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<span class="kw">let </span><font color="Maroon" title="c1">MS</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">MS</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">MS</font> is  <a href="aff_2.html#V2" title="AFF_2:attr.2">Pappian</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">AS</font> =  <a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">MS</font>;<br/>
<div><span class="lab"><font color="Green" title="E17">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:17_1"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
<font color="Maroon" title="c1">MS</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">MS</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="17_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">M</font>, <font color="Maroon" title="c4">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">MS</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">a9</font>, <font color="Olive" title="b6">b9</font>, <font color="Olive" title="b7">c9</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">MS</font></span>)</span>  st <font color="Maroon" title="c3">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c4">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c3">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">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="c3">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="c4">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">a9</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">b9</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">c9</font> &amp; <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">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="c3">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="c3">M</font> &amp; <font color="Olive" title="b5">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> &amp; <font color="Olive" title="b6">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> &amp; <font color="Olive" title="b7">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b6">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">b</font>,<font color="Olive" title="b5">a9</font> &amp; <font color="Olive" title="b3">b</font>,<font color="Olive" title="b7">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">c</font>,<font color="Olive" title="b6">b9</font> holds <br/><font color="Olive" title="b2">a</font>,<font color="Olive" title="b7">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">c</font>,<font color="Olive" title="b5">a9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">o</font>, <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c8">c</font>, <font color="Maroon" title="c9">a9</font>, <font color="Maroon" title="c10">b9</font>, <font color="Maroon" title="c11">c9</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">MS</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="c3">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c4">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c3">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">N</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">a9</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">b9</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">c</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c9</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c9">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> &amp; <font color="Maroon" title="c10">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> &amp; <font color="Maroon" title="c11">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> &amp; <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c10">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">a9</font> &amp; <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c11">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c10">b9</font> implies <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c11">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c9">a9</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:17_1_1"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
( <font color="Maroon" title="c3">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c4">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="E2:17_1_1"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
<font color="Maroon" title="c3">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">N</font>
 <span class="kw">and </span><br/><a NAME="E3:17_1_1"/><span class="lab"><font color="Green" title="E21">A5</font></span>: 
( <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> )
 <span class="kw">and </span><br/><a NAME="E4:17_1_1"/><span class="lab"><font color="Green" title="E22">A6</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a</font>
 <span class="kw">and </span><br/><a NAME="E5:17_1_1"/><span class="lab"><font color="Green" title="E23">A7</font></span>: 
( <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">a9</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b</font> )
 <span class="kw">and </span><br/><a NAME="E6:17_1_1"/><span class="lab"><font color="Green" title="E24">A8</font></span>: 
( <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">b9</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">c</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c9</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> )
 <span class="kw">and </span><br/><a NAME="E7:17_1_1"/><span class="lab"><font color="Green" title="E25">A9</font></span>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font>
 <span class="kw">and </span><br/><a NAME="E8:17_1_1"/><span class="lab"><font color="Green" title="E26">A10</font></span>: 
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font>
 <span class="kw">and </span><br/><a NAME="E9:17_1_1"/><span class="lab"><font color="Green" title="E27">A11</font></span>: 
<font color="Maroon" title="c9">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font>
 <span class="kw">and </span><br/><a NAME="E10:17_1_1"/><span class="lab"><font color="Green" title="E28">A12</font></span>: 
( <font color="Maroon" title="c10">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> &amp; <font color="Maroon" title="c11">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> )
 <span class="kw">and </span><br/><a NAME="E11:17_1_1"/><span class="lab"><font color="Green" title="E29">A13</font></span>: 
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c10">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">a9</font>
 <span class="kw">and </span><br/><a NAME="E12:17_1_1"/><span class="lab"><font color="Green" title="E30">A14</font></span>: 
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c11">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c10">b9</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="c6">a</font>,<font color="Maroon" title="c11">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c9">a9</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c12">M9</font> = <font color="Maroon" title="c3">M</font>, <font color="Maroon" title="c13">N9</font> = <font color="Maroon" title="c4">N</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">MS</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/><span class="kw">reconsider </span><font color="Maroon" title="c14">a1</font> = <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c15">b1</font> = <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c16">c1</font> = <font color="Maroon" title="c8">c</font>, <font color="Maroon" title="c17">a19</font> = <font color="Maroon" title="c9">a9</font>, <font color="Maroon" title="c18">b19</font> = <font color="Maroon" title="c10">b9</font>, <font color="Maroon" title="c19">c19</font> = <font color="Maroon" title="c11">c9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">MS</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/><a NAME="E15:17_1_1"/><span class="lab"><font color="Green" title="E31">A15</font></span>: 
<font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c19">c19</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c16">c1</font>,<font color="Maroon" title="c18">b19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:17_1_1"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="ref" href="analmetr.html#T48" title="ANALMETR:th.48">ANALMETR:48</a></span>;<br/><a NAME="E16:17_1_1"/>
<font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c18">b19</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c17">a19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:17_1_1"><span class="lab"><font color="Green" title="E29">A13</font></span></a>, <a class="ref" href="analmetr.html#T48" title="ANALMETR:th.48">ANALMETR:48</a></span>;<br/><a NAME="E17:17_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A16</font></span>: 
<font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c17">a19</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c18">b19</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T81" title="ANALMETR:th.81">ANALMETR:81</a></span>;<br/><a NAME="E18:17_1_1"/><span class="lab"><font color="Green" title="E33">A17</font></span>: 
( <font color="Maroon" title="c12">M9</font> is  <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a>  &amp; <font color="Maroon" title="c13">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="#E1:17_1_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="analmetr.html#T58" title="ANALMETR:th.58">ANALMETR:58</a></span>;<br/><a NAME="E19:17_1_1"/><span class="kw">then </span>
( not <font color="Maroon" title="c17">a19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">M9</font> &amp; not <font color="Maroon" title="c15">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">N9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1_1"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="txt" href="#E3:17_1_1"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E5:17_1_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E7:17_1_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="txt" href="#E9:17_1_1"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="ref" href="conmetr.html#T5" title="CONMETR:th.5">CONMETR:5</a></span>;<br/><a NAME="E20:17_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">c1</font>,<font color="Maroon" title="c17">a19</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c19">c19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E3:17_1_1"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E4:17_1_1"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E6:17_1_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="txt" href="#E7:17_1_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="txt" href="#E8:17_1_1"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="txt" href="#E9:17_1_1"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="txt" href="#E10:17_1_1"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="txt" href="#E18:17_1_1"><span class="lab"><font color="Green" title="E33">A17</font></span></a>, <a class="txt" href="#E17:17_1_1"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="txt" href="#E15:17_1_1"><span class="lab"><font color="Green" title="E31">A15</font></span></a>, <a class="ref" href="conmetr.html#D2" title="CONMETR:def.2">CONMETR:def 2</a></span>;<br/><a NAME="E21:17_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c19">c19</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c16">c1</font>,<font color="Maroon" title="c17">a19</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T81" title="ANALMETR:th.81">ANALMETR:81</a></span>;<br/><span class="kw">hence </span><a NAME="E22:17_1_1"/>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c11">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c9">a9</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="E3:17_1"/>
 <a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">MS</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:17_2"/><span class="lab"><font color="Green" title="E18">A18</font></span>: 
 <a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">MS</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">MS</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="17_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">o</font>, <font color="Maroon" title="c4">a1</font>, <font color="Maroon" title="c5">a2</font>, <font color="Maroon" title="c6">a3</font>, <font color="Maroon" title="c7">b1</font>, <font color="Maroon" title="c8">b2</font>, <font color="Maroon" title="c9">b3</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">MS</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">M</font>, <font color="Olive" title="b2">N</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">MS</font>  st <font color="Olive" title="b1">M</font> is  <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a>  &amp; <font color="Olive" title="b2">N</font> is  <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a>  &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Maroon" title="c4">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Maroon" title="c6">a3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; <font color="Maroon" title="c7">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; <font color="Maroon" title="c8">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; <font color="Maroon" title="c9">b3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; not <font color="Maroon" title="c8">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; not <font color="Maroon" title="c6">a3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a1</font> &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a2</font> &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a3</font> &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b1</font> &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b2</font> &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b3</font> &amp; <font color="Maroon" title="c6">a3</font>,<font color="Maroon" title="c8">b2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c7">b1</font> &amp; <font color="Maroon" title="c6">a3</font>,<font color="Maroon" title="c9">b3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c7">b1</font> holds <br/><font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c8">b2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c9">b3</font></span><br/><span class="kw">let </span><font color="Maroon" title="c10">M</font>, <font color="Maroon" title="c11">N</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">MS</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="c10">M</font> is  <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a>  &amp; <font color="Maroon" title="c11">N</font> is  <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a>  &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; <font color="Maroon" title="c4">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; <font color="Maroon" title="c6">a3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Maroon" title="c7">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Maroon" title="c8">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Maroon" title="c9">b3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; not <font color="Maroon" title="c8">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; not <font color="Maroon" title="c6">a3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a1</font> &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a2</font> &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a3</font> &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b1</font> &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b2</font> &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b3</font> &amp; <font color="Maroon" title="c6">a3</font>,<font color="Maroon" title="c8">b2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c7">b1</font> &amp; <font color="Maroon" title="c6">a3</font>,<font color="Maroon" title="c9">b3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c7">b1</font> implies <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c8">b2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c9">b3</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:17_2_1"/><span class="lab"><font color="Green" title="E19">A19</font></span>: 
( <font color="Maroon" title="c10">M</font> is  <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a>  &amp; <font color="Maroon" title="c11">N</font> is  <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a>  )
 <span class="kw">and </span><br/><a NAME="E2:17_2_1"/><span class="lab"><font color="Green" title="E20">A20</font></span>: 
( <font color="Maroon" title="c3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; <font color="Maroon" title="c4">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; <font color="Maroon" title="c6">a3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Maroon" title="c7">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Maroon" title="c8">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Maroon" title="c9">b3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; not <font color="Maroon" title="c8">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; not <font color="Maroon" title="c6">a3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a1</font> &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a2</font> )
 <span class="kw">and </span><br/><a NAME="E3:17_2_1"/>
<font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a3</font>
 <span class="kw">and </span><br/><a NAME="E4:17_2_1"/><span class="lab"><font color="Green" title="E21">A21</font></span>: 
<font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b1</font>
 <span class="kw">and </span><br/><a NAME="E5:17_2_1"/>
<font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b2</font>
 <span class="kw">and </span><br/><a NAME="E6:17_2_1"/><span class="lab"><font color="Green" title="E22">A22</font></span>: 
<font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b3</font>
 <span class="kw">and </span><br/><a NAME="E7:17_2_1"/><span class="lab"><font color="Green" title="E23">A23</font></span>: 
<font color="Maroon" title="c6">a3</font>,<font color="Maroon" title="c8">b2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c7">b1</font>
 <span class="kw">and </span><br/><a NAME="E8:17_2_1"/><span class="lab"><font color="Green" title="E24">A24</font></span>: 
<font color="Maroon" title="c6">a3</font>,<font color="Maroon" title="c9">b3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c7">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="c4">a1</font>,<font color="Maroon" title="c8">b2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c9">b3</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c12">M9</font> = <font color="Maroon" title="c10">M</font>, <font color="Maroon" title="c13">N9</font> = <font color="Maroon" title="c11">N</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">MS</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/><span class="kw">reconsider </span><font color="Maroon" title="c14">a</font> = <font color="Maroon" title="c4">a1</font>, <font color="Maroon" title="c15">b</font> = <font color="Maroon" title="c5">a2</font>, <font color="Maroon" title="c16">c</font> = <font color="Maroon" title="c6">a3</font>, <font color="Maroon" title="c17">a9</font> = <font color="Maroon" title="c7">b1</font>, <font color="Maroon" title="c18">b9</font> = <font color="Maroon" title="c8">b2</font>, <font color="Maroon" title="c19">c9</font> = <font color="Maroon" title="c9">b3</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">MS</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/><a NAME="E11:17_2_1"/><span class="lab"><font color="Green" title="E25">A25</font></span>: 
<font color="Maroon" title="c16">c</font>,<font color="Maroon" title="c19">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c17">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:17_2_1"><span class="lab"><font color="Green" title="E24">A24</font></span></a>, <a class="ref" href="analmetr.html#T48" title="ANALMETR:th.48">ANALMETR:48</a></span>;<br/><a NAME="E12:17_2_1"/>
<font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c7">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a3</font>,<font color="Maroon" title="c8">b2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:17_2_1"><span class="lab"><font color="Green" title="E23">A23</font></span></a>, <a class="ref" href="analmetr.html#T81" title="ANALMETR:th.81">ANALMETR:81</a></span>;<br/><a NAME="E13:17_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A26</font></span>: 
<font color="Maroon" title="c15">b</font>,<font color="Maroon" title="c17">a9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c16">c</font>,<font color="Maroon" title="c18">b9</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="E14:17_2_1"/>
( <font color="Maroon" title="c12">M9</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c13">N9</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="#E1:17_2_1"><span class="lab"><font color="Green" title="E19">A19</font></span></a>, <a class="ref" href="analmetr.html#T58" title="ANALMETR:th.58">ANALMETR:58</a></span>;<br/><a NAME="E15:17_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">b</font>,<font color="Maroon" title="c19">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c18">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_2"><span class="lab"><font color="Green" title="E18">A18</font></span></a>, <a class="txt" href="#E2:17_2_1"><span class="lab"><font color="Green" title="E20">A20</font></span></a>, <a class="txt" href="#E4:17_2_1"><span class="lab"><font color="Green" title="E21">A21</font></span></a>, <a class="txt" href="#E6:17_2_1"><span class="lab"><font color="Green" title="E22">A22</font></span></a>, <a class="txt" href="#E13:17_2_1"><span class="lab"><font color="Green" title="E26">A26</font></span></a>, <a class="txt" href="#E11:17_2_1"><span class="lab"><font color="Green" title="E25">A25</font></span></a>, <a class="ref" href="aff_2.html#D2" title="AFF_2:def.2">AFF_2:def 2</a></span>;<br/><a NAME="E16:17_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c9">b3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c8">b2</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/><span class="kw">hence </span><a NAME="E17:17_2_1"/>
<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c8">b2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c9">b3</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T81" title="ANALMETR:th.81">ANALMETR:81</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="E3:17_2"/>
<font color="Maroon" title="c1">MS</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="E3:17"/>
( <font color="Maroon" title="c1">MS</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">MS</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="#E1:17"><span class="lab"><font color="Green" title="E17">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>
