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

<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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <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="E15">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="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <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" 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="E16">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <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" onclick="hs2(this)" href="javascript:()" title="22_1_1"><span class="kw">now </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 <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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 <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font></span>)</span>  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 <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E17">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="E18">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="E19">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="E20">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="E21">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="E22">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="E23">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="E24">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="E25">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="E26">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="E27">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="E28">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="E29">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="E30">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="E31">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="E32">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="E33">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="E34">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="E35">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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> <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T47" title="ANALMETR:th.47">ANALMETR:47</a></span>;<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> <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T57" title="ANALMETR:th.57">ANALMETR:57</a></span>;<br/><a NAME="E22:22_1_1"/><span class="lab"><font color="Green" title="E36">A22</font></span>: 
<font color="Maroon" title="c18">N9</font> is  <a href="analmetr.html#V4" title="ANALMETR:attr.4">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="E18">A4</font></span></a>, <a class="ref" href="analmetr.html#T58" title="ANALMETR:th.58">ANALMETR:58</a></span>;<br/><a NAME="E23:22_1_1"/><span class="lab"><font color="Green" title="E37">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" 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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E38">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="E17">A3</font></span></a>, <a class="txt" href="#E2:22_1_1"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E4:22_1_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E5:22_1_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="ref" href="aff_1.html#T66" title="AFF_1:th.66">AFF_1:66</a></span>;<br/>
<a NAME="E3:22_1_1_1"/><span class="lab"><font color="Green" title="E39">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="E17">A3</font></span></a>, <a class="txt" href="#E4:22_1_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E13:22_1_1"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="ref" href="aff_1.html#T66" title="AFF_1:th.66">AFF_1:66</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="E18">A4</font></span></a>, <a class="txt" href="#E5:22_1_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="txt" href="#E17:22_1_1"><span class="lab"><font color="Green" title="E33">A19</font></span></a>, <a class="ref" href="aff_1.html#T66" title="AFF_1:th.66">AFF_1:66</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="E19">A5</font></span></a>, <a class="txt" href="#E4:22_1_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E5:22_1_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="txt" href="#E8:22_1_1"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="txt" href="#E11:22_1_1"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="txt" href="#E2:22_1_1_1"><span class="lab"><font color="Green" title="E38">A24</font></span></a>, <a class="txt" href="#E3:22_1_1_1"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a>, <a class="ref" href="aff_1.html#T67" title="AFF_1:th.67">AFF_1:67</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E34">A20</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/><a NAME="E25:22_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">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#T48" title="ANALMETR:th.48">ANALMETR:48</a></span>;<br/><a NAME="E26:22_1_1"/><span class="lab"><font color="Green" title="E39">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="E35">A21</font></span></a>, <a class="ref" href="analmetr.html#T48" title="ANALMETR:th.48">ANALMETR:48</a></span>;<br/><a NAME="E27:22_1_1"/>
<font color="Maroon" title="c17">M9</font> is  <a href="analmetr.html#V4" title="ANALMETR:attr.4">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="E17">A3</font></span></a>, <a class="ref" href="analmetr.html#T58" title="ANALMETR:th.58">ANALMETR:58</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="E16">A2</font></span></a>, <a class="txt" href="#E4:22_1_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E5:22_1_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="txt" href="#E6:22_1_1"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E7:22_1_1"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="txt" href="#E9:22_1_1"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="txt" href="#E10:22_1_1"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E12:22_1_1"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="txt" href="#E13:22_1_1"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="txt" href="#E14:22_1_1"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="txt" href="#E15:22_1_1"><span class="lab"><font color="Green" title="E31">A17</font></span></a>, <a class="txt" href="#E16:22_1_1"><span class="lab"><font color="Green" title="E32">A18</font></span></a>, <a class="txt" href="#E17:22_1_1"><span class="lab"><font color="Green" title="E33">A19</font></span></a>, <a class="txt" href="#E22:22_1_1"><span class="lab"><font color="Green" title="E36">A22</font></span></a>, <a class="txt" href="#E25:22_1_1"><span class="lab"><font color="Green" title="E38">A26</font></span></a>, <a class="txt" href="#E26:22_1_1"><span class="lab"><font color="Green" title="E39">A27</font></span></a>, <a class="txt" href="#E23:22_1_1"><span class="lab"><font color="Green" title="E37">A23</font></span></a>, <a class="ref" href="conmetr.html#D2" title="CONMETR:def.2">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#T48" title="ANALMETR:th.48">ANALMETR:48</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <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" title="AFF_2:def.2">AFF_2:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:22"/>
(  <a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <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" 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="E16">A28</font></span>: 
 <a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font> is  <a href="aff_2.html#V2" title="AFF_2:attr.2">Pappian</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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" onclick="hs2(this)" href="javascript:()" title="22_2_1"><span class="kw">now </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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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#V4" title="ANALMETR:attr.4">being_line</a>  &amp; <font color="Olive" title="b2">N9</font> is  <a href="analmetr.html#V4" title="ANALMETR:attr.4">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">M9</font> is  <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a>  &amp; <font color="Maroon" title="c10">N9</font> is  <a href="analmetr.html#V4" title="ANALMETR:attr.4">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="E17">A29</font></span>: 
<font color="Maroon" title="c9">M9</font> is  <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E2:22_2_1"/><span class="lab"><font color="Green" title="E18">A30</font></span>: 
<font color="Maroon" title="c10">N9</font> is  <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E3:22_2_1"/><span class="lab"><font color="Green" title="E19">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="E20">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="E21">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="E22">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="E23">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="E24">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="E25">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="E26">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="E27">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="E28">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="E29">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="E30">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="E31">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="E32">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="E33">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="E34">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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 <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T47" title="ANALMETR:th.47">ANALMETR:47</a></span>;<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 <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T57" title="ANALMETR:th.57">ANALMETR:57</a></span>;<br/><a NAME="E23:22_2_1"/><span class="lab"><font color="Green" title="E35">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="E18">A30</font></span></a>, <a class="ref" href="analmetr.html#T58" title="ANALMETR:th.58">ANALMETR:58</a></span>;<br/><a NAME="E24:22_2_1"/><span class="lab"><font color="Green" title="E36">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="E17">A29</font></span></a>, <a class="ref" href="analmetr.html#T58" title="ANALMETR:th.58">ANALMETR:58</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_1_1"><span class="kw">now </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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E34">A46</font></span></a>, <a class="ref" href="analmetr.html#T48" title="ANALMETR:th.48">ANALMETR:48</a></span>;<br/><a NAME="E3:22_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">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#T13" title="AFF_1:th.13">AFF_1:13</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="E33">A45</font></span></a>, <a class="ref" href="analmetr.html#T48" title="ANALMETR:th.48">ANALMETR:48</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="E16">A28</font></span></a>, <a class="txt" href="#E3:22_2_1"><span class="lab"><font color="Green" title="E19">A31</font></span></a>, <a class="txt" href="#E4:22_2_1"><span class="lab"><font color="Green" title="E20">A32</font></span></a>, <a class="txt" href="#E5:22_2_1"><span class="lab"><font color="Green" title="E21">A33</font></span></a>, <a class="txt" href="#E6:22_2_1"><span class="lab"><font color="Green" title="E22">A34</font></span></a>, <a class="txt" href="#E7:22_2_1"><span class="lab"><font color="Green" title="E23">A35</font></span></a>, <a class="txt" href="#E8:22_2_1"><span class="lab"><font color="Green" title="E24">A36</font></span></a>, <a class="txt" href="#E9:22_2_1"><span class="lab"><font color="Green" title="E25">A37</font></span></a>, <a class="txt" href="#E10:22_2_1"><span class="lab"><font color="Green" title="E26">A38</font></span></a>, <a class="txt" href="#E11:22_2_1"><span class="lab"><font color="Green" title="E27">A39</font></span></a>, <a class="txt" href="#E12:22_2_1"><span class="lab"><font color="Green" title="E28">A40</font></span></a>, <a class="txt" href="#E13:22_2_1"><span class="lab"><font color="Green" title="E29">A41</font></span></a>, <a class="txt" href="#E14:22_2_1"><span class="lab"><font color="Green" title="E30">A42</font></span></a>, <a class="txt" href="#E16:22_2_1"><span class="lab"><font color="Green" title="E31">A43</font></span></a>, <a class="txt" href="#E18:22_2_1"><span class="lab"><font color="Green" title="E32">A44</font></span></a>, <a class="txt" href="#E24:22_2_1"><span class="lab"><font color="Green" title="E36">A48</font></span></a>, <a class="txt" href="#E23:22_2_1"><span class="lab"><font color="Green" title="E35">A47</font></span></a>, <a class="txt" href="#E3:22_2_1_1"><span class="lab"><font color="Green" title="E37">A49</font></span></a>, <a class="ref" href="aff_2.html#D2" title="AFF_2:def.2">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#T48" title="ANALMETR:th.48">ANALMETR:48</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E25">A37</font></span></a>, <a class="txt" href="#E11:22_2_1"><span class="lab"><font color="Green" title="E27">A39</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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" title="CONMETR:def.2">CONMETR:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <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="E15">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
