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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="analmetr.html#NM2">OrtAfPl</a>;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="analmetr.html#K3">Af</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<div><i><font color="Green">E17</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">E18</font></i>: 
 <a href="conmetr.html#NR2" target="_self">PAP_holds_in</a> <font color="Maroon">c<sub>1</sub></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">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></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">c<sub>1</sub></font></span>)</span>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:17_1_1"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>4</sub></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">E20</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E3:17_1_1"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>and </b><br/><a NAME="E4:17_1_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>and </b><br/><a NAME="E5:17_1_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>and </b><br/><a NAME="E6:17_1_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font> )
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></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">E25</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="analmetr.html#NR6" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>13</sub></font> <a href="analmetr.html#NR6" target="_self">is_line</a>  )
 <b>by </b><i><a class="txt" href="euclmetr.html#E1:17_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="analmetr.html#T58">ANALMETR:58</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></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">E26</font></i>: 
( not <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; not <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="euclmetr.html#E2:17_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="euclmetr.html#E3:17_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="euclmetr.html#E4:17_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="euclmetr.html#E5:17_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="euclmetr.html#E8:17_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="conmetr.html#T5">CONMETR:5</a></i>;<br/><a NAME="E11:17_1_1"/>
( <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>18</sub></font> )
 <b>by </b><i><a class="txt" href="euclmetr.html#E6:17_1_1"><i><font color="Green">E24</font></i></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">c<sub>15</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>18</sub></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">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="euclmetr.html#E1:17_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="euclmetr.html#E3:17_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="euclmetr.html#E4:17_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="euclmetr.html#E5:17_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="euclmetr.html#E8:17_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="euclmetr.html#E10:17_1_1"><i><font color="Green">E26</font></i></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">c<sub>14</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></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">E18</font></i>: 
 <a href="analmetr.html#K3">Af</a> <font color="Maroon">c<sub>1</sub></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">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:17_2_1"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="analmetr.html#NR6" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>11</sub></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">E20</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>and </b><br/><a NAME="E3:17_2_1"/><i><font color="Green">E21</font></i>: 
( not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>and </b><br/><a NAME="E4:17_2_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>and </b><br/><a NAME="E5:17_2_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 ;<br/><a NAME="E6:17_2_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="euclmetr.html#E5:17_2_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></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">c<sub>1</sub></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">E25</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>13</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>by </b><i><a class="txt" href="euclmetr.html#E1:17_2_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="analmetr.html#T58">ANALMETR:58</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></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">c<sub>15</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="euclmetr.html#E6:17_2_1"><i><font color="Green">E24</font></i></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">c<sub>15</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="euclmetr.html#E1:17_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="euclmetr.html#E2:17_2_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="euclmetr.html#E3:17_2_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="euclmetr.html#E4:17_2_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="euclmetr.html#E8:17_2_1"><i><font color="Green">E25</font></i></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">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>8</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></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">c<sub>1</sub></font> iff  <a href="analmetr.html#K3">Af</a> <font color="Maroon">c<sub>1</sub></font> <a href="papdesaf.html#NR1" target="_self">satisfies_PAP'</a>  )
 <b>by </b><i><a class="txt" href="euclmetr.html#E1:17"><i><font color="Green">E17</font></i></a></i>;<br/>


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