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<div class="add">

<b>let </b><font color="Maroon">POS</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="analmetr.html#L1" target="_self">ParOrtStr</a> ;<br/>

<b>set </b><font color="Maroon">P</font> =  <a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:64_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">POS</font> is <a href="analmetr.html#V3" target="_self">OrtAfPl-like</a>
 ;<br/><a NAME="E2:64_1"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font> is   <a href="diraf.html#NM2">AffinPlane</a>
 <b>by </b><i><a class="txt" href="analmetr.html#E10:64_1"><i><font color="Green">E46</font></i></a></i>;<br/><b>hence </b><a NAME="E3:64_1"/>
 ex <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font> st <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font>
 <b>by </b><i><a class="ref" href="diraf.html#T54">DIRAF:54</a></i>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font>, <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font>, <font color="Maroon">d</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font>, <font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font>, <font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font>, <font color="Maroon">r</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font>, <font color="Maroon">s</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font>;<br/><b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</font>, <font color="Maroon">c'</font> = <font color="Maroon">c</font>, <font color="Maroon">d'</font> = <font color="Maroon">d</font>, <font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">q'</font> = <font color="Maroon">q</font>, <font color="Maroon">r'</font> = <font color="Maroon">r</font>, <font color="Maroon">s'</font> = <font color="Maroon">s</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span> ;<br/><a NAME="E5:64_1"/>
( <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b'</font>,<font color="Maroon">a'</font> &amp; <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c'</font>,<font color="Maroon">c'</font> )
 <b>by </b><i><a class="txt" href="analmetr.html#E20:64_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="diraf.html#T54">DIRAF:54</a></i>;<br/><b>hence </b><a NAME="E6:64_1"/>
( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">c</font> )
 <b>by </b><i/>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:64_1_1"/>
( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> )
 ;<br/><a NAME="E2:64_1_1"/><b>then </b>
( <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font> &amp; <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">r'</font>,<font color="Maroon">s'</font> )
 <b>by </b><i/>;<br/><a NAME="E3:64_1_1"/><b>then </b>
( <font color="Maroon">p'</font>,<font color="Maroon">q'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">r'</font>,<font color="Maroon">s'</font> or <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font> )
 <b>by </b><i><a class="txt" href="analmetr.html#E20:64_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="diraf.html#T54">DIRAF:54</a></i>;<br/><b>hence </b><a NAME="E4:64_1_1"/>
( <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><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">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>
 ;<br/><a NAME="E2:64_1_2"/><b>then </b>
<font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a'</font>,<font color="Maroon">c'</font>
 <b>by </b><i/>;<br/><a NAME="E3:64_1_2"/><b>then </b>
<font color="Maroon">b'</font>,<font color="Maroon">a'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font>
 <b>by </b><i><a class="txt" href="analmetr.html#E20:64_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="diraf.html#T54">DIRAF:54</a></i>;<br/><b>hence </b><a NAME="E4:64_1_2"/>
<font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div><b>consider </b><font color="Maroon">x'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span><b> such that </b><br/><a NAME="E10:64_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c'</font>,<font color="Maroon">x'</font> &amp; <font color="Maroon">a'</font>,<font color="Maroon">c'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b'</font>,<font color="Maroon">x'</font> )
 <b>by </b><i><a class="txt" href="analmetr.html#E20:64_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="diraf.html#T54">DIRAF:54</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x'</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">POS</font> ;<br/><a NAME="E12:64_1"/>
( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">x</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">x</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E13:64_1"/>
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font> st <br/>( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Olive">x</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Olive">x</font> )
 ;<br/><b>consider </b><font color="Maroon">x'</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span>, <font color="Maroon">y'</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span>, <font color="Maroon">z'</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span><b> such that </b><br/><a NAME="E15:64_1"/><i><font color="Green">E51</font></i>: 
not <font color="Maroon">x'</font>,<font color="Maroon">y'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x'</font>,<font color="Maroon">z'</font>
 <b>by </b><i><a class="txt" href="analmetr.html#E20:64_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="diraf.html#T54">DIRAF:54</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x'</font>, <font color="Maroon">y</font> = <font color="Maroon">y'</font>, <font color="Maroon">z</font> = <font color="Maroon">z'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font> ;<br/><a NAME="E17:64_1"/>
not <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="analmetr.html#E12"><i><font color="Green">Lemma54</font></i></a>, </i>;<br/><b>hence </b><a NAME="E18:64_1"/>
not for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font> holds  <font color="Olive">x</font>,<font color="Olive">y</font> <a href="analoaf.html#R2">//</a> <font color="Olive">x</font>,<font color="Olive">z</font>
 ;<br/><b>consider </b><font color="Maroon">x'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span><b> such that </b><br/><a NAME="E20:64_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c'</font>,<font color="Maroon">x'</font> &amp; <font color="Maroon">c'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x'</font> )
 <b>by </b><i><a class="txt" href="analmetr.html#E20:64_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="diraf.html#T54">DIRAF:54</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font> ;<br/><a NAME="E22:64_1"/>
( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">x</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T7" target="_self">Th7</a>, </i>;<br/><b>hence </b><a NAME="E23:64_1"/>
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font> st <br/>( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Olive">x</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">x</font> )
 ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:64_1_3"/>
( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">d</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> )
 ;<br/><a NAME="E2:64_1_3"/><b>then </b>
( <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b'</font>,<font color="Maroon">d'</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a'</font> )
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">x'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span><b> such that </b><br/><a NAME="E4:64_1_3"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c'</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b'</font>,<font color="Maroon">x'</font> &amp; <font color="Maroon">c'</font>,<font color="Maroon">a'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">d'</font>,<font color="Maroon">x'</font> )
 <b>by </b><i><a class="txt" href="analmetr.html#E20:64_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="diraf.html#T54">DIRAF:54</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font> ;<br/><a NAME="E6:64_1_3"/>
( <font color="Maroon">c</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">x</font> &amp; <font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">d</font>,<font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T8" target="_self">Th8</a>, </i>;<br/><b>hence </b><a NAME="E7:64_1_3"/>
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font> st <br/>( <font color="Maroon">c</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Olive">x</font> &amp; <font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">d</font>,<font color="Olive">x</font> )
 ;<br/></div>
<b>end;</b></div><b>thus </b><a NAME="E25:64_1"/>
( ( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> implies <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> ) &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">c</font>,<font color="Maroon">c</font> &amp; ( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">c</font>,<font color="Maroon">d</font> implies ( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">d</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> ) ) &amp; ( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> &amp; not <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> implies <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> ) &amp; ( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> &amp; not <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> implies <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> ) &amp;  ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font> st <br/>( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Maroon">c</font>,<font color="Olive">x</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">x</font> ) )
 <b>by </b><i><a class="txt" href="analmetr.html#E15:64_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="analmetr.html#E10:64_1"><i><font color="Green">E46</font></i></a></i>;<br/><b>assume </b><a NAME="E26:64_1"/>
not <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">d</font>
 ;<br/><a NAME="E27:64_1"/><b>then </b>
not <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c'</font>,<font color="Maroon">d'</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">x'</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span><b> such that </b><br/><a NAME="E29:64_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a'</font>,<font color="Maroon">x'</font> &amp; <font color="Maroon">c'</font>,<font color="Maroon">d'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c'</font>,<font color="Maroon">x'</font> )
 <b>by </b><i><a class="txt" href="analmetr.html#E20:64_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="diraf.html#T54">DIRAF:54</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font> ;<br/><a NAME="E31:64_1"/>
( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">x</font> &amp; <font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T8" target="_self">Th8</a>, </i>;<br/><b>hence </b><a NAME="E32:64_1"/>
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font> st <br/>( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Olive">x</font> &amp; <font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Olive">x</font> )
 ;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font>, <font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font><b> such that </b><a NAME="E2:64"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/>

<b>assume </b><a NAME="E3:64"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">d</font>, <font color="Olive">p</font>, <font color="Olive">q</font>, <font color="Olive">r</font>, <font color="Olive">s</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font> holds <br/> ( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">b</font>,<font color="Olive">a</font> &amp; <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">c</font>,<font color="Olive">c</font> &amp; ( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">p</font>,<font color="Olive">q</font> &amp; <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">r</font>,<font color="Olive">s</font> &amp; not <font color="Olive">p</font>,<font color="Olive">q</font> <a href="analoaf.html#R2">//</a> <font color="Olive">r</font>,<font color="Olive">s</font> implies <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> ) &amp; ( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">a</font>,<font color="Olive">c</font> implies <font color="Olive">b</font>,<font color="Olive">a</font> <a href="analoaf.html#R2">//</a> <font color="Olive">b</font>,<font color="Olive">c</font> ) &amp;  ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font> st <br/>( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">c</font>,<font color="Olive">x</font> &amp; <font color="Olive">a</font>,<font color="Olive">c</font> <a href="analoaf.html#R2">//</a> <font color="Olive">b</font>,<font color="Olive">x</font> ) &amp; not for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font> holds  <font color="Olive">x</font>,<font color="Olive">y</font> <a href="analoaf.html#R2">//</a> <font color="Olive">x</font>,<font color="Olive">z</font> &amp;  ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font> st <br/>( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">c</font>,<font color="Olive">x</font> &amp; <font color="Olive">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">x</font> ) &amp; ( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">b</font>,<font color="Olive">d</font> &amp; <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">a</font> implies  ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font> st <br/>( <font color="Olive">c</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">b</font>,<font color="Olive">x</font> &amp; <font color="Olive">c</font>,<font color="Olive">a</font> <a href="analoaf.html#R2">//</a> <font color="Olive">d</font>,<font color="Olive">x</font> ) ) &amp; ( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">a</font>,<font color="Olive">b</font> implies <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> ) &amp; <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">c</font>,<font color="Olive">c</font> &amp; ( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">c</font>,<font color="Olive">d</font> implies ( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">d</font>,<font color="Olive">c</font> &amp; <font color="Olive">c</font>,<font color="Olive">d</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">a</font>,<font color="Olive">b</font> ) ) &amp; ( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">p</font>,<font color="Olive">q</font> &amp; <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">r</font>,<font color="Olive">s</font> &amp; not <font color="Olive">p</font>,<font color="Olive">q</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">r</font>,<font color="Olive">s</font> implies <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> ) &amp; ( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">p</font>,<font color="Olive">q</font> &amp; <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">r</font>,<font color="Olive">s</font> &amp; not <font color="Olive">p</font>,<font color="Olive">q</font> <a href="analoaf.html#R2">//</a> <font color="Olive">r</font>,<font color="Olive">s</font> implies <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> ) &amp;  ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font> st <br/>( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">c</font>,<font color="Olive">x</font> &amp; <font color="Olive">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">x</font> ) &amp; ( not <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">c</font>,<font color="Olive">d</font> implies  ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font> st <br/>( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">a</font>,<font color="Olive">x</font> &amp; <font color="Olive">c</font>,<font color="Olive">d</font> <a href="analoaf.html#R2">//</a> <font color="Olive">c</font>,<font color="Olive">x</font> ) ) )
 ;<br/>

<a NAME="E4:64"/><i><font color="Green">E65</font></i>: 
 ex <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span> st <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font>
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E66</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">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span>, <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span>, <font color="Maroon">t</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span>, <font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span>, <font color="Maroon">w</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font>, <font color="Maroon">b</font> = <font color="Maroon">y</font>, <font color="Maroon">c</font> = <font color="Maroon">z</font>, <font color="Maroon">d</font> = <font color="Maroon">t</font>, <font color="Maroon">e</font> = <font color="Maroon">u</font>, <font color="Maroon">f</font> = <font color="Maroon">w</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font> ;<br/><b>thus </b><a NAME="E2:64_2"/>
( <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">y</font>,<font color="Maroon">x</font> &amp; <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">z</font>,<font color="Maroon">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">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">c</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:64_2_1"/>
( <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">y</font>,<font color="Maroon">x</font> &amp; <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">z</font>,<font color="Maroon">z</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:64_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">z</font>,<font color="Maroon">t</font> &amp; <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">u</font>,<font color="Maroon">w</font> implies <font color="Maroon">z</font>,<font color="Maroon">t</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">u</font>,<font color="Maroon">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">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">z</font>,<font color="Maroon">t</font> &amp; <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">u</font>,<font color="Maroon">w</font> )
 ;<br/>

<a NAME="E2:64_2_2"/><b>then </b>
( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">d</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">e</font>,<font color="Maroon">f</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:64_2_2"/><b>then </b>
<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">e</font>,<font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:64_2_2"/>
<font color="Maroon">z</font>,<font color="Maroon">t</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">u</font>,<font color="Maroon">w</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E4:64_2"/>
( <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> implies <font color="Maroon">y</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">y</font>,<font color="Maroon">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">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 ;<br/>

<a NAME="E2:64_2_3"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:64_2_3"/><b>then </b>
<font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:64_2_3"/>
<font color="Maroon">y</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">y</font>,<font color="Maroon">z</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E6:64"/><i><font color="Green">E67</font></i>: 
not for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span> holds  <font color="Olive">x</font>,<font color="Olive">y</font> <a href="analoaf.html#R2">//</a> <font color="Olive">x</font>,<font color="Olive">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">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font>, <font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font>, <font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font><b> such that </b><br/><a NAME="E2:64_3"/><i><font color="Green">E68</font></i>: 
not <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font>, <font color="Maroon">z'</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span> ;<br/>
<a NAME="E4:64_3"/>
not <font color="Maroon">x'</font>,<font color="Maroon">y'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x'</font>,<font color="Maroon">z'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T8" target="_self">Th8</a>, </i>;<br/>
<b>hence </b><a NAME="E5:64_3"/>
not for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span> holds  <font color="Olive">x</font>,<font color="Olive">y</font> <a href="analoaf.html#R2">//</a> <font color="Olive">x</font>,<font color="Olive">z</font>
 ;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E69</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">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span>, <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font>, <font color="Maroon">z'</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font> ;<br/><b>consider </b><font color="Maroon">t'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font><b> such that </b><br/><a NAME="E3:64_4"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">x'</font>,<font color="Maroon">z'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">y'</font>,<font color="Maroon">t'</font> &amp; <font color="Maroon">y'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">t'</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">t'</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span> ;<br/><a NAME="E5:64_4"/>
( <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">y</font>,<font color="Maroon">t</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">t</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T9" target="_self">Th9</a>, </i>;<br/><b>hence </b><a NAME="E6:64_4"/>
 ex <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span> st <br/>( <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">y</font>,<font color="Olive">t</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">t</font> )
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E71</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">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span>, <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font>, <font color="Maroon">z'</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font> ;<br/><b>consider </b><font color="Maroon">t'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font><b> such that </b><br/><a NAME="E3:64_5"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">x'</font>,<font color="Maroon">y'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">z'</font>,<font color="Maroon">t'</font> &amp; <font color="Maroon">x'</font>,<font color="Maroon">z'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">y'</font>,<font color="Maroon">t'</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">t'</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span> ;<br/><a NAME="E5:64_5"/>
( <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">z</font>,<font color="Maroon">t</font> &amp; <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">y</font>,<font color="Maroon">t</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T10" target="_self">Th10</a>, </i>;<br/><b>hence </b><a NAME="E6:64_5"/>
 ex <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span> st <br/>( <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">z</font>,<font color="Olive">t</font> &amp; <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">y</font>,<font color="Olive">t</font> )
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E73</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">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span>, <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span>, <font color="Maroon">t</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span>;<br/><b>assume </b><a NAME="E1:64_6"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">z</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">t</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font> )
 ;<br/><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font>, <font color="Maroon">z'</font> = <font color="Maroon">z</font>, <font color="Maroon">t'</font> = <font color="Maroon">t</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">POS</font> ;<br/><a NAME="E3:64_6"/>
( <font color="Maroon">z'</font>,<font color="Maroon">x'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x'</font>,<font color="Maroon">t'</font> &amp; <font color="Maroon">x'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z'</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T11" target="_self">Th11</a>, </i>;<br/><b>then consider </b><font color="Maroon">u'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font><b> such that </b><br/><a NAME="E5:64_6"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">y'</font>,<font color="Maroon">x'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x'</font>,<font color="Maroon">u'</font> &amp; <font color="Maroon">y'</font>,<font color="Maroon">z'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">t'</font>,<font color="Maroon">u'</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">u'</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span> ;<br/><a NAME="E7:64_6"/>
( <font color="Maroon">y</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">u</font> &amp; <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">t</font>,<font color="Maroon">u</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T12" target="_self">Th12</a>, </i>;<br/><b>hence </b><a NAME="E8:64_6"/>
 ex <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span> st <br/>( <font color="Maroon">y</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Olive">u</font> &amp; <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">t</font>,<font color="Olive">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">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span>, <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span>, <font color="Maroon">t</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span>;<br/><b>assume </b><a NAME="E1:64_7"/><i><font color="Green">E76</font></i>: 
not <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">z</font>,<font color="Maroon">t</font>
 ;<br/><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font>, <font color="Maroon">z'</font> = <font color="Maroon">z</font>, <font color="Maroon">t'</font> = <font color="Maroon">t</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">POS</font> ;<br/><a NAME="E3:64_7"/>
not <font color="Maroon">x'</font>,<font color="Maroon">y'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">z'</font>,<font color="Maroon">t'</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T11" target="_self">Th11</a>, </i>;<br/><b>then consider </b><font color="Maroon">u'</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">POS</font><b> such that </b><br/><a NAME="E5:64_7"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">x'</font>,<font color="Maroon">y'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x'</font>,<font color="Maroon">u'</font> &amp; <font color="Maroon">z'</font>,<font color="Maroon">t'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">z'</font>,<font color="Maroon">u'</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">u'</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span> ;<br/><a NAME="E7:64_7"/>
( <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">u</font> &amp; <font color="Maroon">z</font>,<font color="Maroon">t</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">z</font>,<font color="Maroon">u</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T12" target="_self">Th12</a>, </i>;<br/><b>hence </b><a NAME="E8:64_7"/>
 ex <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" target="_self">Af</a> <font color="Maroon">POS</font></span>)</span> st <br/>( <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Olive">u</font> &amp; <font color="Maroon">z</font>,<font color="Maroon">t</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">z</font>,<font color="Olive">u</font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:64"/>
 <a href="analoaf.html#G1">AffinStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">POS</font>,the <a href="analoaf.html#U1">CONGR</a> of <font color="Maroon">POS</font> #) is   <a href="diraf.html#NM2">AffinPlane</a>
 <b>by </b><i>, <a class="txt" href="analmetr.html#E12"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="analmetr.html#T7" target="_self">Th7</a>, <a class="ref" href="analmetr.html#T8" target="_self">Th8</a>, <a class="ref" href="analmetr.html#T9" target="_self">Th9</a>, <a class="ref" href="analmetr.html#T10" target="_self">Th10</a>, <a class="ref" href="diraf.html#T54">DIRAF:54</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="analmetr.html#D10" target="_self">ANALMETR:def 10</a><br/>

<b>thus </b><a NAME="E12:64"/>
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">d</font>, <font color="Olive">p</font>, <font color="Olive">q</font>, <font color="Olive">r</font>, <font color="Olive">s</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font> holds <br/> ( ( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">a</font>,<font color="Olive">b</font> implies <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> ) &amp; <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">c</font>,<font color="Olive">c</font> &amp; ( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">c</font>,<font color="Olive">d</font> implies ( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">d</font>,<font color="Olive">c</font> &amp; <font color="Olive">c</font>,<font color="Olive">d</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">a</font>,<font color="Olive">b</font> ) ) &amp; ( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">p</font>,<font color="Olive">q</font> &amp; <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">r</font>,<font color="Olive">s</font> &amp; not <font color="Olive">p</font>,<font color="Olive">q</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">r</font>,<font color="Olive">s</font> implies <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> ) &amp; ( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">p</font>,<font color="Olive">q</font> &amp; <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">r</font>,<font color="Olive">s</font> &amp; not <font color="Olive">p</font>,<font color="Olive">q</font> <a href="analoaf.html#R2">//</a> <font color="Olive">r</font>,<font color="Olive">s</font> implies <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> ) )
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E13:64"/>
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font>  ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">POS</font> st <br/>( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analmetr.html#R4" target="_self">_|_</a> <font color="Olive">c</font>,<font color="Olive">x</font> &amp; <font color="Olive">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">x</font> )
 <b>by </b><i/>;<br/>


</div>
