
<a NAME="E4:63_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">r</font>,<font color="Maroon" title="c15">s</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</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#T36" target="_self" onmouseover="rs('analmetr/T36')" onmouseout="rh()">Th36</a></span>;<br/>
<a NAME="E5:63_1_1_3_1"/><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="c8">a</font>,<font color="Maroon" title="c9">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:63_1_1"><span class="lab"><font color="Green" title="E57">A12</font></span></a>, <a class="txt" href="#E5:63_1_1"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="ref" href="analmetr.html#D8" target="_self" onmouseover="rs('analmetr/D8')" onmouseout="rh()">Def7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:63_1_1_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:63_1_1"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="ref" href="analmetr.html#D8" target="_self" onmouseover="rs('analmetr/D8')" onmouseout="rh()">Def7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">z9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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> #)<span class="kw"> such that </span><br/><a NAME="E7:63_1_1_3"/><span class="lab"><font color="Green" title="E70">A41</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c18">p9</font>,<font color="Maroon" title="c24">y9</font>,<font color="Maroon" title="c25">z9</font>
 <span class="kw">and </span><br/><a NAME="E8:63_1_1_3"/><span class="lab"><font color="Green" title="E71">A42</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c21">s9</font>,<font color="Maroon" title="c25">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="aff_1.html#T60" onmouseover="rs('aff_1/T60')" onmouseout="rh()">AFF_1:60</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c26">z</font> = <font color="Maroon" title="c25">z9</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="E10:63_1_1_3"/>
<font color="Maroon" title="c18">p9</font>,<font color="Maroon" title="c24">y9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c18">p9</font>,<font color="Maroon" title="c25">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:63_1_1_3"><span class="lab"><font color="Green" title="E70">A41</font></span></a>, <a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/><a NAME="E11:63_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c16">a9</font>,<font color="Maroon" title="c17">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c18">p9</font>,<font color="Maroon" title="c25">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63_1_1_3"><span class="lab"><font color="Green" title="E67">A38</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/><a NAME="E12:63_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A43</font></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="c12">p</font>,<font color="Maroon" title="c26">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T36" target="_self" onmouseover="rs('analmetr/T36')" onmouseout="rh()">Th36</a></span>;<br/><a NAME="E13:63_1_1_3"/><span class="lab"><font color="Green" title="E73">A44</font></span>: 
<font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c21">s9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c25">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:63_1_1_3"><span class="lab"><font color="Green" title="E71">A42</font></span></a>, <a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/><a NAME="E14:63_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c23">x</font>,<font color="Maroon" title="c15">s</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c23">x</font>,<font color="Maroon" title="c26">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T36" target="_self" onmouseover="rs('analmetr/T36')" onmouseout="rh()">Th36</a></span>;<br/><a NAME="E15:63_1_1_3"/><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="c23">x</font>,<font color="Maroon" title="c26">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:63_1_1"><span class="lab"><font color="Green" title="E59">A14</font></span></a>, <a class="txt" href="#E4:63_1_1_3"><span class="lab"><font color="Green" title="E69">A40</font></span></a>, <a class="ref" href="analmetr.html#D8" target="_self" onmouseover="rs('analmetr/D8')" onmouseout="rh()">Def7</a></span>;<br/><a NAME="E16:63_1_1_3"/><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="c12">p</font>,<font color="Maroon" title="c26">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:63_1_1"><span class="lab"><font color="Green" title="E65">A36</font></span></a>, <a class="ref" href="analmetr.html#D8" target="_self" onmouseover="rs('analmetr/D8')" onmouseout="rh()">Def7</a></span>;<br/><a NAME="E17:63_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c26">z</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c26">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:63_1_1"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="txt" href="#E12:63_1_1_3"><span class="lab"><font color="Green" title="E72">A43</font></span></a>, <a class="ref" href="analmetr.html#D8" target="_self" onmouseover="rs('analmetr/D8')" onmouseout="rh()">Def7</a></span>;<br/><a NAME="E18:63_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c21">s9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c18">p9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:63_1_1_3"><span class="lab"><font color="Green" title="E73">A44</font></span></a>, <a class="ref" href="analmetr.html#D8" target="_self" onmouseover="rs('analmetr/D8')" onmouseout="rh()">Def7</a></span>;<br/><a NAME="E19:63_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A45</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c21">s9</font>,<font color="Maroon" title="c18">p9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/><a NAME="E20:63_1_1_3"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c21">s9</font>,<font color="Maroon" title="c22">x9</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c18">p9</font>,<font color="Maroon" title="c19">q9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:63_1_1"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="ref" href="aff_1.html#T6" onmouseover="rs('aff_1/T6')" onmouseout="rh()">AFF_1:6</a>, <a class="ref" href="aff_1.html#T7" onmouseover="rs('aff_1/T7')" onmouseout="rh()">AFF_1:7</a></span>;<br/><a NAME="E21:63_1_1_3"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c21">s9</font>,<font color="Maroon" title="c19">q9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_1_1_3"><span class="lab"><font color="Green" title="E68">A39</font></span></a>, <a class="txt" href="#E19:63_1_1_3"><span class="lab"><font color="Green" title="E74">A45</font></span></a>, <a class="ref" href="aff_1.html#T11" onmouseover="rs('aff_1/T11')" onmouseout="rh()">AFF_1:11</a></span>;<br/><a NAME="E22:63_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c21">s9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c18">p9</font>,<font color="Maroon" title="c19">q9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:63_1_1_3"><span class="lab"><font color="Green" title="E74">A45</font></span></a>, <a class="ref" href="aff_1.html#T10" onmouseover="rs('aff_1/T10')" onmouseout="rh()">AFF_1:10</a></span>;<br/><a NAME="E23:63_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c18">p9</font>,<font color="Maroon" title="c19">q9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">r9</font>,<font color="Maroon" title="c21">s9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:63_1_1"><span class="lab"><font color="Green" title="E58">A13</font></span></a>, <a class="txt" href="#E4:63_1_1_3"><span class="lab"><font color="Green" title="E69">A40</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E24:63_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:63_1_1"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="ref" href="analmetr.html#T36" target="_self" onmouseover="rs('analmetr/T36')" onmouseout="rh()">Th36</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_1_4"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">p</font> implies  not <font color="Maroon" title="c23">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">r</font> )</span></a><div class="add"><span class="kw">consider </span><font color="Maroon" title="c24">y9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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> #)<span class="kw"> such that </span><br/><a NAME="E2:63_1_1_4"/><span class="lab"><font color="Green" title="E67">A46</font></span>: 
( <font color="Maroon" title="c16">a9</font>,<font color="Maroon" title="c17">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c18">p9</font>,<font color="Maroon" title="c24">y9</font> &amp; <font color="Maroon" title="c18">p9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">y9</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T40" onmouseover="rs('diraf/T40')" onmouseout="rh()">DIRAF:40</a></span>;<br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:63_1_1_4"/><span class="lab"><font color="Green" title="E68">A47</font></span>: 
<font color="Maroon" title="c23">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">p</font>
 <span class="kw">and </span><br/><a NAME="E4:63_1_1_4"/><span class="lab"><font color="Green" title="E69">A48</font></span>: 
<font color="Maroon" title="c23">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">r</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E5:63_1_1_4"/>
 not <font color="Maroon" title="c18">p9</font>,<font color="Maroon" title="c24">y9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c20">r9</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_1_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:63_1_1_4_1"/>
<font color="Maroon" title="c18">p9</font>,<font color="Maroon" title="c24">y9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c20">r9</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:63_1_1_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">p9</font>,<font color="Maroon" title="c24">y9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">r9</font>,<font color="Maroon" title="c21">s9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:63_1_1"><span class="lab"><font color="Green" title="E62">A25</font></span></a>, <a class="txt" href="#E4:63_1_1_4"><span class="lab"><font color="Green" title="E69">A48</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/>
<a NAME="E3:63_1_1_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">r9</font>,<font color="Maroon" title="c21">s9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c16">a9</font>,<font color="Maroon" title="c17">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63_1_1_4"><span class="lab"><font color="Green" title="E67">A46</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/>
<a NAME="E4:63_1_1_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">r</font>,<font color="Maroon" title="c15">s</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</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#T36" target="_self" onmouseover="rs('analmetr/T36')" onmouseout="rh()">Th36</a></span>;<br/>
<a NAME="E5:63_1_1_4_1"/><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="c8">a</font>,<font color="Maroon" title="c9">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:63_1_1"><span class="lab"><font color="Green" title="E57">A12</font></span></a>, <a class="txt" href="#E5:63_1_1"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="ref" href="analmetr.html#D8" target="_self" onmouseover="rs('analmetr/D8')" onmouseout="rh()">Def7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:63_1_1_4_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:63_1_1"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="ref" href="analmetr.html#D8" target="_self" onmouseover="rs('analmetr/D8')" onmouseout="rh()">Def7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">z9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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> #)<span class="kw"> such that </span><br/><a NAME="E7:63_1_1_4"/><span class="lab"><font color="Green" title="E70">A49</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c18">p9</font>,<font color="Maroon" title="c24">y9</font>,<font color="Maroon" title="c25">z9</font>
 <span class="kw">and </span><br/><a NAME="E8:63_1_1_4"/><span class="lab"><font color="Green" title="E71">A50</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c20">r9</font>,<font color="Maroon" title="c25">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="aff_1.html#T60" onmouseover="rs('aff_1/T60')" onmouseout="rh()">AFF_1:60</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c26">z</font> = <font color="Maroon" title="c25">z9</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="E10:63_1_1_4"/>
<font color="Maroon" title="c18">p9</font>,<font color="Maroon" title="c24">y9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c18">p9</font>,<font color="Maroon" title="c25">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:63_1_1_4"><span class="lab"><font color="Green" title="E70">A49</font></span></a>, <a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/><a NAME="E11:63_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c16">a9</font>,<font color="Maroon" title="c17">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c18">p9</font>,<font color="Maroon" title="c25">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63_1_1_4"><span class="lab"><font color="Green" title="E67">A46</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/><a NAME="E12:63_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A51</font></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="c12">p</font>,<font color="Maroon" title="c26">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T36" target="_self" onmouseover="rs('analmetr/T36')" onmouseout="rh()">Th36</a></span>;<br/><a NAME="E13:63_1_1_4"/><span class="lab"><font color="Green" title="E73">A52</font></span>: 
<font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c20">r9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c25">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:63_1_1_4"><span class="lab"><font color="Green" title="E71">A50</font></span></a>, <a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/><a NAME="E14:63_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c23">x</font>,<font color="Maroon" title="c14">r</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c23">x</font>,<font color="Maroon" title="c26">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T36" target="_self" onmouseover="rs('analmetr/T36')" onmouseout="rh()">Th36</a></span>;<br/><a NAME="E15:63_1_1_4"/><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="c23">x</font>,<font color="Maroon" title="c26">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:63_1_1"><span class="lab"><font color="Green" title="E63">A26</font></span></a>, <a class="txt" href="#E4:63_1_1_4"><span class="lab"><font color="Green" title="E69">A48</font></span></a>, <a class="ref" href="analmetr.html#D8" target="_self" onmouseover="rs('analmetr/D8')" onmouseout="rh()">Def7</a></span>;<br/><a NAME="E16:63_1_1_4"/><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="c12">p</font>,<font color="Maroon" title="c26">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:63_1_1"><span class="lab"><font color="Green" title="E65">A36</font></span></a>, <a class="ref" href="analmetr.html#D8" target="_self" onmouseover="rs('analmetr/D8')" onmouseout="rh()">Def7</a></span>;<br/><a NAME="E17:63_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c26">z</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c26">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:63_1_1"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="txt" href="#E12:63_1_1_4"><span class="lab"><font color="Green" title="E72">A51</font></span></a>, <a class="ref" href="analmetr.html#D8" target="_self" onmouseover="rs('analmetr/D8')" onmouseout="rh()">Def7</a></span>;<br/><a NAME="E18:63_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c20">r9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c18">p9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:63_1_1_4"><span class="lab"><font color="Green" title="E73">A52</font></span></a>, <a class="ref" href="analmetr.html#D8" target="_self" onmouseover="rs('analmetr/D8')" onmouseout="rh()">Def7</a></span>;<br/><a NAME="E19:63_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A53</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c20">r9</font>,<font color="Maroon" title="c18">p9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/><a NAME="E20:63_1_1_4"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c20">r9</font>,<font color="Maroon" title="c22">x9</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c18">p9</font>,<font color="Maroon" title="c19">q9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:63_1_1"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="ref" href="aff_1.html#T6" onmouseover="rs('aff_1/T6')" onmouseout="rh()">AFF_1:6</a>, <a class="ref" href="aff_1.html#T7" onmouseover="rs('aff_1/T7')" onmouseout="rh()">AFF_1:7</a></span>;<br/><a NAME="E21:63_1_1_4"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c20">r9</font>,<font color="Maroon" title="c19">q9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_1_1_4"><span class="lab"><font color="Green" title="E68">A47</font></span></a>, <a class="txt" href="#E19:63_1_1_4"><span class="lab"><font color="Green" title="E74">A53</font></span></a>, <a class="ref" href="aff_1.html#T11" onmouseover="rs('aff_1/T11')" onmouseout="rh()">AFF_1:11</a></span>;<br/><a NAME="E22:63_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c22">x9</font>,<font color="Maroon" title="c20">r9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c18">p9</font>,<font color="Maroon" title="c19">q9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:63_1_1_4"><span class="lab"><font color="Green" title="E74">A53</font></span></a>, <a class="ref" href="aff_1.html#T10" onmouseover="rs('aff_1/T10')" onmouseout="rh()">AFF_1:10</a></span>;<br/><a NAME="E23:63_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c18">p9</font>,<font color="Maroon" title="c19">q9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">r9</font>,<font color="Maroon" title="c21">s9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:63_1_1"><span class="lab"><font color="Green" title="E62">A25</font></span></a>, <a class="txt" href="#E4:63_1_1_4"><span class="lab"><font color="Green" title="E69">A48</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E24:63_1_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:63_1_1"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="ref" href="analmetr.html#T36" target="_self" onmouseover="rs('analmetr/T36')" onmouseout="rh()">Th36</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E38:63_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:63_1_1"><span class="lab"><font color="Green" title="E53">A8</font></span></a>, <a class="txt" href="#E36:63_1_1"><span class="lab"><font color="Green" title="E66">A37</font></span></a>, <a class="txt" href="#E31:63_1_1"><span class="lab"><font color="Green" title="E64">A27</font></span></a>, <a class="txt" href="#E25:63_1_1"><span class="lab"><font color="Green" title="E61">A16</font></span></a>, <a class="ref" href="aff_1.html#T3" onmouseover="rs('aff_1/T3')" onmouseout="rh()">AFF_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E3:63"/>
 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#D8" target="_self" onmouseover="rs('analmetr/D8')" onmouseout="rh()">Def7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:63"/>
<font color="Maroon" title="c7">POS</font> is   <a href="analmetr.html#NM2" title="ANALMETR:NM.2">OrtAfPl</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E2:63"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="ref" href="analmetr.html#D9" target="_self" onmouseover="rs('analmetr/D9')" onmouseout="rh()">Def8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
