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

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

<a NAME="E1:13"/><i><font color="Green">E8</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="conmetr1.html#NR4" target="_self">satisfies_SCH*</a>  implies ( <font color="Maroon">c<sub>1</sub></font> <a href="conmetr1.html#NR5" target="_self">satisfies_minor_SCH*</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="conmetr1.html#NR6" target="_self">satisfies_major_SCH*</a>  ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<a NAME="E1:13_1"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="conmetr1.html#NR4" target="_self">satisfies_SCH*</a>  implies <font color="Maroon">c<sub>1</sub></font> <a href="conmetr1.html#NR5" target="_self">satisfies_minor_SCH*</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_1_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="conmetr1.html#NR4" target="_self">satisfies_SCH*</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font>, <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="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:13_1_1_1"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>2</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>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</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>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>6</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; not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; not <font color="Maroon">c<sub>3</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>10</sub></font> &amp; 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>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font> )
 ;<br/><a NAME="E2:13_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>10</sub></font> is <a href="aff_1.html#V1">being_line</a> &amp; <font color="Maroon">c<sub>11</sub></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/><b>hence </b><a NAME="E3:13_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</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="txt" href="conmetr1.html#E1:13_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="conmetr1.html#E1:13_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="conmetr1.html#D4" target="_self">Def4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:13_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="conmetr1.html#NR5" target="_self">satisfies_minor_SCH*</a> 
 <b>by </b><i><a class="ref" href="conmetr1.html#D5" target="_self">Def5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:13_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="conmetr1.html#NR4" target="_self">satisfies_SCH*</a>  implies <font color="Maroon">c<sub>1</sub></font> <a href="conmetr1.html#NR6" target="_self">satisfies_major_SCH*</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="conmetr1.html#NR4" target="_self">satisfies_SCH*</a> 
 ;<br/>

<a NAME="E2:13_1_2"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font>, <font color="Olive">b<sub>6</sub></font>, <font color="Olive">b<sub>7</sub></font>, <font color="Olive">b<sub>8</sub></font>, <font color="Olive">b<sub>9</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>10</sub></font>, <font color="Olive">b<sub>11</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>10</sub></font> is <a href="aff_1.html#V1">being_line</a> &amp; <font color="Olive">b<sub>11</sub></font> is <a href="aff_1.html#V1">being_line</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>10</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>11</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>10</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>10</sub></font> &amp; <font color="Olive">b<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>10</sub></font> &amp; <font color="Olive">b<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>10</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>11</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>11</sub></font> &amp; <font color="Olive">b<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>11</sub></font> &amp; <font color="Olive">b<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>11</sub></font> &amp; not <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>10</sub></font> &amp; not <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>10</sub></font> &amp; not <font color="Olive">b<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>10</sub></font> &amp; not <font color="Olive">b<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>10</sub></font> &amp; not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>11</sub></font> &amp; not <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>11</sub></font> &amp; not <font color="Olive">b<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>11</sub></font> &amp; not <font color="Olive">b<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>11</sub></font> &amp; <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Olive">b<sub>8</sub></font>,<font color="Olive">b<sub>7</sub></font> &amp; <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="analoaf.html#R2">//</a> <font color="Olive">b<sub>7</sub></font>,<font color="Olive">b<sub>6</sub></font> &amp; <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Olive">b<sub>6</sub></font>,<font color="Olive">b<sub>9</sub></font> holds <br/><font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Olive">b<sub>8</sub></font>,<font color="Olive">b<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="conmetr1.html#D4" target="_self">Def4</a></i>;<br/>
<b>hence </b><a NAME="E3:13_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="conmetr1.html#NR6" target="_self">satisfies_major_SCH*</a> 
 <b>by </b><i><a class="ref" href="conmetr1.html#D6" target="_self">Def6</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:13_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="conmetr1.html#NR4" target="_self">satisfies_SCH*</a>  implies ( <font color="Maroon">c<sub>1</sub></font> <a href="conmetr1.html#NR5" target="_self">satisfies_minor_SCH*</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="conmetr1.html#NR6" target="_self">satisfies_major_SCH*</a>  ) )
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:13_1"><i><font color="Green">E9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:13"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="conmetr1.html#NR5" target="_self">satisfies_minor_SCH*</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="conmetr1.html#NR6" target="_self">satisfies_major_SCH*</a>  implies <font color="Maroon">c<sub>1</sub></font> <a href="conmetr1.html#NR4" target="_self">satisfies_SCH*</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:13_2"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="conmetr1.html#NR5" target="_self">satisfies_minor_SCH*</a> 
 <b>and </b><br/><a NAME="E2:13_2"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="conmetr1.html#NR6" target="_self">satisfies_major_SCH*</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font>, <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="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:13_2_1"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> is <a href="aff_1.html#V1">being_line</a> &amp; <font color="Maroon">c<sub>11</sub></font> is <a href="aff_1.html#V1">being_line</a> &amp; <font color="Maroon">c<sub>2</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>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</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>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>6</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; not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; not <font color="Maroon">c<sub>3</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>10</sub></font> &amp; 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>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2_1_1"/>
not <font color="Maroon">c<sub>10</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><a NAME="E2:13_2_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:13_2_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="aff_1.html#T72">AFF_1:72</a></i>;<br/><b>hence </b><a NAME="E3:13_2_1_1"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</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="txt" href="conmetr1.html#E2:13_2"><i><font color="Green">E10</font></i></a>, <a class="txt" href="conmetr1.html#E1:13_2_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="conmetr1.html#D6" target="_self">Def6</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:13_2_1"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</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="txt" href="conmetr1.html#E1:13_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="conmetr1.html#E1:13_2_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="conmetr1.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:13_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="conmetr1.html#NR4" target="_self">satisfies_SCH*</a> 
 <b>by </b><i><a class="ref" href="conmetr1.html#D4" target="_self">Def4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:13"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="conmetr1.html#NR4" target="_self">satisfies_SCH*</a>  iff ( <font color="Maroon">c<sub>1</sub></font> <a href="conmetr1.html#NR5" target="_self">satisfies_minor_SCH*</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="conmetr1.html#NR6" target="_self">satisfies_major_SCH*</a>  ) )
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:13"><i><font color="Green">E8</font></i></a></i>;<br/>


</div>
