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<div class="add">

<b>let </b><font color="Maroon" title="c7">POS</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="analmetr.html#L1" title="ANALMETR:struct.1">ParOrtStr</a> ;<br/>

<b>set </b><font color="Maroon" title="c8">P</font> =  <a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c7">POS</font>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:64_1"/><i><font color="Green" title="E51">A1</font></i>: 
<font color="Maroon" title="c7">POS</font> is <a href="analmetr.html#V3" title="ANALMETR:attr.3">OrtAfPl-like</a>
 ;<br/><a NAME="E2:64_1"/><b>then </b><i><font color="Green" title="E52">A2</font></i>: 
 <a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c7">POS</font> is   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</a>
 <b>by </b><i><a class="ref" href="analmetr.html#D10" target="_self" title="ANALMETR:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E3:64_1"/>
 ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font> st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font>
 <b>by </b><i><a class="ref" href="diraf.html#T54" title="DIRAF:th.54">DIRAF:54</a></i>;<br/><b>let </b><font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c10">b</font>, <font color="Maroon" title="c11">c</font>, <font color="Maroon" title="c12">d</font>, <font color="Maroon" title="c13">p</font>, <font color="Maroon" title="c14">q</font>, <font color="Maroon" title="c15">r</font>, <font color="Maroon" title="c16">s</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font>;<br/><b>reconsider </b><font color="Maroon" title="c17">a'</font> = <font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c18">b'</font> = <font color="Maroon" title="c10">b</font>, <font color="Maroon" title="c19">c'</font> = <font color="Maroon" title="c11">c</font>, <font color="Maroon" title="c20">d'</font> = <font color="Maroon" title="c12">d</font>, <font color="Maroon" title="c21">p'</font> = <font color="Maroon" title="c13">p</font>, <font color="Maroon" title="c22">q'</font> = <font color="Maroon" title="c14">q</font>, <font color="Maroon" title="c23">r'</font> = <font color="Maroon" title="c15">r</font>, <font color="Maroon" title="c24">s'</font> = <font color="Maroon" title="c16">s</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c7">POS</font></span>)</span> ;<br/><a NAME="E5:64_1"/>
( <font color="Maroon" title="c17">a'</font>,<font color="Maroon" title="c18">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c18">b'</font>,<font color="Maroon" title="c17">a'</font> &amp; <font color="Maroon" title="c17">a'</font>,<font color="Maroon" title="c18">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c19">c'</font>,<font color="Maroon" title="c19">c'</font> )
 <b>by </b><i><a class="txt" href="analmetr.html#E2:64_1"><i><font color="Green" title="E52">A2</font></i></a>, <a class="ref" href="diraf.html#T54" title="DIRAF:th.54">DIRAF:54</a></i>;<br/><b>hence </b><a NAME="E6:64_1"/>
( <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b</font>,<font color="Maroon" title="c9">a</font> &amp; <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c11">c</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T48" target="_self" title="ANALMETR:th.48">Th48</a></i>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:64_1_1"/>
( <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> &amp; <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c15">r</font>,<font color="Maroon" title="c16">s</font> )
 ;<br/><a NAME="E2:64_1_1"/><b>then </b>
( <font color="Maroon" title="c17">a'</font>,<font color="Maroon" title="c18">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c21">p'</font>,<font color="Maroon" title="c22">q'</font> &amp; <font color="Maroon" title="c17">a'</font>,<font color="Maroon" title="c18">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c23">r'</font>,<font color="Maroon" title="c24">s'</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T48" target="_self" title="ANALMETR:th.48">Th48</a></i>;<br/><a NAME="E3:64_1_1"/><b>then </b>
( <font color="Maroon" title="c21">p'</font>,<font color="Maroon" title="c22">q'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c23">r'</font>,<font color="Maroon" title="c24">s'</font> or <font color="Maroon" title="c17">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b'</font> )
 <b>by </b><i><a class="txt" href="analmetr.html#E2:64_1"><i><font color="Green" title="E52">A2</font></i></a>, <a class="ref" href="diraf.html#T54" title="DIRAF:th.54">DIRAF:54</a></i>;<br/><b>hence </b><a NAME="E4:64_1_1"/>
( <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c15">r</font>,<font color="Maroon" title="c16">s</font> or <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T48" target="_self" title="ANALMETR:th.48">Th48</a></i>;<br/></div>
<b>end;</b></div><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:64_1_2"/>
<font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c11">c</font>
 ;<br/><a NAME="E2:64_1_2"/><b>then </b>
<font color="Maroon" title="c17">a'</font>,<font color="Maroon" title="c18">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c17">a'</font>,<font color="Maroon" title="c19">c'</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T48" target="_self" title="ANALMETR:th.48">Th48</a></i>;<br/><a NAME="E3:64_1_2"/><b>then </b>
<font color="Maroon" title="c18">b'</font>,<font color="Maroon" title="c17">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c18">b'</font>,<font color="Maroon" title="c19">c'</font>
 <b>by </b><i><a class="txt" href="analmetr.html#E2:64_1"><i><font color="Green" title="E52">A2</font></i></a>, <a class="ref" href="diraf.html#T54" title="DIRAF:th.54">DIRAF:54</a></i>;<br/><b>hence </b><a NAME="E4:64_1_2"/>
<font color="Maroon" title="c10">b</font>,<font color="Maroon" title="c9">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b</font>,<font color="Maroon" title="c11">c</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T48" target="_self" title="ANALMETR:th.48">Th48</a></i>;<br/></div>
<b>end;</b></div><b>consider </b><font color="Maroon" title="c25">x'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c7">POS</font></span>)</span><b> such that </b><br/><a NAME="E10:64_1"/><i><font color="Green" title="E53">A3</font></i>: 
( <font color="Maroon" title="c17">a'</font>,<font color="Maroon" title="c18">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c19">c'</font>,<font color="Maroon" title="c25">x'</font> &amp; <font color="Maroon" title="c17">a'</font>,<font color="Maroon" title="c19">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c18">b'</font>,<font color="Maroon" title="c25">x'</font> )
 <b>by </b><i><a class="txt" href="analmetr.html#E2:64_1"><i><font color="Green" title="E52">A2</font></i></a>, <a class="ref" href="diraf.html#T54" title="DIRAF:th.54">DIRAF:54</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c26">x</font> = <font color="Maroon" title="c25">x'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">POS</font> ;<br/><a NAME="E12:64_1"/>
( <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c26">x</font> &amp; <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c11">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b</font>,<font color="Maroon" title="c26">x</font> )
 <b>by </b><i><a class="txt" href="analmetr.html#E10:64_1"><i><font color="Green" title="E53">A3</font></i></a>, <a class="ref" href="analmetr.html#T48" target="_self" title="ANALMETR:th.48">Th48</a></i>;<br/><b>hence </b><a NAME="E13:64_1"/>
 ex <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font> st <br/>( <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c</font>,<font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c11">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b</font>,<font color="Olive" title="b1">x</font> )
 ;<br/><b>consider </b><font color="Maroon" title="c27">x'</font>, <font color="Maroon" title="c28">y'</font>, <font color="Maroon" title="c29">z'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c7">POS</font></span>)</span><b> such that </b><br/><a NAME="E15:64_1"/><i><font color="Green" title="E54">A4</font></i>: 
not <font color="Maroon" title="c27">x'</font>,<font color="Maroon" title="c28">y'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c27">x'</font>,<font color="Maroon" title="c29">z'</font>
 <b>by </b><i><a class="txt" href="analmetr.html#E2:64_1"><i><font color="Green" title="E52">A2</font></i></a>, <a class="ref" href="diraf.html#T54" title="DIRAF:th.54">DIRAF:54</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c30">x</font> = <font color="Maroon" title="c27">x'</font>, <font color="Maroon" title="c31">y</font> = <font color="Maroon" title="c28">y'</font>, <font color="Maroon" title="c32">z</font> = <font color="Maroon" title="c29">z'</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="E17:64_1"/>
not <font color="Maroon" title="c30">x</font>,<font color="Maroon" title="c31">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c30">x</font>,<font color="Maroon" title="c32">z</font>
 <b>by </b><i><a class="txt" href="analmetr.html#E15:64_1"><i><font color="Green" title="E54">A4</font></i></a>, <a class="ref" href="analmetr.html#T48" target="_self" title="ANALMETR:th.48">Th48</a></i>;<br/><b>hence </b><a NAME="E18:64_1"/>
not for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font> holds  <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">z</font>
 ;<br/><b>consider </b><font color="Maroon" title="c33">x'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c7">POS</font></span>)</span><b> such that </b><br/><a NAME="E20:64_1"/><i><font color="Green" title="E55">A5</font></i>: 
( <font color="Maroon" title="c17">a'</font>,<font color="Maroon" title="c18">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c19">c'</font>,<font color="Maroon" title="c33">x'</font> &amp; <font color="Maroon" title="c19">c'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c33">x'</font> )
 <b>by </b><i><a class="txt" href="analmetr.html#E2:64_1"><i><font color="Green" title="E52">A2</font></i></a>, <a class="ref" href="diraf.html#T54" title="DIRAF:th.54">DIRAF:54</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c34">x</font> = <font color="Maroon" title="c33">x'</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="E22:64_1"/>
( <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c34">x</font> &amp; <font color="Maroon" title="c11">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c34">x</font> )
 <b>by </b><i><a class="txt" href="analmetr.html#E20:64_1"><i><font color="Green" title="E55">A5</font></i></a>, <a class="ref" href="analmetr.html#T48" target="_self" title="ANALMETR:th.48">Th48</a></i>;<br/><b>hence </b><a NAME="E23:64_1"/>
 ex <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font> st <br/>( <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c</font>,<font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c11">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">x</font> )
 ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:64_1_3"/>
( <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b</font>,<font color="Maroon" title="c12">d</font> &amp; <font color="Maroon" title="c10">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">a</font> )
 ;<br/><a NAME="E2:64_1_3"/><b>then </b>
( <font color="Maroon" title="c17">a'</font>,<font color="Maroon" title="c18">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c18">b'</font>,<font color="Maroon" title="c20">d'</font> &amp; <font color="Maroon" title="c18">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">a'</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T48" target="_self" title="ANALMETR:th.48">Th48</a></i>;<br/><b>then consider </b><font color="Maroon" title="c35">x'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c7">POS</font></span>)</span><b> such that </b><br/><a NAME="E4:64_1_3"/><i><font color="Green" title="E56">A6</font></i>: 
( <font color="Maroon" title="c19">c'</font>,<font color="Maroon" title="c18">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c18">b'</font>,<font color="Maroon" title="c35">x'</font> &amp; <font color="Maroon" title="c19">c'</font>,<font color="Maroon" title="c17">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">d'</font>,<font color="Maroon" title="c35">x'</font> )
 <b>by </b><i><a class="txt" href="analmetr.html#E2:64_1"><i><font color="Green" title="E52">A2</font></i></a>, <a class="ref" href="diraf.html#T54" title="DIRAF:th.54">DIRAF:54</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c36">x</font> = <font color="Maroon" title="c35">x'</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="E6:64_1_3"/>
( <font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c10">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b</font>,<font color="Maroon" title="c36">x</font> &amp; <font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c9">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c12">d</font>,<font color="Maroon" title="c36">x</font> )
 <b>by </b><i><a class="txt" href="analmetr.html#E4:64_1_3"><i><font color="Green" title="E56">A6</font></i></a>, <a class="ref" href="analmetr.html#T48" target="_self" title="ANALMETR:th.48">Th48</a></i>;<br/><b>hence </b><a NAME="E7:64_1_3"/>
 ex <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font> st <br/>( <font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c10">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b</font>,<font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c9">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c12">d</font>,<font color="Olive" title="b1">x</font> )
 ;<br/></div>
<b>end;</b></div><b>thus </b><a NAME="E25:64_1"/>
( ( <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font> implies <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> ) &amp; <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c11">c</font> &amp; ( <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c12">d</font> implies ( <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c12">d</font>,<font color="Maroon" title="c11">c</font> &amp; <font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c12">d</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font> ) ) &amp; ( <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> &amp; <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c15">r</font>,<font color="Maroon" title="c16">s</font> &amp; not <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c15">r</font>,<font color="Maroon" title="c16">s</font> implies <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> ) &amp; ( <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> &amp; <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c15">r</font>,<font color="Maroon" title="c16">s</font> &amp; not <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c15">r</font>,<font color="Maroon" title="c16">s</font> implies <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> ) &amp;  ex <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font> st <br/>( <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c11">c</font>,<font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c11">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">x</font> ) )
 <b>by </b><i><a class="txt" href="analmetr.html#E1:64_1"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="analmetr.html#D10" target="_self" title="ANALMETR:def.10">Def10</a></i>;<br/><b>assume </b><a NAME="E26:64_1"/>
not <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c12">d</font>
 ;<br/><a NAME="E27:64_1"/><b>then </b>
not <font color="Maroon" title="c17">a'</font>,<font color="Maroon" title="c18">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c19">c'</font>,<font color="Maroon" title="c20">d'</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T48" target="_self" title="ANALMETR:th.48">Th48</a></i>;<br/><b>then consider </b><font color="Maroon" title="c35">x'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c7">POS</font></span>)</span><b> such that </b><br/><a NAME="E29:64_1"/><i><font color="Green" title="E56">A7</font></i>: 
( <font color="Maroon" title="c17">a'</font>,<font color="Maroon" title="c18">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c17">a'</font>,<font color="Maroon" title="c35">x'</font> &amp; <font color="Maroon" title="c19">c'</font>,<font color="Maroon" title="c20">d'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c19">c'</font>,<font color="Maroon" title="c35">x'</font> )
 <b>by </b><i><a class="txt" href="analmetr.html#E2:64_1"><i><font color="Green" title="E52">A2</font></i></a>, <a class="ref" href="diraf.html#T54" title="DIRAF:th.54">DIRAF:54</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c36">x</font> = <font color="Maroon" title="c35">x'</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="E31:64_1"/>
( <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c36">x</font> &amp; <font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c12">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c36">x</font> )
 <b>by </b><i><a class="txt" href="analmetr.html#E29:64_1"><i><font color="Green" title="E56">A7</font></i></a>, <a class="ref" href="analmetr.html#T48" target="_self" title="ANALMETR:th.48">Th48</a></i>;<br/><b>hence </b><a NAME="E32:64_1"/>
 ex <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font> st <br/>( <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a</font>,<font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c12">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c</font>,<font color="Olive" title="b1">x</font> )
 ;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c10">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font><b> such that </b><a NAME="E2:64"/><i><font color="Green" title="E51">A8</font></i>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">b</font>
 ;<br/>

<b>assume </b><a NAME="E3:64"/><i><font color="Green" title="E52">A9</font></i>: 
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="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b3">c</font> &amp; ( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="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="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">r</font>,<font color="Olive" title="b8">s</font> implies <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> ) &amp; ( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font> implies <font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> ) &amp;  ex <font color="Olive" title="b9">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font> st <br/>( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b9">x</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b9">x</font> ) &amp; not for <font color="Olive" title="b9">x</font>, <font color="Olive" title="b10">y</font>, <font color="Olive" title="b11">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font> holds  <font color="Olive" title="b9">x</font>,<font color="Olive" title="b10">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b9">x</font>,<font color="Olive" title="b11">z</font> &amp;  ex <font color="Olive" title="b9">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font> st <br/>( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b9">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="b9">x</font> ) &amp; ( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b4">d</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> implies  ex <font color="Olive" title="b9">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="b3">c</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b9">x</font> &amp; <font color="Olive" title="b3">c</font>,<font color="Olive" title="b1">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">d</font>,<font color="Olive" title="b9">x</font> ) ) &amp; ( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> implies <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> ) &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b3">c</font> &amp; ( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font> implies ( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b4">d</font>,<font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> ) ) &amp; ( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b5">p</font>,<font color="Olive" title="b6">q</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">r</font>,<font color="Olive" title="b8">s</font> &amp; not <font color="Olive" title="b5">p</font>,<font color="Olive" title="b6">q</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b7">r</font>,<font color="Olive" title="b8">s</font> implies <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> ) &amp; ( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b5">p</font>,<font color="Olive" title="b6">q</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b7">r</font>,<font color="Olive" title="b8">s</font> &amp; not <font color="Olive" title="b5">p</font>,<font color="Olive" title="b6">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">r</font>,<font color="Olive" title="b8">s</font> implies <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> ) &amp;  ex <font color="Olive" title="b9">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="b9">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="b9">x</font> ) &amp; ( not <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font> implies  ex <font color="Olive" title="b9">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font> st <br/>( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b9">x</font> &amp; <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b9">x</font> ) ) )
 ;<br/>

<a NAME="E4:64"/><i><font color="Green" title="E53">A10</font></i>: 
 ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c7">POS</font></span>)</span> st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font>
 
<b>by </b><i><a class="txt" href="analmetr.html#E2:64"><i><font color="Green" title="E51">A8</font></i></a></i>;<br/>
<div><i><font color="Green" title="E54">A11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c12">y</font>, <font color="Maroon" title="c13">z</font>, <font color="Maroon" title="c14">t</font>, <font color="Maroon" title="c15">u</font>, <font color="Maroon" title="c16">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c7">POS</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c17">a</font> = <font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c18">b</font> = <font color="Maroon" title="c12">y</font>, <font color="Maroon" title="c19">c</font> = <font color="Maroon" title="c13">z</font>, <font color="Maroon" title="c20">d</font> = <font color="Maroon" title="c14">t</font>, <font color="Maroon" title="c21">e</font> = <font color="Maroon" title="c15">u</font>, <font color="Maroon" title="c22">f</font> = <font color="Maroon" title="c16">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font> ;<br/><b>thus </b><a NAME="E2:64_2"/>
( <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c11">x</font> &amp; <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">z</font>,<font color="Maroon" title="c13">z</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:64_2_1"/>
( <font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c18">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c18">b</font>,<font color="Maroon" title="c17">a</font> &amp; <font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c18">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c19">c</font>,<font color="Maroon" title="c19">c</font> )
 
<b>by </b><i><a class="txt" href="analmetr.html#E3:64"><i><font color="Green" title="E52">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:64_2_1"/>
( <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c11">x</font> &amp; <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">z</font>,<font color="Maroon" title="c13">z</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T48" target="_self" title="ANALMETR:th.48">Th48</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:64_2"/>
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">y</font> &amp; <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">z</font>,<font color="Maroon" title="c14">t</font> &amp; <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">w</font> implies <font color="Maroon" title="c13">z</font>,<font color="Maroon" title="c14">t</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">w</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:64_2_2"/>
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">y</font> &amp; <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">z</font>,<font color="Maroon" title="c14">t</font> &amp; <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">w</font> )
 ;<br/>

<a NAME="E2:64_2_2"/><b>then </b>
( <font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c18">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c19">c</font>,<font color="Maroon" title="c20">d</font> &amp; <font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c18">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c21">e</font>,<font color="Maroon" title="c22">f</font> &amp; <font color="Maroon" title="c17">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">b</font> )
 
<b>by </b><i><a class="ref" href="analmetr.html#T48" target="_self" title="ANALMETR:th.48">Th48</a></i>;<br/>
<a NAME="E3:64_2_2"/><b>then </b>
<font color="Maroon" title="c19">c</font>,<font color="Maroon" title="c20">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c21">e</font>,<font color="Maroon" title="c22">f</font>
 
<b>by </b><i><a class="txt" href="analmetr.html#E3:64"><i><font color="Green" title="E52">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:64_2_2"/>
<font color="Maroon" title="c13">z</font>,<font color="Maroon" title="c14">t</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">w</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T48" target="_self" title="ANALMETR:th.48">Th48</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E4:64_2"/>
( <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font> implies <font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c11">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:64_2_3"/>
<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font>
 ;<br/>

<a NAME="E2:64_2_3"/><b>then </b>
<font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c18">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c19">c</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T48" target="_self" title="ANALMETR:th.48">Th48</a></i>;<br/>
<a NAME="E3:64_2_3"/><b>then </b>
<font color="Maroon" title="c18">b</font>,<font color="Maroon" title="c17">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c18">b</font>,<font color="Maroon" title="c19">c</font>
 
<b>by </b><i><a class="txt" href="analmetr.html#E3:64"><i><font color="Green" title="E52">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:64_2_3"/>
<font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c11">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T48" target="_self" title="ANALMETR:th.48">Th48</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E6:64"/><i><font color="Green" title="E55">A12</font></i>: 
not for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c7">POS</font></span>)</span> holds  <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_3"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c12">y</font>, <font color="Maroon" title="c13">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font><b> such that </b><br/><a NAME="E2:64_3"/><i><font color="Green" title="E56">A13</font></i>: 
not <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font>
 <b>by </b><i><a class="txt" href="analmetr.html#E3:64"><i><font color="Green" title="E52">A9</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">x'</font> = <font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c15">y'</font> = <font color="Maroon" title="c12">y</font>, <font color="Maroon" title="c16">z'</font> = <font color="Maroon" title="c13">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c7">POS</font></span>)</span> ;<br/>
<a NAME="E4:64_3"/>
not <font color="Maroon" title="c14">x'</font>,<font color="Maroon" title="c15">y'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c14">x'</font>,<font color="Maroon" title="c16">z'</font>
 
<b>by </b><i><a class="txt" href="analmetr.html#E2:64_3"><i><font color="Green" title="E56">A13</font></i></a>, <a class="ref" href="analmetr.html#T48" target="_self" title="ANALMETR:th.48">Th48</a></i>;<br/>
<b>hence </b><a NAME="E5:64_3"/>
not for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c7">POS</font></span>)</span> holds  <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">z</font>
 ;<br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E56">A14</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c12">y</font>, <font color="Maroon" title="c13">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c7">POS</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c14">x'</font> = <font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c15">y'</font> = <font color="Maroon" title="c12">y</font>, <font color="Maroon" title="c16">z'</font> = <font color="Maroon" title="c13">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font> ;<br/><b>consider </b><font color="Maroon" title="c17">t'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font><b> such that </b><br/><a NAME="E3:64_4"/><i><font color="Green" title="E57">A15</font></i>: 
( <font color="Maroon" title="c14">x'</font>,<font color="Maroon" title="c16">z'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c15">y'</font>,<font color="Maroon" title="c17">t'</font> &amp; <font color="Maroon" title="c15">y'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">t'</font> )
 <b>by </b><i><a class="txt" href="analmetr.html#E3:64"><i><font color="Green" title="E52">A9</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">t</font> = <font color="Maroon" title="c17">t'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c7">POS</font></span>)</span> ;<br/><a NAME="E5:64_4"/>
( <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c18">t</font> &amp; <font color="Maroon" title="c12">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">t</font> )
 <b>by </b><i><a class="txt" href="analmetr.html#E3:64_4"><i><font color="Green" title="E57">A15</font></i></a>, <a class="ref" href="analmetr.html#T48" target="_self" title="ANALMETR:th.48">Th48</a></i>;<br/><b>hence </b><a NAME="E6:64_4"/>
 ex <font color="Olive" title="b1">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c7">POS</font></span>)</span> st <br/>( <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c12">y</font>,<font color="Olive" title="b1">t</font> &amp; <font color="Maroon" title="c12">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">t</font> )
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E57">A16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c12">y</font>, <font color="Maroon" title="c13">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c7">POS</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c14">x'</font> = <font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c15">y'</font> = <font color="Maroon" title="c12">y</font>, <font color="Maroon" title="c16">z'</font> = <font color="Maroon" title="c13">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font> ;<br/><b>consider </b><font color="Maroon" title="c17">t'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font><b> such that </b><br/><a NAME="E3:64_5"/><i><font color="Green" title="E58">A17</font></i>: 
( <font color="Maroon" title="c14">x'</font>,<font color="Maroon" title="c15">y'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c16">z'</font>,<font color="Maroon" title="c17">t'</font> &amp; <font color="Maroon" title="c14">x'</font>,<font color="Maroon" title="c16">z'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c15">y'</font>,<font color="Maroon" title="c17">t'</font> )
 <b>by </b><i><a class="txt" href="analmetr.html#E3:64"><i><font color="Green" title="E52">A9</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">t</font> = <font color="Maroon" title="c17">t'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c7">POS</font></span>)</span> ;<br/><a NAME="E5:64_5"/>
( <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">z</font>,<font color="Maroon" title="c18">t</font> &amp; <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c18">t</font> )
 <b>by </b><i><a class="txt" href="analmetr.html#E3:64_5"><i><font color="Green" title="E58">A17</font></i></a>, <a class="ref" href="analmetr.html#T48" target="_self" title="ANALMETR:th.48">Th48</a></i>;<br/><b>hence </b><a NAME="E6:64_5"/>
 ex <font color="Olive" title="b1">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c7">POS</font></span>)</span> st <br/>( <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">z</font>,<font color="Olive" title="b1">t</font> &amp; <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c12">y</font>,<font color="Olive" title="b1">t</font> )
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E58">A18</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c12">y</font>, <font color="Maroon" title="c13">z</font>, <font color="Maroon" title="c14">t</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c7">POS</font></span>)</span>;<br/><b>assume </b><a NAME="E1:64_6"/><i><font color="Green" title="E59">A19</font></i>: 
( <font color="Maroon" title="c13">z</font>,<font color="Maroon" title="c11">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c14">t</font> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">z</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c15">x'</font> = <font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c16">y'</font> = <font color="Maroon" title="c12">y</font>, <font color="Maroon" title="c17">z'</font> = <font color="Maroon" title="c13">z</font>, <font color="Maroon" title="c18">t'</font> = <font color="Maroon" title="c14">t</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">POS</font> ;<br/><a NAME="E3:64_6"/>
( <font color="Maroon" title="c17">z'</font>,<font color="Maroon" title="c15">x'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c15">x'</font>,<font color="Maroon" title="c18">t'</font> &amp; <font color="Maroon" title="c15">x'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">z'</font> )
 <b>by </b><i><a class="txt" href="analmetr.html#E1:64_6"><i><font color="Green" title="E59">A19</font></i></a>, <a class="ref" href="analmetr.html#T48" target="_self" title="ANALMETR:th.48">Th48</a></i>;<br/><b>then consider </b><font color="Maroon" title="c19">u'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font><b> such that </b><br/><a NAME="E5:64_6"/><i><font color="Green" title="E60">A20</font></i>: 
( <font color="Maroon" title="c16">y'</font>,<font color="Maroon" title="c15">x'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c15">x'</font>,<font color="Maroon" title="c19">u'</font> &amp; <font color="Maroon" title="c16">y'</font>,<font color="Maroon" title="c17">z'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c18">t'</font>,<font color="Maroon" title="c19">u'</font> )
 <b>by </b><i><a class="txt" href="analmetr.html#E3:64"><i><font color="Green" title="E52">A9</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">u</font> = <font color="Maroon" title="c19">u'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c7">POS</font></span>)</span> ;<br/><a NAME="E7:64_6"/>
( <font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c11">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c20">u</font> &amp; <font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c14">t</font>,<font color="Maroon" title="c20">u</font> )
 <b>by </b><i><a class="txt" href="analmetr.html#E5:64_6"><i><font color="Green" title="E60">A20</font></i></a>, <a class="ref" href="analmetr.html#T48" target="_self" title="ANALMETR:th.48">Th48</a></i>;<br/><b>hence </b><a NAME="E8:64_6"/>
 ex <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c7">POS</font></span>)</span> st <br/>( <font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c11">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">x</font>,<font color="Olive" title="b1">u</font> &amp; <font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c14">t</font>,<font color="Olive" title="b1">u</font> )
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c12">y</font>, <font color="Maroon" title="c13">z</font>, <font color="Maroon" title="c14">t</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c7">POS</font></span>)</span>;<br/><b>assume </b><a NAME="E1:64_7"/><i><font color="Green" title="E59">A21</font></i>: 
not <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">z</font>,<font color="Maroon" title="c14">t</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c15">x'</font> = <font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c16">y'</font> = <font color="Maroon" title="c12">y</font>, <font color="Maroon" title="c17">z'</font> = <font color="Maroon" title="c13">z</font>, <font color="Maroon" title="c18">t'</font> = <font color="Maroon" title="c14">t</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">POS</font> ;<br/><a NAME="E3:64_7"/>
not <font color="Maroon" title="c15">x'</font>,<font color="Maroon" title="c16">y'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c17">z'</font>,<font color="Maroon" title="c18">t'</font>
 <b>by </b><i><a class="txt" href="analmetr.html#E1:64_7"><i><font color="Green" title="E59">A21</font></i></a>, <a class="ref" href="analmetr.html#T48" target="_self" title="ANALMETR:th.48">Th48</a></i>;<br/><b>then consider </b><font color="Maroon" title="c19">u'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">POS</font><b> such that </b><br/><a NAME="E5:64_7"/><i><font color="Green" title="E60">A22</font></i>: 
( <font color="Maroon" title="c15">x'</font>,<font color="Maroon" title="c16">y'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c15">x'</font>,<font color="Maroon" title="c19">u'</font> &amp; <font color="Maroon" title="c17">z'</font>,<font color="Maroon" title="c18">t'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c17">z'</font>,<font color="Maroon" title="c19">u'</font> )
 <b>by </b><i><a class="txt" href="analmetr.html#E3:64"><i><font color="Green" title="E52">A9</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">u</font> = <font color="Maroon" title="c19">u'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c7">POS</font></span>)</span> ;<br/><a NAME="E7:64_7"/>
( <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c20">u</font> &amp; <font color="Maroon" title="c13">z</font>,<font color="Maroon" title="c14">t</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">z</font>,<font color="Maroon" title="c20">u</font> )
 <b>by </b><i><a class="txt" href="analmetr.html#E5:64_7"><i><font color="Green" title="E60">A22</font></i></a>, <a class="ref" href="analmetr.html#T48" target="_self" title="ANALMETR:th.48">Th48</a></i>;<br/><b>hence </b><a NAME="E8:64_7"/>
 ex <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c7">POS</font></span>)</span> st <br/>( <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">x</font>,<font color="Olive" title="b1">u</font> &amp; <font color="Maroon" title="c13">z</font>,<font color="Maroon" title="c14">t</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">z</font>,<font color="Olive" title="b1">u</font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:64"/>
 <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>
 <b>by </b><i><a class="txt" href="analmetr.html#E4:64"><i><font color="Green" title="E53">A10</font></i></a>, <a class="txt" href="analmetr.html#E5:64"><i><font color="Green" title="E54">A11</font></i></a>, <a class="txt" href="analmetr.html#E6:64"><i><font color="Green" title="E55">A12</font></i></a>, <a class="txt" href="analmetr.html#E7:64"><i><font color="Green" title="E56">A14</font></i></a>, <a class="txt" href="analmetr.html#E8:64"><i><font color="Green" title="E57">A16</font></i></a>, <a class="txt" href="analmetr.html#E9:64"><i><font color="Green" title="E58">A18</font></i></a>, <a class="ref" href="diraf.html#T54" title="DIRAF:th.54">DIRAF:54</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="analmetr.html#D10" target="_self" title="ANALMETR:def.10">ANALMETR:def 10</a><br/>

<b>thus </b><a NAME="E12:64"/>
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> ) )
 <b>by </b><i><a class="txt" href="analmetr.html#E3:64"><i><font color="Green" title="E52">A9</font></i></a></i>;<br/>

<b>thus </b><a NAME="E13:64"/>
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> )
 <b>by </b><i><a class="txt" href="analmetr.html#E3:64"><i><font color="Green" title="E52">A9</font></i></a></i>;<br/>


</div>
