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

<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</a>;<br/>

<a NAME="E1:21"/><i><font color="Green" title="E14">A1</font></i>: 
( <font color="Maroon" title="c1">X</font> <a href="aff_2.html#NR13" title="AFF_2:NR.13">satisfies_pap</a>  implies <font color="Maroon" title="c1">X</font> <a href="conmetr1.html#NR5" title="CONMETR1:NR.5">satisfies_minor_SCH*</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1"/><i><font color="Green" title="E15">A2</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="aff_2.html#NR13" title="AFF_2:NR.13">satisfies_pap</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">a1</font>, <font color="Maroon" title="c3">a2</font>, <font color="Maroon" title="c4">a3</font>, <font color="Maroon" title="c5">a4</font>, <font color="Maroon" title="c6">b1</font>, <font color="Maroon" title="c7">b2</font>, <font color="Maroon" title="c8">b3</font>, <font color="Maroon" title="c9">b4</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c10">M</font>, <font color="Maroon" title="c11">N</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/><b>assume </b><a NAME="E1:21_1_1"/><i><font color="Green" title="E16">A3</font></i>: 
( <font color="Maroon" title="c10">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Maroon" title="c2">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="c4">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="c7">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; <font color="Maroon" title="c9">b4</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">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Maroon" title="c5">a4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Maroon" title="c6">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">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="c5">a4</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="c3">a2</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">b1</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="c8">b3</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="c2">a1</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="c4">a3</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="c7">b2</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="c9">b4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c3">a2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c7">b2</font> &amp; <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c2">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c6">b1</font> &amp; <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c9">b4</font> )
 ;<br/><a NAME="E2:21_1_1"/><b>then </b><i><font color="Green" title="E17">A4</font></i>: 
( <font color="Maroon" title="c10">M</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c11">N</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> )
 <b>by </b><i><a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></i>;<br/><a NAME="E3:21_1_1"/>
( <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">a2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c6">b1</font> &amp; <font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c8">b3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c3">a2</font> )
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:21_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/><a NAME="E4:21_1_1"/><b>then </b>
<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c8">b3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c6">b1</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:21_1"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="conmetr1.html#E1:21_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="conmetr1.html#E2:21_1_1"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="aff_2.html#D13" title="AFF_2:def.13">AFF_2:def 13</a></i>;<br/><a NAME="E5:21_1_1"/><b>then </b><i><font color="Green" title="E18">A5</font></i>: 
<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c4">a3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c2">a1</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/><a NAME="E6:21_1_1"/>
<font color="Maroon" title="c5">a4</font>,<font color="Maroon" title="c2">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c9">b4</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:21_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/><a NAME="E7:21_1_1"/><b>then </b>
<font color="Maroon" title="c5">a4</font>,<font color="Maroon" title="c4">a3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c9">b4</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:21_1"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="conmetr1.html#E1:21_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="conmetr1.html#E2:21_1_1"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="conmetr1.html#E5:21_1_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="aff_2.html#D13" title="AFF_2:def.13">AFF_2:def 13</a></i>;<br/><b>hence </b><a NAME="E8:21_1_1"/>
<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c9">b4</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:21_1"/>
<font color="Maroon" title="c1">X</font> <a href="conmetr1.html#NR5" title="CONMETR1:NR.5">satisfies_minor_SCH*</a> 
 <b>by </b><i><a class="ref" href="conmetr1.html#D5" target="_self" title="CONMETR1:def.5">Def5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:21"/>
( <font color="Maroon" title="c1">X</font> <a href="conmetr1.html#NR5" title="CONMETR1:NR.5">satisfies_minor_SCH*</a>  implies <font color="Maroon" title="c1">X</font> <a href="aff_2.html#NR13" title="AFF_2:NR.13">satisfies_pap</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_2"/><i><font color="Green" title="E15">A6</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="conmetr1.html#NR5" title="CONMETR1:NR.5">satisfies_minor_SCH*</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">M</font>, <font color="Maroon" title="c3">N</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font>, <font color="Maroon" title="c7">a1</font>, <font color="Maroon" title="c8">b1</font>, <font color="Maroon" title="c9">c1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/><b>assume </b><a NAME="E1:21_2_1"/><i><font color="Green" title="E16">A7</font></i>: 
( <font color="Maroon" title="c2">M</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c3">N</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c2">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c2">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c7">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c8">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c9">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c8">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c7">a1</font> &amp; <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c9">c1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">b1</font> )
 ;<br/><a NAME="E2:21_2_1"/><b>then </b><i><font color="Green" title="E17">A8</font></i>: 
( not <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; not <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; not <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/><a NAME="E3:21_2_1"/><i><font color="Green" title="E18">A9</font></i>: 
( not <font color="Maroon" title="c7">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; not <font color="Maroon" title="c8">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; not <font color="Maroon" title="c9">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> )
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:21_2_1"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/><a NAME="E4:21_2_1"/><i><font color="Green" title="E19">A10</font></i>: 
( <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c8">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c9">c1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b1</font>,<font color="Maroon" title="c6">c</font> )
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:21_2_1"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/><a NAME="E5:21_2_1"/>
<font color="Maroon" title="c8">b1</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c8">b1</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T11" title="AFF_1:th.11">AFF_1:11</a></i>;<br/><a NAME="E6:21_2_1"/><b>then </b>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c9">c1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c6">c</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:21_2"><i><font color="Green" title="E15">A6</font></i></a>, <a class="txt" href="conmetr1.html#E1:21_2_1"><i><font color="Green" title="E16">A7</font></i></a>, <a class="txt" href="conmetr1.html#E2:21_2_1"><i><font color="Green" title="E17">A8</font></i></a>, <a class="txt" href="conmetr1.html#E3:21_2_1"><i><font color="Green" title="E18">A9</font></i></a>, <a class="txt" href="conmetr1.html#E4:21_2_1"><i><font color="Green" title="E19">A10</font></i></a>, <a class="ref" href="conmetr1.html#D5" target="_self" title="CONMETR1:def.5">Def5</a></i>;<br/><b>hence </b><a NAME="E7:21_2_1"/>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c9">c1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">a1</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:21_2"/>
<font color="Maroon" title="c1">X</font> <a href="aff_2.html#NR13" title="AFF_2:NR.13">satisfies_pap</a> 
 <b>by </b><i><a class="ref" href="aff_2.html#D13" title="AFF_2:def.13">AFF_2:def 13</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:21"/>
( <font color="Maroon" title="c1">X</font> <a href="aff_2.html#NR13" title="AFF_2:NR.13">satisfies_pap</a>  iff <font color="Maroon" title="c1">X</font> <a href="conmetr1.html#NR5" title="CONMETR1:NR.5">satisfies_minor_SCH*</a>  )
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:21"><i><font color="Green" title="E14">A1</font></i></a></i>;<br/>


</div>
