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

<b>let </b><font color="Maroon">AP</font> be   <a href="diraf.html#NM2">AffinPlane</a>;<br/>

<a NAME="E1:38"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">AP</font> <a href="aff_2.html#NR1" target="_self">satisfies_PPAP</a>  implies ( <font color="Maroon">AP</font> <a href="aff_2.html#NR2" target="_self">satisfies_PAP</a>  &amp; <font color="Maroon">AP</font> <a href="aff_2.html#NR13" target="_self">satisfies_pap</a>  ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">AP</font> <a href="aff_2.html#NR1" target="_self">satisfies_PPAP</a> 
 ;<br/>

<b>thus </b><a NAME="E2:38_1"/>
<font color="Maroon">AP</font> <a href="aff_2.html#NR2" target="_self">satisfies_PAP</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">M</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AP</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="aff_2.html#D2" target="_self">AFF_2:def 2</a><br/><b>let </b><font color="Maroon">N</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AP</font>;<br/><b>let </b><font color="Maroon">o</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font>;<br/><b>let </b><font color="Maroon">a'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font>;<br/><b>let </b><font color="Maroon">b'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font>;<br/><b>let </b><font color="Maroon">c'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font>;<br/>

















<b>assume </b><a NAME="E1:38_1_1"/>
( <font color="Maroon">M</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">N</font> <a href="aff_1.html#NR2" target="_self">is_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">a'</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">b'</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">c'</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">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">c'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">a'</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">c'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">b'</font> )
 ;<br/>

<b>hence </b><a NAME="E2:38_1_1"/>
<font color="Maroon">a</font>,<font color="Maroon">c'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">a'</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:38_1"/>
<font color="Maroon">AP</font> <a href="aff_2.html#NR13" target="_self">satisfies_pap</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">M</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AP</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="aff_2.html#D13" target="_self">AFF_2:def 13</a><br/><b>let </b><font color="Maroon">N</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AP</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font>;<br/><b>let </b><font color="Maroon">a'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font>;<br/><b>let </b><font color="Maroon">b'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font>;<br/><b>let </b><font color="Maroon">c'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font>;<br/>















<b>assume </b><a NAME="E1:38_1_2"/>
( <font color="Maroon">M</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">N</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &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">M</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N</font> &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">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">c'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">a'</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">c'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">b'</font> )
 ;<br/>

<b>hence </b><a NAME="E2:38_1_2"/>
<font color="Maroon">a</font>,<font color="Maroon">c'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">a'</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E2:38"/>
( <font color="Maroon">AP</font> <a href="aff_2.html#NR2" target="_self">satisfies_PAP</a>  &amp; <font color="Maroon">AP</font> <a href="aff_2.html#NR13" target="_self">satisfies_pap</a>  implies <font color="Maroon">AP</font> <a href="aff_2.html#NR1" target="_self">satisfies_PPAP</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">AP</font> <a href="aff_2.html#NR2" target="_self">satisfies_PAP</a>  &amp; <font color="Maroon">AP</font> <a href="aff_2.html#NR13" target="_self">satisfies_pap</a>  )
 ;<br/>

<b>thus </b><a NAME="E2:38_2"/>
<font color="Maroon">AP</font> <a href="aff_2.html#NR1" target="_self">satisfies_PPAP</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">M</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AP</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="aff_2.html#D1" target="_self">AFF_2:def 1</a><br/><b>let </b><font color="Maroon">N</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AP</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font>;<br/><b>let </b><font color="Maroon">a'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font>;<br/><b>let </b><font color="Maroon">b'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font>;<br/><b>let </b><font color="Maroon">c'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font>;<br/>















<b>assume </b><a NAME="E1:38_2_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">M</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">N</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &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">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">c'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">a'</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">c'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">b'</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_2_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">N</font>
 ;<br/><b>assume </b><a NAME="E2:38_2_1_1"/><i><font color="Green">E47</font></i>: 
not <font color="Maroon">a</font>,<font color="Maroon">c'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">a'</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_2_1_1_1"/>
not <font color="Maroon">M</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N</font>
 ;<br/><b>then consider </b><font color="Maroon">o</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font><b> such that </b><br/><a NAME="E3:38_2_1_1_1"/><i><font color="Green">E48</font></i>: 
( <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> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T72">AFF_1:72</a></i>;<br/><a NAME="E4:38_2_1_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_2_1_1_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E2:38_2_1_1_1_1"/><b>then </b>
<font color="Maroon">o</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a'</font>,<font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E3:38_2_1_1_1_1"/><b>then </b>
( <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font> or <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> )
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T62">AFF_1:62</a></i>;<br/>
<a NAME="E4:38_2_1_1_1_1"/><b>then </b>
( <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font> or <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<a NAME="E5:38_2_1_1_1_1"/><b>then </b>
( <font color="Maroon">c</font>,<font color="Maroon">o</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">c'</font> or <font color="Maroon">o</font>,<font color="Maroon">c'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b'</font>,<font color="Maroon">c</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E6:38_2_1_1_1_1"/><b>then </b>
( <font color="Maroon">c'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> or <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> or <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> or <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c'</font> )
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T62">AFF_1:62</a></i>;<br/>
<a NAME="E7:38_2_1_1_1_1"/><b>then </b>
( <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> or <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c'</font> )
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<b>hence </b><a NAME="E8:38_2_1_1_1_1"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="aff_1.html#T12">AFF_1:12</a>, <a class="ref" href="aff_1.html#T65">AFF_1:65</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:38_2_1_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_1_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_2_1_1_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:38_2_1_1_1_2"/><b>then </b>
<font color="Maroon">o</font>,<font color="Maroon">a'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b'</font>,<font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E3:38_2_1_1_1_2"/><b>then </b><i><font color="Green">E53</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> )
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T62">AFF_1:62</a></i>;<br/>
<a NAME="E4:38_2_1_1_1_2"/>
<font color="Maroon">o</font>,<font color="Maroon">c'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b'</font>,<font color="Maroon">c</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E5:38_2_1_1_1_2"/><b>then </b>
( <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c'</font> or <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> )
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T62">AFF_1:62</a></i>;<br/>
<a NAME="E6:38_2_1_1_1_2"/><b>then </b>
( <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c'</font> or <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<b>hence </b><a NAME="E7:38_2_1_1_1_2"/>
contradiction
 <b>by </b><i>, , , , , , <a class="ref" href="aff_1.html#T12">AFF_1:12</a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a>, <a class="ref" href="aff_1.html#T65">AFF_1:65</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:38_2_1_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_1_1_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_2_1_1_1_3"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 ;<br/>

<a NAME="E2:38_2_1_1_1_3"/><b>then </b>
<font color="Maroon">o</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c'</font>,<font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E3:38_2_1_1_1_3"/><b>then </b>
( <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font> or <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> )
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T62">AFF_1:62</a></i>;<br/>
<a NAME="E4:38_2_1_1_1_3"/><b>then </b>
<font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i>, , , , <a class="ref" href="aff_1.html#T30">AFF_1:30</a>, <a class="ref" href="aff_1.html#T62">AFF_1:62</a></i>;<br/>
<a NAME="E5:38_2_1_1_1_3"/><b>then </b>
<font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font>
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<b>hence </b><a NAME="E6:38_2_1_1_1_3"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T12">AFF_1:12</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:38_2_1_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_1_1_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_2_1_1_1_4"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font>
 ;<br/>

<a NAME="E2:38_2_1_1_1_4"/><b>then </b>
<font color="Maroon">o</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">b'</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E3:38_2_1_1_1_4"/><b>then </b>
<font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T62">AFF_1:62</a></i>;<br/>
<a NAME="E4:38_2_1_1_1_4"/><b>then </b>
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font>
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<a NAME="E5:38_2_1_1_1_4"/><b>then </b>
<font color="Maroon">c</font>,<font color="Maroon">o</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">c'</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E6:38_2_1_1_1_4"/><b>then </b>
( <font color="Maroon">a'</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> )
 
<b>by </b><i>, , , , <a class="ref" href="aff_1.html#T62">AFF_1:62</a></i>;<br/>
<b>hence </b><a NAME="E7:38_2_1_1_1_4"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T65">AFF_1:65</a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:38_2_1_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_1_1_1_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_2_1_1_1_5"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font>
 ;<br/>

<a NAME="E2:38_2_1_1_1_5"/><b>then </b>
<font color="Maroon">o</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">c'</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E3:38_2_1_1_1_5"/><b>then </b>
( <font color="Maroon">a'</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> )
 
<b>by </b><i>, , , , <a class="ref" href="aff_2.html#D1" target="_self">Def1</a>, <a class="ref" href="aff_1.html#T62">AFF_1:62</a></i>;<br/>
<b>hence </b><a NAME="E4:38_2_1_1_1_5"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T65">AFF_1:65</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:38_2_1_1_1"/>
<font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_1_1_1_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_2_1_1_1_6"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c'</font>
 ;<br/>

<a NAME="E2:38_2_1_1_1_6"/><b>then </b>
<font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T62">AFF_1:62</a></i>;<br/>
<a NAME="E3:38_2_1_1_1_6"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">o</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">a'</font>
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<a NAME="E4:38_2_1_1_1_6"/><b>then </b>
( <font color="Maroon">a'</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> )
 
<b>by </b><i>, , , <a class="ref" href="aff_2.html#D1" target="_self">Def1</a>, <a class="ref" href="aff_1.html#T62">AFF_1:62</a></i>;<br/>
<b>hence </b><a NAME="E5:38_2_1_1_1_6"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T65">AFF_1:65</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E10:38_2_1_1_1"/>
<font color="Maroon">a</font>,<font color="Maroon">c'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">a'</font>
 <b>by </b><i>, , , , , , , , , </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:38_2_1_1"/>
<font color="Maroon">a</font>,<font color="Maroon">c'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">a'</font>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:38_2_1"/>
<font color="Maroon">a</font>,<font color="Maroon">c'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">a'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T65">AFF_1:65</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:38"/>
( <font color="Maroon">AP</font> <a href="aff_2.html#NR1" target="_self">satisfies_PPAP</a>  iff ( <font color="Maroon">AP</font> <a href="aff_2.html#NR2" target="_self">satisfies_PAP</a>  &amp; <font color="Maroon">AP</font> <a href="aff_2.html#NR13" target="_self">satisfies_pap</a>  ) )
 <b>by </b><i/>;<br/>


</div>
