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

<b>let </b><font color="Maroon">X</font> be   <a href="analmetr.html#NM2">OrtAfPl</a>;<br/>

<a NAME="E1:28"/><i><font color="Green">E45</font></i>: 
(  <a href="conmetr.html#NR2" target="_self">PAP_holds_in</a> <font color="Maroon">X</font> implies  <a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font> <a href="aff_2.html#NR2" target="_self">satisfies_PAP</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_1"/><i><font color="Green">E46</font></i>: 
 <a href="conmetr.html#NR2" target="_self">PAP_holds_in</a> <font color="Maroon">X</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">M</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span>;<br/><b>let </b><font color="Maroon">N</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span>;<br/><b>let </b><font color="Maroon">o</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span>;<br/><b>let </b><font color="Maroon">a1</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span>;<br/><b>let </b><font color="Maroon">b1</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span>;<br/><b>let </b><font color="Maroon">c1</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span>;<br/><b>assume </b><a NAME="E1:28_1_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">M</font> is <a href="aff_1.html#V1">being_line</a> &amp; <font color="Maroon">N</font> is <a href="aff_1.html#V1">being_line</a> &amp; <font color="Maroon">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">N</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c1</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">c1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">a1</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">c1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">b1</font> )
 ;<br/><b>reconsider </b><font color="Maroon">M'</font> = <font color="Maroon">M</font>, <font color="Maroon">N'</font> = <font color="Maroon">N</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="analmetr.html#T57">ANALMETR:57</a></i>;<br/><b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</font>, <font color="Maroon">c'</font> = <font color="Maroon">c</font>, <font color="Maroon">a1'</font> = <font color="Maroon">a1</font>, <font color="Maroon">b1'</font> = <font color="Maroon">b1</font>, <font color="Maroon">c1'</font> = <font color="Maroon">c1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="analmetr.html#T47">ANALMETR:47</a></i>;<br/><a NAME="E4:28_1_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">M'</font> <a href="analmetr.html#NR6" target="_self">is_line</a>  &amp; <font color="Maroon">N'</font> <a href="analmetr.html#NR6" target="_self">is_line</a>  )
 <b>by </b><i>, <a class="ref" href="analmetr.html#T58">ANALMETR:58</a></i>;<br/><a NAME="E5:28_1_1"/>
<font color="Maroon">b</font>,<font color="Maroon">a1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">b1</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><a NAME="E6:28_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
( <font color="Maroon">b'</font>,<font color="Maroon">a1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a'</font>,<font color="Maroon">b1'</font> &amp; <font color="Maroon">b'</font>,<font color="Maroon">c1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c'</font>,<font color="Maroon">b1'</font> )
 <b>by </b><i>, <a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/><a NAME="E7:28_1_1"/>
( not <font color="Maroon">c1'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M'</font> &amp; not <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N'</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_1_1_1"/>
( <font color="Maroon">c1'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M'</font> or <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N'</font> )
 ;<br/>

<a NAME="E2:28_1_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
( <font color="Maroon">o</font>,<font color="Maroon">c1</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">M</font> or <font color="Maroon">o</font>,<font color="Maroon">b</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">N</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T66">AFF_1:66</a></i>;<br/>
<a NAME="E3:28_1_1_1"/>
( <font color="Maroon">o</font>,<font color="Maroon">c1</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">N</font> &amp; <font color="Maroon">o</font>,<font color="Maroon">b</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">M</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T66">AFF_1:66</a></i>;<br/>
<a NAME="E4:28_1_1_1"/><b>then </b>
<font color="Maroon">M</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T67">AFF_1:67</a></i>;<br/>
<b>hence </b><a NAME="E5:28_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:28_1_1"/><b>then </b>
<font color="Maroon">a'</font>,<font color="Maroon">c1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c'</font>,<font color="Maroon">a1'</font>
 <b>by </b><i>, , , , <a class="ref" href="conmetr.html#D2">CONMETR:def 2</a></i>;<br/><b>hence </b><a NAME="E9:28_1_1"/>
<font color="Maroon">a</font>,<font color="Maroon">c1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">a1</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:28_1"/>
 <a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font> <a href="aff_2.html#NR2" target="_self">satisfies_PAP</a> 
 <b>by </b><i><a class="ref" href="aff_2.html#D2">AFF_2:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:28"/>
(  <a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font> <a href="aff_2.html#NR2" target="_self">satisfies_PAP</a>  implies  <a href="conmetr.html#NR2" target="_self">PAP_holds_in</a> <font color="Maroon">X</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_2"/><i><font color="Green">E51</font></i>: 
 <a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font> <a href="aff_2.html#NR2" target="_self">satisfies_PAP</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">o'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">a1'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">a2'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">a3'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">b1'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">b2'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">b3'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">M'</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">N'</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:28_2_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">M'</font> <a href="analmetr.html#NR6" target="_self">is_line</a>  &amp; <font color="Maroon">N'</font> <a href="analmetr.html#NR6" target="_self">is_line</a>  &amp; <font color="Maroon">o'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M'</font> &amp; <font color="Maroon">a1'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M'</font> &amp; <font color="Maroon">a2'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M'</font> &amp; <font color="Maroon">a3'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M'</font> &amp; <font color="Maroon">o'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N'</font> &amp; <font color="Maroon">b1'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N'</font> &amp; <font color="Maroon">b2'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N'</font> &amp; <font color="Maroon">b3'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N'</font> &amp; not <font color="Maroon">b2'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M'</font> &amp; not <font color="Maroon">a3'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N'</font> &amp; <font color="Maroon">o'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a1'</font> &amp; <font color="Maroon">o'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2'</font> &amp; <font color="Maroon">o'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a3'</font> &amp; <font color="Maroon">o'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b1'</font> &amp; <font color="Maroon">o'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2'</font> &amp; <font color="Maroon">o'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b3'</font> &amp; <font color="Maroon">a3'</font>,<font color="Maroon">b2'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2'</font>,<font color="Maroon">b1'</font> &amp; <font color="Maroon">a3'</font>,<font color="Maroon">b3'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1'</font>,<font color="Maroon">b1'</font> )
 ;<br/><b>reconsider </b><font color="Maroon">M</font> = <font color="Maroon">M'</font>, <font color="Maroon">N</font> = <font color="Maroon">N'</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i><a class="ref" href="analmetr.html#T57">ANALMETR:57</a></i>;<br/><b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a1'</font>, <font color="Maroon">a2</font> = <font color="Maroon">a2'</font>, <font color="Maroon">a3</font> = <font color="Maroon">a3'</font>, <font color="Maroon">b1</font> = <font color="Maroon">b1'</font>, <font color="Maroon">b2</font> = <font color="Maroon">b2'</font>, <font color="Maroon">b3</font> = <font color="Maroon">b3'</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i><a class="ref" href="analmetr.html#T47">ANALMETR:47</a></i>;<br/><a NAME="E4:28_2_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">M</font> is <a href="aff_1.html#V1">being_line</a> &amp; <font color="Maroon">N</font> is <a href="aff_1.html#V1">being_line</a> )
 <b>by </b><i>, <a class="ref" href="analmetr.html#T58">ANALMETR:58</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_2_1_1"/>
<font color="Maroon">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">N</font>
 ;<br/><a NAME="E2:28_2_1_1"/>
( <font color="Maroon">a3</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">b1</font> &amp; <font color="Maroon">a3</font>,<font color="Maroon">b3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font> )
 <b>by </b><i>, <a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/><a NAME="E3:28_2_1_1"/><b>then </b>
( <font color="Maroon">a3</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">b1</font> &amp; <font color="Maroon">a1</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a3</font>,<font color="Maroon">b3</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><a NAME="E4:28_2_1_1"/><b>then </b>
<font color="Maroon">a1</font>,<font color="Maroon">b2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2</font>,<font color="Maroon">b3</font>
 <b>by </b><i>, , , <a class="ref" href="aff_2.html#D2">AFF_2:def 2</a></i>;<br/><b>hence </b><a NAME="E5:28_2_1_1"/>
<font color="Maroon">a1'</font>,<font color="Maroon">b2'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2'</font>,<font color="Maroon">b3'</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:28_2_1"/>
<font color="Maroon">a1'</font>,<font color="Maroon">b2'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a2'</font>,<font color="Maroon">b3'</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:28_2"/>
 <a href="conmetr.html#NR2" target="_self">PAP_holds_in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="conmetr.html#D2">CONMETR:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:28"/>
(  <a href="conmetr.html#NR2" target="_self">PAP_holds_in</a> <font color="Maroon">X</font> iff  <a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font> <a href="aff_2.html#NR2" target="_self">satisfies_PAP</a>  )
 <b>by </b><i/>;<br/>


</div>
