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

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

<a NAME="E1:21"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">X</font> <a href="aff_2.html#NR13" target="_self">satisfies_pap</a>  implies <font color="Maroon">X</font> <a href="conmetr1.html#NR5" target="_self">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">E46</font></i>: 
<font color="Maroon">X</font> <a href="aff_2.html#NR13" target="_self">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">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">a4</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">b4</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="E1:21_1_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">M</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N</font> &amp; <font color="Maroon">a1</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">b2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">b4</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">N</font> &amp; <font color="Maroon">a4</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">b3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; not <font color="Maroon">a4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; not <font color="Maroon">a2</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">M</font> &amp; not <font color="Maroon">b3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; not <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; not <font color="Maroon">a3</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">N</font> &amp; not <font color="Maroon">b4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">a3</font>,<font color="Maroon">a2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3</font>,<font color="Maroon">b2</font> &amp; <font color="Maroon">a2</font>,<font color="Maroon">a1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2</font>,<font color="Maroon">b1</font> &amp; <font color="Maroon">a1</font>,<font color="Maroon">a4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">b4</font> )
 ;<br/><a NAME="E2:21_1_1"/><b>then </b><i><font color="Green">E48</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="aff_1.html#T50">AFF_1:50</a></i>;<br/><a NAME="E3:21_1_1"/>
( <font color="Maroon">a1</font>,<font color="Maroon">a2</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b2</font>,<font color="Maroon">b1</font> &amp; <font color="Maroon">b2</font>,<font color="Maroon">b3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a3</font>,<font color="Maroon">a2</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><a NAME="E4:21_1_1"/><b>then </b>
<font color="Maroon">a1</font>,<font color="Maroon">b3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a3</font>,<font color="Maroon">b1</font>
 <b>by </b><i>, , , <a class="ref" href="aff_2.html#D13">AFF_2:def 13</a></i>;<br/><a NAME="E5:21_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">b1</font>,<font color="Maroon">a3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3</font>,<font color="Maroon">a1</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><a NAME="E6:21_1_1"/>
<font color="Maroon">a4</font>,<font color="Maroon">a1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">b4</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><a NAME="E7:21_1_1"/><b>then </b>
<font color="Maroon">a4</font>,<font color="Maroon">a3</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3</font>,<font color="Maroon">b4</font>
 <b>by </b><i>, , , , <a class="ref" href="aff_2.html#D13">AFF_2:def 13</a></i>;<br/><b>hence </b><a NAME="E8:21_1_1"/>
<font color="Maroon">a3</font>,<font color="Maroon">a4</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b3</font>,<font color="Maroon">b4</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:21_1"/>
<font color="Maroon">X</font> <a href="conmetr1.html#NR5" target="_self">satisfies_minor_SCH*</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:21"/>
( <font color="Maroon">X</font> <a href="conmetr1.html#NR5" target="_self">satisfies_minor_SCH*</a>  implies <font color="Maroon">X</font> <a href="aff_2.html#NR13" target="_self">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">E50</font></i>: 
<font color="Maroon">X</font> <a href="conmetr1.html#NR5" target="_self">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">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>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">c</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">b1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">c1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:21_2_1"/><i><font color="Green">E51</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">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">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/><a NAME="E2:21_2_1"/><b>then </b><i><font color="Green">E54</font></i>: 
( not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; not <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#T59">AFF_1:59</a></i>;<br/><a NAME="E3:21_2_1"/><i><font color="Green">E55</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">M</font> &amp; not <font color="Maroon">c1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/><a NAME="E4:21_2_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">a</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">c1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</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="E5:21_2_1"/>
<font color="Maroon">b1</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">b1</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T11">AFF_1:11</a></i>;<br/><a NAME="E6:21_2_1"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">c1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">c</font>
 <b>by </b><i>, , , , , </i>;<br/><b>hence </b><a NAME="E7:21_2_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="aff_1.html#T13">AFF_1:13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:21_2"/>
<font color="Maroon">X</font> <a href="aff_2.html#NR13" target="_self">satisfies_pap</a> 
 <b>by </b><i><a class="ref" href="aff_2.html#D13">AFF_2:def 13</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:21"/>
( <font color="Maroon">X</font> <a href="aff_2.html#NR13" target="_self">satisfies_pap</a>  iff <font color="Maroon">X</font> <a href="conmetr1.html#NR5" target="_self">satisfies_minor_SCH*</a>  )
 <b>by </b><i/>;<br/>


</div>
