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<div class="add">

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

<b>set </b><font color="Maroon">AS</font> =  <a href="analmetr.html#K3">Af</a> <font color="Maroon">MS</font>;<br/>
<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_1"/><i><font color="Green">E35</font></i>: 
 <a href="conmetr.html#NR2" target="_self">PAP_holds_in</a> <font color="Maroon">MS</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_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">MS</font></span>)</span>, <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">MS</font></span>)</span>;<br/><b>let </b><font color="Maroon">o</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">MS</font></span>)</span>, <font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">MS</font></span>)</span>, <font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">MS</font></span>)</span>, <font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">MS</font></span>)</span>, <font color="Maroon">a'</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">MS</font></span>)</span>, <font color="Maroon">b'</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">MS</font></span>)</span>, <font color="Maroon">c'</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">MS</font></span>)</span>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:17_1_1"/><i><font color="Green">E44</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>  )
 <b>and </b><br/><a NAME="E2:17_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E3:17_1_1"/><i><font color="Green">E51</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>and </b><br/><a NAME="E4:17_1_1"/><i><font color="Green">E64</font></i>: 
( <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> )
 <b>and </b><br/><a NAME="E5:17_1_1"/><i><font color="Green">E66</font></i>: 
( <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> )
 <b>and </b><br/><a NAME="E6:17_1_1"/><i><font color="Green">E67</font></i>: 
( <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>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">MS</font> <b>by </b><i><a class="ref" href="analmetr.html#T57">ANALMETR:57</a></i>;<br/><a NAME="E8:17_1_1"/><i><font color="Green">E69</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/><b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font>, <font color="Maroon">b1</font> = <font color="Maroon">b</font>, <font color="Maroon">c1</font> = <font color="Maroon">c</font>, <font color="Maroon">a1'</font> = <font color="Maroon">a'</font>, <font color="Maroon">b1'</font> = <font color="Maroon">b'</font>, <font color="Maroon">c1'</font> = <font color="Maroon">c'</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">MS</font> <b>by </b><i><a class="ref" href="analmetr.html#T47">ANALMETR:47</a></i>;<br/><a NAME="E10:17_1_1"/><i><font color="Green">E77</font></i>: 
( not <font color="Maroon">a1'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M'</font> &amp; not <font color="Maroon">b1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N'</font> )
 <b>by </b><i>, <a class="ref" href="euclmetr.html#D2" target="_self">Def2</a>, <a class="ref" href="euclmetr.html#D3" target="_self">Def3</a>, <a class="ref" href="euclmetr.html#D4" target="_self">Def4</a>, <a class="ref" href="euclmetr.html#D6" target="_self">Def6</a>, <a class="ref" href="conmetr.html#T5">CONMETR:5</a></i>;<br/><a NAME="E11:17_1_1"/>
( <font color="Maroon">a1</font>,<font color="Maroon">b1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">a1'</font> &amp; <font color="Maroon">b1</font>,<font color="Maroon">c1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c1</font>,<font color="Maroon">b1'</font> )
 <b>by </b><i><a class="ref" href="euclmetr.html#D5" target="_self">Def5</a>, <a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/><a NAME="E12:17_1_1"/><b>then </b>
( <font color="Maroon">b1</font>,<font color="Maroon">a1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">b1'</font> &amp; <font color="Maroon">b1</font>,<font color="Maroon">c1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c1</font>,<font color="Maroon">b1'</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/><a NAME="E13:17_1_1"/><b>then </b>
<font color="Maroon">c1</font>,<font color="Maroon">a1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">c1'</font>
 <b>by </b><i>, <a class="ref" href="euclmetr.html#D2" target="_self">Def2</a>, <a class="ref" href="euclmetr.html#D3" target="_self">Def3</a>, <a class="ref" href="euclmetr.html#D4" target="_self">Def4</a>, <a class="ref" href="euclmetr.html#D6" target="_self">Def6</a>, <a class="ref" href="euclmetr.html#D7" target="_self">Def7</a>, <a class="ref" href="conmetr.html#D2">CONMETR:def 2</a></i>;<br/><a NAME="E14:17_1_1"/><b>then </b>
<font color="Maroon">a1</font>,<font color="Maroon">c1'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c1</font>,<font color="Maroon">a1'</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/><b>hence </b><a NAME="E15:17_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><a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:17_1"/>
 <a href="analmetr.html#K3">Af</a> <font color="Maroon">MS</font> <a href="papdesaf.html#NR1" 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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_2"/><i><font color="Green">E80</font></i>: 
 <a href="analmetr.html#K3">Af</a> <font color="Maroon">MS</font> <a href="papdesaf.html#NR1" target="_self">satisfies_PAP'</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_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">MS</font>, <font color="Maroon">a1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">MS</font>, <font color="Maroon">a2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">MS</font>, <font color="Maroon">a3</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">MS</font>, <font color="Maroon">b1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">MS</font>, <font color="Maroon">b2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">MS</font>, <font color="Maroon">b3</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">MS</font>;<br/><b>let </b><font color="Maroon">M</font> be   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">MS</font>, <font color="Maroon">N</font> be   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">MS</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:17_2_1"/><i><font color="Green">E81</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>and </b><br/><a NAME="E2:17_2_1"/><i><font color="Green">E82</font></i>: 
( <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> )
 <b>and </b><br/><a NAME="E3:17_2_1"/><i><font color="Green">E97</font></i>: 
( 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> )
 <b>and </b><br/><a NAME="E4:17_2_1"/><i><font color="Green">E98</font></i>: 
( <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> )
 <b>and </b><br/><a NAME="E5:17_2_1"/><i><font color="Green">E99</font></i>: 
( <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/><a NAME="E6:17_2_1"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">a2</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a3</font>,<font color="Maroon">b2</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="euclmetr.html#D4" target="_self">Def4</a>, <a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<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">MS</font></span>)</span> <b>by </b><i><a class="ref" href="analmetr.html#T57">ANALMETR:57</a></i>;<br/><a NAME="E8:17_2_1"/><i><font color="Green">E101</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>  )
 <b>by </b><i>, <a class="ref" href="analmetr.html#T58">ANALMETR:58</a></i>;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a1</font>, <font color="Maroon">b</font> = <font color="Maroon">a2</font>, <font color="Maroon">c</font> = <font color="Maroon">a3</font>, <font color="Maroon">a'</font> = <font color="Maroon">b1</font>, <font color="Maroon">b'</font> = <font color="Maroon">b2</font>, <font color="Maroon">c'</font> = <font color="Maroon">b3</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">MS</font></span>)</span> <b>by </b><i><a class="ref" href="analmetr.html#T47">ANALMETR:47</a></i>;<br/><a NAME="E10:17_2_1"/>
( <font color="Maroon">b</font>,<font color="Maroon">a'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">b'</font> &amp; <font color="Maroon">c</font>,<font color="Maroon">c'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">a'</font> )
 <b>by </b><i><a class="ref" href="euclmetr.html#D5" target="_self">Def5</a>, <a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/><a NAME="E11:17_2_1"/><b>then </b>
<font color="Maroon">b</font>,<font color="Maroon">c'</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="euclmetr.html#D2" target="_self">Def2</a>, <a class="ref" href="euclmetr.html#D3" target="_self">Def3</a>, <a class="ref" href="euclmetr.html#D6" target="_self">Def6</a>, <a class="ref" href="aff_2.html#D2">AFF_2:def 2</a></i>;<br/><a NAME="E12:17_2_1"/><b>then </b>
<font color="Maroon">a2</font>,<font color="Maroon">b3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">b2</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/><b>hence </b><a NAME="E13:17_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><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:17_2"/>
 <a href="conmetr.html#NR2" target="_self">PAP_holds_in</a> <font color="Maroon">MS</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="E3:17"/>
(  <a href="conmetr.html#NR2" target="_self">PAP_holds_in</a> <font color="Maroon">MS</font> iff  <a href="analmetr.html#K3">Af</a> <font color="Maroon">MS</font> <a href="papdesaf.html#NR1" target="_self">satisfies_PAP'</a>  )
 <b>by </b><i/>;<br/>


</div>
