<?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/>

<a NAME="E1:25"/><i><font color="Green">E15</font></i>: 
(  <a href="analmetr.html#K3">Af</a> <font color="Maroon">c<sub>1</sub></font> <a href="conmetr1.html#NR3" target="_self">satisfies_SCH</a>  implies  <a href="conmetr.html#NR6" target="_self">SCH_holds_in</a> <font color="Maroon">c<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1"/><i><font color="Green">E16</font></i>: 
 <a href="analmetr.html#K3">Af</a> <font color="Maroon">c<sub>1</sub></font> <a href="conmetr1.html#NR3" target="_self">satisfies_SCH</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_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="E1:25_1_1"/><i><font color="Green">E17</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>  &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>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>8</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>7</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> &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>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; not <font color="Maroon">c<sub>9</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>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; not <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>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/><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="E3:25_1_1"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> is <a href="aff_1.html#V1">being_line</a> &amp; <font color="Maroon">c<sub>13</sub></font> is <a href="aff_1.html#V1">being_line</a> )
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:25_1_1"><i><font color="Green">E17</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>2</sub></font>, <font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="struct_0.html#NM1">Element</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="E5:25_1_1"/>
( <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:25_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/><a NAME="E6:25_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>20</sub></font>,<font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:25_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="conmetr1.html#E1:25_1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="conmetr1.html#E3:25_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="conmetr1.html#D3" target="_self">Def3</a></i>;<br/><b>hence </b><a NAME="E7:25_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="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:25_1"/>
 <a href="conmetr.html#NR6" target="_self">SCH_holds_in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="conmetr.html#D6">CONMETR:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:25"/>
(  <a href="conmetr.html#NR6" target="_self">SCH_holds_in</a> <font color="Maroon">c<sub>1</sub></font> implies  <a href="analmetr.html#K3">Af</a> <font color="Maroon">c<sub>1</sub></font> <a href="conmetr1.html#NR3" target="_self">satisfies_SCH</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_2"/><i><font color="Green">E16</font></i>: 
 <a href="conmetr.html#NR6" target="_self">SCH_holds_in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_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 <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>10</sub></font>, <font color="Maroon">c<sub>11</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>assume </b><a NAME="E1:25_2_1"/><i><font color="Green">E17</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>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>8</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>7</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> &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>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; not <font color="Maroon">c<sub>9</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>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; not <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>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/><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 <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="E3:25_2_1"/><i><font color="Green">E18</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="conmetr1.html#E1:25_2_1"><i><font color="Green">E17</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>2</sub></font>, <font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="struct_0.html#NM1">Element</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="E5:25_2_1"/>
( <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:25_2_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/><a NAME="E6:25_2_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>20</sub></font>,<font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:25_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="conmetr1.html#E1:25_2_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="conmetr1.html#E3:25_2_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="conmetr.html#D6">CONMETR:def 6</a></i>;<br/><b>hence </b><a NAME="E7:25_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="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:25_2"/>
 <a href="analmetr.html#K3">Af</a> <font color="Maroon">c<sub>1</sub></font> <a href="conmetr1.html#NR3" target="_self">satisfies_SCH</a> 
 <b>by </b><i><a class="ref" href="conmetr1.html#D3" target="_self">Def3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:25"/>
(  <a href="analmetr.html#K3">Af</a> <font color="Maroon">c<sub>1</sub></font> <a href="conmetr1.html#NR3" target="_self">satisfies_SCH</a>  iff  <a href="conmetr.html#NR6" target="_self">SCH_holds_in</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:25"><i><font color="Green">E15</font></i></a></i>;<br/>


</div>
