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<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="ami_1.html#NM3">Instruction</a> of <a href="scmpds_2.html#K1" target="_self">SCMPDS</a> ;<br/>

<b>assume </b><a NAME="E1:56"/><i><font color="Green">E21</font></i>: 
 <a href="ami_5.html#K1">InsCode</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 9
 ;<br/>

<a NAME="E2:56"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  : verum </span>}</span>  or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,<span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a>  : verum </span>}</span>  or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<span class="p3"><a href="scmpds_1.html#K9">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="scmpds_1.html#K9">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> 14, B is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a> , B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">2,3</span><a href="tarski.html#K2">}</a></span> </span>}</span>  or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<span class="p3"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="finseq_1.html#K11">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> 14, B is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a> , B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a> , B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K3">{</a><span class="default">4,5,6,7,8</span><a href="enumset1.html#K3">}</a></span> </span>}</span>  or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<span class="p3"><a href="finseq_4.html#K8">&lt;*</a><span class="default"><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></span><a href="finseq_4.html#K8">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> 14, B is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a> , B is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a> , B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a> , B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K3">{</a><span class="default">9,10,11,12,13</span><a href="enumset1.html#K3">}</a></span> </span>}</span>  )
 
<b>by </b><i><a class="txt" href="scmpds_2.html#E17"><i><font color="Green">Lemma8</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> 14, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a> , <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a> <b> such that </b><br/><a NAME="E4:56"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<span class="p2"><a href="finseq_4.html#K8">&lt;*</a><span class="default"><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></span><a href="finseq_4.html#K8">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K3">{</a><span class="default">9,10,11,12,13</span><a href="enumset1.html#K3">}</a></span> )
 <b>by </b><i><a class="txt" href="scmpds_2.html#E1:56"><i><font color="Green">E21</font></i></a>, <a class="txt" href="scmpds_2.html#E69"><i><font color="Green">Lemma20</font></i></a></i>;<br/>
<a NAME="E5:56"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 9 or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 10 or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 11 or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 12 or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 13 )
 
<b>by </b><i><a class="txt" href="scmpds_2.html#E4:56"><i><font color="Green">E22</font></i></a>, <a class="ref" href="enumset1.html#D3">ENUMSET1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:56_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 10 or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 11 or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 12 or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 13 )
 ;<br/><a NAME="E2:56_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 10 or <font color="Maroon">c<sub>6</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 11 or <font color="Maroon">c<sub>6</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 12 or <font color="Maroon">c<sub>6</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 13 )
 <b>by </b><i><a class="txt" href="scmpds_2.html#E4:56"><i><font color="Green">E22</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/><b>hence </b><a NAME="E3:56_1"/>
contradiction
 <b>by </b><i><a class="txt" href="scmpds_2.html#E1:56"><i><font color="Green">E21</font></i></a>, <a class="ref" href="ami_5.html#D1">AMI_5:def 1</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a> , <font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E8:56"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">9,<span class="p2"><a href="finseq_4.html#K8">&lt;*</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="finseq_4.html#K8">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="scmpds_2.html#E4:56"><i><font color="Green">E22</font></i></a>, <a class="txt" href="scmpds_2.html#E5:56"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>13</sub></font> as    <a href="scmpds_2.html#M1" target="_self">Int_position</a>  <b>by </b><i><a class="ref" href="scmpds_2.html#D2" target="_self">Def2</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>16</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>17</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>14</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>15</sub></font>
;<br/>

<b>thus </b><a NAME="E10:56"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K13" target="_self">AddTo</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="scmpds_2.html#E8:56"><i><font color="Green">E24</font></i></a></i>;<br/>


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