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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="analmetr.html#NM2" title="ANALMETR:NM.2">OrtAfPl</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> is  <a href="conmetr.html#V2" title="CONMETR:attr.2">satisfying_PAP</a>  iff  <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">X</font> #) is  <a href="aff_2.html#V2" title="AFF_2:attr.2">Pappian</a>  )</span><br/>

<a NAME="E1:22"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
( <font color="Maroon" title="c1">X</font> is  <a href="conmetr.html#V2" title="CONMETR:attr.2">satisfying_PAP</a>  implies  <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">X</font> #) is  <a href="aff_2.html#V2" title="AFF_2:attr.2">Pappian</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:22_1"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<font color="Maroon" title="c1">X</font> is  <a href="conmetr.html#V2" title="CONMETR:attr.2">satisfying_PAP</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">X</font> #) is  <a href="aff_2.html#V2" title="AFF_2:attr.2">Pappian</a> </span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</font>, <font color="Olive" title="b2">N</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">X</font> #)<br/>  for <font color="Olive" title="b3">o</font>, <font color="Olive" title="b4">a</font>, <font color="Olive" title="b5">b</font>, <font color="Olive" title="b6">c</font>, <font color="Olive" title="b7">a1</font>, <font color="Olive" title="b8">b1</font>, <font color="Olive" title="b9">c1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">X</font> #)  st <font color="Olive" title="b1">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b2">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b1">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">a</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b7">a1</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">b</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b8">b1</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">c</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b9">c1</font> &amp; <font color="Olive" title="b4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b7">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b8">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b9">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b4">a</font>,<font color="Olive" title="b8">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">b</font>,<font color="Olive" title="b7">a1</font> &amp; <font color="Olive" title="b5">b</font>,<font color="Olive" title="b9">c1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">c</font>,<font color="Olive" title="b8">b1</font> holds <br/><font color="Olive" title="b4">a</font>,<font color="Olive" title="b9">c1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">c</font>,<font color="Olive" title="b7">a1</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">M</font>, <font color="Maroon" title="c3">N</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">X</font> #); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">o</font>, <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</font>, <font color="Olive" title="b5">a1</font>, <font color="Olive" title="b6">b1</font>, <font color="Olive" title="b7">c1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">X</font> #)  st <font color="Maroon" title="c2">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c3">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c2">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">a</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">a1</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">b</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">b1</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">c</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b7">c1</font> &amp; <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b5">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Olive" title="b6">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Olive" title="b7">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b6">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">b</font>,<font color="Olive" title="b5">a1</font> &amp; <font color="Olive" title="b3">b</font>,<font color="Olive" title="b7">c1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">c</font>,<font color="Olive" title="b6">b1</font> holds <br/><font color="Olive" title="b2">a</font>,<font color="Olive" title="b7">c1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">c</font>,<font color="Olive" title="b5">a1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">o</font>, <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">c</font>, <font color="Maroon" title="c8">a1</font>, <font color="Maroon" title="c9">b1</font>, <font color="Maroon" title="c10">c1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">X</font> #); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c3">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c2">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a1</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b1</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c1</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c8">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c9">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c10">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">a1</font> &amp; <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c10">c1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c9">b1</font> implies <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c10">c1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">a1</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:22_1_1"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
<font color="Maroon" title="c2">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E2:22_1_1"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
<font color="Maroon" title="c3">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E3:22_1_1"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
<font color="Maroon" title="c2">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">and </span><br/><a NAME="E4:22_1_1"/><span class="lab"><font color="Green" title="E14">A6</font></span>: 
<font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 <span class="kw">and </span><br/><a NAME="E5:22_1_1"/><span class="lab"><font color="Green" title="E15">A7</font></span>: 
<font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">and </span><br/><a NAME="E6:22_1_1"/><span class="lab"><font color="Green" title="E16">A8</font></span>: 
<font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">and </span><br/><a NAME="E7:22_1_1"/><span class="lab"><font color="Green" title="E17">A9</font></span>: 
<font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a1</font>
 <span class="kw">and </span><br/><a NAME="E8:22_1_1"/><span class="lab"><font color="Green" title="E18">A10</font></span>: 
<font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font>
 <span class="kw">and </span><br/><a NAME="E9:22_1_1"/><span class="lab"><font color="Green" title="E19">A11</font></span>: 
<font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b1</font>
 <span class="kw">and </span><br/><a NAME="E10:22_1_1"/><span class="lab"><font color="Green" title="E20">A12</font></span>: 
<font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c</font>
 <span class="kw">and </span><br/><a NAME="E11:22_1_1"/><span class="lab"><font color="Green" title="E21">A13</font></span>: 
<font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c1</font>
 <span class="kw">and </span><br/><a NAME="E12:22_1_1"/><span class="lab"><font color="Green" title="E22">A14</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 <span class="kw">and </span><br/><a NAME="E13:22_1_1"/><span class="lab"><font color="Green" title="E23">A15</font></span>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 <span class="kw">and </span><br/><a NAME="E14:22_1_1"/><span class="lab"><font color="Green" title="E24">A16</font></span>: 
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 <span class="kw">and </span><br/><a NAME="E15:22_1_1"/><span class="lab"><font color="Green" title="E25">A17</font></span>: 
<font color="Maroon" title="c8">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">and </span><br/><a NAME="E16:22_1_1"/><span class="lab"><font color="Green" title="E26">A18</font></span>: 
<font color="Maroon" title="c9">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">and </span><br/><a NAME="E17:22_1_1"/><span class="lab"><font color="Green" title="E27">A19</font></span>: 
<font color="Maroon" title="c10">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">and </span><br/><a NAME="E18:22_1_1"/><span class="lab"><font color="Green" title="E28">A20</font></span>: 
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">a1</font>
 <span class="kw">and </span><br/><a NAME="E19:22_1_1"/><span class="lab"><font color="Green" title="E29">A21</font></span>: 
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c10">c1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c9">b1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c10">c1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">a1</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c11">a9</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c12">b9</font> = <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c13">c9</font> = <font color="Maroon" title="c7">c</font>, <font color="Maroon" title="c14">a19</font> = <font color="Maroon" title="c8">a1</font>, <font color="Maroon" title="c15">b19</font> = <font color="Maroon" title="c9">b1</font>, <font color="Maroon" title="c16">c19</font> = <font color="Maroon" title="c10">c1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">M9</font> = <font color="Maroon" title="c2">M</font>, <font color="Maroon" title="c18">N9</font> = <font color="Maroon" title="c3">N</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ;<br/><a NAME="E22:22_1_1"/><span class="lab"><font color="Green" title="E30">A22</font></span>: 
<font color="Maroon" title="c18">N9</font> is  <a href="analmetr.html#V5" title="ANALMETR:attr.5">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_1_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="analmetr.html#T43" onmouseover="rs('analmetr/T43')" onmouseout="rh()">ANALMETR:43</a></span>;<br/><a NAME="E23:22_1_1"/><span class="lab"><font color="Green" title="E31">A23</font></span>: 
(  not <font color="Maroon" title="c16">c19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">M9</font> &amp;  not <font color="Maroon" title="c12">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">N9</font> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:22_1_1_1"/>
( <font color="Maroon" title="c16">c19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">M9</font> or <font color="Maroon" title="c12">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">N9</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:22_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A24</font></span>: 
( <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c10">c1</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">M</font> or <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c6">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c3">N</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E2:22_1_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E4:22_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E5:22_1_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="ref" href="aff_1.html#T52" onmouseover="rs('aff_1/T52')" onmouseout="rh()">AFF_1:52</a></span>;<br/>
<a NAME="E3:22_1_1_1"/><span class="lab"><font color="Green" title="E33">A25</font></span>: 
<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c6">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E4:22_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E13:22_1_1"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="ref" href="aff_1.html#T52" onmouseover="rs('aff_1/T52')" onmouseout="rh()">AFF_1:52</a></span>;<br/>
<a NAME="E4:22_1_1_1"/>
<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c10">c1</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c3">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_1_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E5:22_1_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E17:22_1_1"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="ref" href="aff_1.html#T52" onmouseover="rs('aff_1/T52')" onmouseout="rh()">AFF_1:52</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:22_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E4:22_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E5:22_1_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E8:22_1_1"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E11:22_1_1"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E2:22_1_1_1"><span class="lab"><font color="Green" title="E32">A24</font></span></a>, <a class="txt" href="#E3:22_1_1_1"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="ref" href="aff_1.html#T45" onmouseover="rs('aff_1/T45')" onmouseout="rh()">AFF_1:45</a>, <a class="ref" href="aff_1.html#T53" onmouseover="rs('aff_1/T53')" onmouseout="rh()">AFF_1:53</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E24:22_1_1"/>
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:22_1_1"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="ref" href="aff_1.html#T4" onmouseover="rs('aff_1/T4')" onmouseout="rh()">AFF_1:4</a></span>;<br/><a NAME="E25:22_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A26</font></span>: 
<font color="Maroon" title="c12">b9</font>,<font color="Maroon" title="c14">a19</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">a9</font>,<font color="Maroon" title="c15">b19</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T36" onmouseover="rs('analmetr/T36')" onmouseout="rh()">ANALMETR:36</a></span>;<br/><a NAME="E26:22_1_1"/><span class="lab"><font color="Green" title="E33">A27</font></span>: 
<font color="Maroon" title="c12">b9</font>,<font color="Maroon" title="c16">c19</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">c9</font>,<font color="Maroon" title="c15">b19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:22_1_1"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="ref" href="analmetr.html#T36" onmouseover="rs('analmetr/T36')" onmouseout="rh()">ANALMETR:36</a></span>;<br/><a NAME="E27:22_1_1"/>
<font color="Maroon" title="c17">M9</font> is  <a href="analmetr.html#V5" title="ANALMETR:attr.5">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="analmetr.html#T43" onmouseover="rs('analmetr/T43')" onmouseout="rh()">ANALMETR:43</a></span>;<br/><a NAME="E28:22_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">a9</font>,<font color="Maroon" title="c16">c19</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">c9</font>,<font color="Maroon" title="c14">a19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E4:22_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E5:22_1_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E6:22_1_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E7:22_1_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E9:22_1_1"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E10:22_1_1"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E12:22_1_1"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E13:22_1_1"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E14:22_1_1"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E15:22_1_1"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E16:22_1_1"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="txt" href="#E17:22_1_1"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="txt" href="#E22:22_1_1"><span class="lab"><font color="Green" title="E30">A22</font></span></a>, <a class="txt" href="#E25:22_1_1"><span class="lab"><font color="Green" title="E32">A26</font></span></a>, <a class="txt" href="#E26:22_1_1"><span class="lab"><font color="Green" title="E33">A27</font></span></a>, <a class="txt" href="#E23:22_1_1"><span class="lab"><font color="Green" title="E31">A23</font></span></a>, <a class="ref" href="conmetr.html#D2" onmouseover="rs('conmetr/D2')" onmouseout="rh()">CONMETR:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E29:22_1_1"/>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c10">c1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T36" onmouseover="rs('analmetr/T36')" onmouseout="rh()">ANALMETR:36</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:22_1"/>
 <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">X</font> #) is  <a href="aff_2.html#V2" title="AFF_2:attr.2">Pappian</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_2.html#D2" onmouseover="rs('aff_2/D2')" onmouseout="rh()">AFF_2:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:22"/>
(  <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">X</font> #) is  <a href="aff_2.html#V2" title="AFF_2:attr.2">Pappian</a>  implies <font color="Maroon" title="c1">X</font> is  <a href="conmetr.html#V2" title="CONMETR:attr.2">satisfying_PAP</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:22_2"/><span class="lab"><font color="Green" title="E10">A28</font></span>: 
 <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">X</font> #) is  <a href="aff_2.html#V2" title="AFF_2:attr.2">Pappian</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">X</font> is  <a href="conmetr.html#V2" title="CONMETR:attr.2">satisfying_PAP</a> </span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">o9</font>, <font color="Olive" title="b2">a19</font>, <font color="Olive" title="b3">a29</font>, <font color="Olive" title="b4">a39</font>, <font color="Olive" title="b5">b19</font>, <font color="Olive" title="b6">b29</font>, <font color="Olive" title="b7">b39</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b8">M9</font>, <font color="Olive" title="b9">N9</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b8">M9</font> is  <a href="analmetr.html#V5" title="ANALMETR:attr.5">being_line</a>  &amp; <font color="Olive" title="b9">N9</font> is  <a href="analmetr.html#V5" title="ANALMETR:attr.5">being_line</a>  &amp; <font color="Olive" title="b1">o9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b8">M9</font> &amp; <font color="Olive" title="b2">a19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b8">M9</font> &amp; <font color="Olive" title="b3">a29</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b8">M9</font> &amp; <font color="Olive" title="b4">a39</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b8">M9</font> &amp; <font color="Olive" title="b1">o9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b9">N9</font> &amp; <font color="Olive" title="b5">b19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b9">N9</font> &amp; <font color="Olive" title="b6">b29</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b9">N9</font> &amp; <font color="Olive" title="b7">b39</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b9">N9</font> &amp;  not <font color="Olive" title="b6">b29</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b8">M9</font> &amp;  not <font color="Olive" title="b4">a39</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b9">N9</font> &amp; <font color="Olive" title="b1">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">a19</font> &amp; <font color="Olive" title="b1">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">a29</font> &amp; <font color="Olive" title="b1">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">a39</font> &amp; <font color="Olive" title="b1">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">b19</font> &amp; <font color="Olive" title="b1">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">b29</font> &amp; <font color="Olive" title="b1">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b7">b39</font> &amp; <font color="Olive" title="b4">a39</font>,<font color="Olive" title="b6">b29</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">a29</font>,<font color="Olive" title="b5">b19</font> &amp; <font color="Olive" title="b4">a39</font>,<font color="Olive" title="b7">b39</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">a19</font>,<font color="Olive" title="b5">b19</font> holds <br/><font color="Olive" title="b2">a19</font>,<font color="Olive" title="b6">b29</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">a29</font>,<font color="Olive" title="b7">b39</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">o9</font>, <font color="Maroon" title="c3">a19</font>, <font color="Maroon" title="c4">a29</font>, <font color="Maroon" title="c5">a39</font>, <font color="Maroon" title="c6">b19</font>, <font color="Maroon" title="c7">b29</font>, <font color="Maroon" title="c8">b39</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M9</font>, <font color="Olive" title="b2">N9</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">M9</font> is  <a href="analmetr.html#V5" title="ANALMETR:attr.5">being_line</a>  &amp; <font color="Olive" title="b2">N9</font> is  <a href="analmetr.html#V5" title="ANALMETR:attr.5">being_line</a>  &amp; <font color="Maroon" title="c2">o9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M9</font> &amp; <font color="Maroon" title="c3">a19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M9</font> &amp; <font color="Maroon" title="c4">a29</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M9</font> &amp; <font color="Maroon" title="c5">a39</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M9</font> &amp; <font color="Maroon" title="c2">o9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N9</font> &amp; <font color="Maroon" title="c6">b19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N9</font> &amp; <font color="Maroon" title="c7">b29</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N9</font> &amp; <font color="Maroon" title="c8">b39</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N9</font> &amp;  not <font color="Maroon" title="c7">b29</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M9</font> &amp;  not <font color="Maroon" title="c5">a39</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N9</font> &amp; <font color="Maroon" title="c2">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a19</font> &amp; <font color="Maroon" title="c2">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a29</font> &amp; <font color="Maroon" title="c2">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a39</font> &amp; <font color="Maroon" title="c2">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b19</font> &amp; <font color="Maroon" title="c2">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b29</font> &amp; <font color="Maroon" title="c2">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b39</font> &amp; <font color="Maroon" title="c5">a39</font>,<font color="Maroon" title="c7">b29</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a29</font>,<font color="Maroon" title="c6">b19</font> &amp; <font color="Maroon" title="c5">a39</font>,<font color="Maroon" title="c8">b39</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a19</font>,<font color="Maroon" title="c6">b19</font> holds <br/><font color="Maroon" title="c3">a19</font>,<font color="Maroon" title="c7">b29</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a29</font>,<font color="Maroon" title="c8">b39</font></span><br/><span class="kw">let </span><font color="Maroon" title="c9">M9</font>, <font color="Maroon" title="c10">N9</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">M9</font> is  <a href="analmetr.html#V5" title="ANALMETR:attr.5">being_line</a>  &amp; <font color="Maroon" title="c10">N9</font> is  <a href="analmetr.html#V5" title="ANALMETR:attr.5">being_line</a>  &amp; <font color="Maroon" title="c2">o9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">M9</font> &amp; <font color="Maroon" title="c3">a19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">M9</font> &amp; <font color="Maroon" title="c4">a29</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">M9</font> &amp; <font color="Maroon" title="c5">a39</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">M9</font> &amp; <font color="Maroon" title="c2">o9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">N9</font> &amp; <font color="Maroon" title="c6">b19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">N9</font> &amp; <font color="Maroon" title="c7">b29</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">N9</font> &amp; <font color="Maroon" title="c8">b39</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">N9</font> &amp;  not <font color="Maroon" title="c7">b29</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">M9</font> &amp;  not <font color="Maroon" title="c5">a39</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">N9</font> &amp; <font color="Maroon" title="c2">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a19</font> &amp; <font color="Maroon" title="c2">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a29</font> &amp; <font color="Maroon" title="c2">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a39</font> &amp; <font color="Maroon" title="c2">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b19</font> &amp; <font color="Maroon" title="c2">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b29</font> &amp; <font color="Maroon" title="c2">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b39</font> &amp; <font color="Maroon" title="c5">a39</font>,<font color="Maroon" title="c7">b29</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a29</font>,<font color="Maroon" title="c6">b19</font> &amp; <font color="Maroon" title="c5">a39</font>,<font color="Maroon" title="c8">b39</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a19</font>,<font color="Maroon" title="c6">b19</font> implies <font color="Maroon" title="c3">a19</font>,<font color="Maroon" title="c7">b29</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a29</font>,<font color="Maroon" title="c8">b39</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:22_2_1"/><span class="lab"><font color="Green" title="E11">A29</font></span>: 
<font color="Maroon" title="c9">M9</font> is  <a href="analmetr.html#V5" title="ANALMETR:attr.5">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E2:22_2_1"/><span class="lab"><font color="Green" title="E12">A30</font></span>: 
<font color="Maroon" title="c10">N9</font> is  <a href="analmetr.html#V5" title="ANALMETR:attr.5">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E3:22_2_1"/><span class="lab"><font color="Green" title="E13">A31</font></span>: 
<font color="Maroon" title="c2">o9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">M9</font>
 <span class="kw">and </span><br/><a NAME="E4:22_2_1"/><span class="lab"><font color="Green" title="E14">A32</font></span>: 
<font color="Maroon" title="c3">a19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">M9</font>
 <span class="kw">and </span><br/><a NAME="E5:22_2_1"/><span class="lab"><font color="Green" title="E15">A33</font></span>: 
<font color="Maroon" title="c4">a29</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">M9</font>
 <span class="kw">and </span><br/><a NAME="E6:22_2_1"/><span class="lab"><font color="Green" title="E16">A34</font></span>: 
<font color="Maroon" title="c5">a39</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">M9</font>
 <span class="kw">and </span><br/><a NAME="E7:22_2_1"/><span class="lab"><font color="Green" title="E17">A35</font></span>: 
<font color="Maroon" title="c2">o9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">N9</font>
 <span class="kw">and </span><br/><a NAME="E8:22_2_1"/><span class="lab"><font color="Green" title="E18">A36</font></span>: 
<font color="Maroon" title="c6">b19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">N9</font>
 <span class="kw">and </span><br/><a NAME="E9:22_2_1"/><span class="lab"><font color="Green" title="E19">A37</font></span>: 
<font color="Maroon" title="c7">b29</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">N9</font>
 <span class="kw">and </span><br/><a NAME="E10:22_2_1"/><span class="lab"><font color="Green" title="E20">A38</font></span>: 
<font color="Maroon" title="c8">b39</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">N9</font>
 <span class="kw">and </span><br/><a NAME="E11:22_2_1"/><span class="lab"><font color="Green" title="E21">A39</font></span>: 
 not <font color="Maroon" title="c7">b29</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">M9</font>
 <span class="kw">and </span><br/><a NAME="E12:22_2_1"/><span class="lab"><font color="Green" title="E22">A40</font></span>: 
 not <font color="Maroon" title="c5">a39</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">N9</font>
 <span class="kw">and </span><br/><a NAME="E13:22_2_1"/><span class="lab"><font color="Green" title="E23">A41</font></span>: 
<font color="Maroon" title="c2">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a19</font>
 <span class="kw">and </span><br/><a NAME="E14:22_2_1"/><span class="lab"><font color="Green" title="E24">A42</font></span>: 
<font color="Maroon" title="c2">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a29</font>
 <span class="kw">and </span><br/><a NAME="E15:22_2_1"/>
<font color="Maroon" title="c2">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a39</font>
 <span class="kw">and </span><br/><a NAME="E16:22_2_1"/><span class="lab"><font color="Green" title="E25">A43</font></span>: 
<font color="Maroon" title="c2">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b19</font>
 <span class="kw">and </span><br/><a NAME="E17:22_2_1"/>
<font color="Maroon" title="c2">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b29</font>
 <span class="kw">and </span><br/><a NAME="E18:22_2_1"/><span class="lab"><font color="Green" title="E26">A44</font></span>: 
<font color="Maroon" title="c2">o9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b39</font>
 <span class="kw">and </span><br/><a NAME="E19:22_2_1"/><span class="lab"><font color="Green" title="E27">A45</font></span>: 
<font color="Maroon" title="c5">a39</font>,<font color="Maroon" title="c7">b29</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a29</font>,<font color="Maroon" title="c6">b19</font>
 <span class="kw">and </span><br/><a NAME="E20:22_2_1"/><span class="lab"><font color="Green" title="E28">A46</font></span>: 
<font color="Maroon" title="c5">a39</font>,<font color="Maroon" title="c8">b39</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a19</font>,<font color="Maroon" title="c6">b19</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">a19</font>,<font color="Maroon" title="c7">b29</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a29</font>,<font color="Maroon" title="c8">b39</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c11">a1</font> = <font color="Maroon" title="c3">a19</font>, <font color="Maroon" title="c12">a2</font> = <font color="Maroon" title="c4">a29</font>, <font color="Maroon" title="c13">a3</font> = <font color="Maroon" title="c5">a39</font>, <font color="Maroon" title="c14">b1</font> = <font color="Maroon" title="c6">b19</font>, <font color="Maroon" title="c15">b2</font> = <font color="Maroon" title="c7">b29</font>, <font color="Maroon" title="c16">b3</font> = <font color="Maroon" title="c8">b39</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">X</font> #) ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">M</font> = <font color="Maroon" title="c9">M9</font>, <font color="Maroon" title="c18">N</font> = <font color="Maroon" title="c10">N9</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">X</font> #) ;<br/><a NAME="E23:22_2_1"/><span class="lab"><font color="Green" title="E29">A47</font></span>: 
<font color="Maroon" title="c18">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_2_1"><span class="lab"><font color="Green" title="E12">A30</font></span></a>, <a class="ref" href="analmetr.html#T43" onmouseover="rs('analmetr/T43')" onmouseout="rh()">ANALMETR:43</a></span>;<br/><a NAME="E24:22_2_1"/><span class="lab"><font color="Green" title="E30">A48</font></span>: 
<font color="Maroon" title="c17">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_2_1"><span class="lab"><font color="Green" title="E11">A29</font></span></a>, <a class="ref" href="analmetr.html#T43" onmouseover="rs('analmetr/T43')" onmouseout="rh()">ANALMETR:43</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_2_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">N</font> implies <font color="Maroon" title="c3">a19</font>,<font color="Maroon" title="c7">b29</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a29</font>,<font color="Maroon" title="c8">b39</font> )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:22_2_1_1"/>
<font color="Maroon" title="c17">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">N</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">a19</font>,<font color="Maroon" title="c7">b29</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a29</font>,<font color="Maroon" title="c8">b39</font></span><br/><a NAME="E2:22_2_1_1"/>
<font color="Maroon" title="c13">a3</font>,<font color="Maroon" title="c16">b3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">a1</font>,<font color="Maroon" title="c14">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:22_2_1"><span class="lab"><font color="Green" title="E28">A46</font></span></a>, <a class="ref" href="analmetr.html#T36" onmouseover="rs('analmetr/T36')" onmouseout="rh()">ANALMETR:36</a></span>;<br/><a NAME="E3:22_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A49</font></span>: 
<font color="Maroon" title="c11">a1</font>,<font color="Maroon" title="c14">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">a3</font>,<font color="Maroon" title="c16">b3</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T4" onmouseover="rs('aff_1/T4')" onmouseout="rh()">AFF_1:4</a></span>;<br/><a NAME="E4:22_2_1_1"/>
<font color="Maroon" title="c13">a3</font>,<font color="Maroon" title="c15">b2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c12">a2</font>,<font color="Maroon" title="c14">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:22_2_1"><span class="lab"><font color="Green" title="E27">A45</font></span></a>, <a class="ref" href="analmetr.html#T36" onmouseover="rs('analmetr/T36')" onmouseout="rh()">ANALMETR:36</a></span>;<br/><a NAME="E5:22_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">a1</font>,<font color="Maroon" title="c15">b2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c12">a2</font>,<font color="Maroon" title="c16">b3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_2"><span class="lab"><font color="Green" title="E10">A28</font></span></a>, <a class="txt" href="#E3:22_2_1"><span class="lab"><font color="Green" title="E13">A31</font></span></a>, <a class="txt" href="#E4:22_2_1"><span class="lab"><font color="Green" title="E14">A32</font></span></a>, <a class="txt" href="#E5:22_2_1"><span class="lab"><font color="Green" title="E15">A33</font></span></a>, <a class="txt" href="#E6:22_2_1"><span class="lab"><font color="Green" title="E16">A34</font></span></a>, <a class="txt" href="#E7:22_2_1"><span class="lab"><font color="Green" title="E17">A35</font></span></a>, <a class="txt" href="#E8:22_2_1"><span class="lab"><font color="Green" title="E18">A36</font></span></a>, <a class="txt" href="#E9:22_2_1"><span class="lab"><font color="Green" title="E19">A37</font></span></a>, <a class="txt" href="#E10:22_2_1"><span class="lab"><font color="Green" title="E20">A38</font></span></a>, <a class="txt" href="#E11:22_2_1"><span class="lab"><font color="Green" title="E21">A39</font></span></a>, <a class="txt" href="#E12:22_2_1"><span class="lab"><font color="Green" title="E22">A40</font></span></a>, <a class="txt" href="#E13:22_2_1"><span class="lab"><font color="Green" title="E23">A41</font></span></a>, <a class="txt" href="#E14:22_2_1"><span class="lab"><font color="Green" title="E24">A42</font></span></a>, <a class="txt" href="#E16:22_2_1"><span class="lab"><font color="Green" title="E25">A43</font></span></a>, <a class="txt" href="#E18:22_2_1"><span class="lab"><font color="Green" title="E26">A44</font></span></a>, <a class="txt" href="#E24:22_2_1"><span class="lab"><font color="Green" title="E30">A48</font></span></a>, <a class="txt" href="#E23:22_2_1"><span class="lab"><font color="Green" title="E29">A47</font></span></a>, <a class="txt" href="#E3:22_2_1_1"><span class="lab"><font color="Green" title="E31">A49</font></span></a>, <a class="ref" href="aff_2.html#D2" onmouseover="rs('aff_2/D2')" onmouseout="rh()">AFF_2:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E6:22_2_1_1"/>
<font color="Maroon" title="c3">a19</font>,<font color="Maroon" title="c7">b29</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a29</font>,<font color="Maroon" title="c8">b39</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T36" onmouseover="rs('analmetr/T36')" onmouseout="rh()">ANALMETR:36</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E26:22_2_1"/>
<font color="Maroon" title="c3">a19</font>,<font color="Maroon" title="c7">b29</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a29</font>,<font color="Maroon" title="c8">b39</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_2_1"><span class="lab"><font color="Green" title="E19">A37</font></span></a>, <a class="txt" href="#E11:22_2_1"><span class="lab"><font color="Green" title="E21">A39</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:22_2"/>
<font color="Maroon" title="c1">X</font> is  <a href="conmetr.html#V2" title="CONMETR:attr.2">satisfying_PAP</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="conmetr.html#D2" onmouseover="rs('conmetr/D2')" onmouseout="rh()">CONMETR:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:22"/>
( <font color="Maroon" title="c1">X</font> is  <a href="conmetr.html#V2" title="CONMETR:attr.2">satisfying_PAP</a>  iff  <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">X</font> #) is  <a href="aff_2.html#V2" title="AFF_2:attr.2">Pappian</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E9">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
