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<b>let </b><font color="Maroon">k1</font> be   <a href="int_1.html#NM1">Integer</a>;<br/>

<b>assume </b><a NAME="E1:96"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">k1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<b>set </b><font color="Maroon">n</font> =  <a href="int_2.html#K1">abs</a> <font color="Maroon">k1</font>;<br/>
<a NAME="E2:96"/>
<span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">k1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>
<a NAME="E3:96"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">k1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="ami_2.html#K3">SCM-Instr-Loc</a> 
 
<b>by </b><i><a class="ref" href="ami_2.html#D3">AMI_2:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">loc</font> = 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">k1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="ami_1.html#NM2">Instruction-Location</a> of <a href="scmpds_2.html#K1" target="_self">SCMPDS</a>  ;<br/>
<b>consider </b><font color="Maroon">s</font> being   <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1" target="_self">SCMPDS</a> <b> such that </b><br/><a NAME="E6:96"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">loc</font> &amp; ( for <font color="Olive">d</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  holds  <font color="Maroon">s</font> <a href="scmpds_2.html#K2" target="_self">.</a> <font color="Olive">d</font> <a href="hidden.html#R1">=</a> 0 ) )
 <b>by </b><i/>;<br/>
<a NAME="E7:96"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">loc</font> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K6">IC</a> <font color="Maroon">s</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:96"/>
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">k1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k1</font>
 
<b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a></i>;<br/>
<a NAME="E9:96"/><b>then </b>
0 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">k1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k1</font>
 
;<br/>
<a NAME="E10:96"/><b>then </b><i><font color="Green">E26</font></i>: 
<span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">k1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<b>consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E12:96"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K6">IC</a> <font color="Maroon">s</font> &amp;  <a href="scmpds_2.html#K19" target="_self">ICplusConst</a> <font color="Maroon">s</font>,<font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k1</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 )
 <b>by </b><i/>;<br/>
<i><font color="Green">E62</font></i>: <span class="p1">(<span class="default"><a href="ami_1.html#K4">Exec</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K4" target="_self">goto</a> <font color="Maroon">k1</font></span>)</span>,<font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmpds_2.html#K1" target="_self">SCMPDS</a> </span>)</span> = 
<span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">k1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k1</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> 2</span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">k1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k1</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">k1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k1</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">k1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
<b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">k1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k1</font></span>)</span>

;<br/>
<b>assume </b><a NAME="E14:96"/>
 <a href="scmpds_2.html#K4" target="_self">goto</a> <font color="Maroon">k1</font> is <a href="ami_1.html#V3">halting</a>
 ;<br/>

<a NAME="E15:96"/><b>then </b>
<span class="p1">(<span class="default"><a href="ami_1.html#K4">Exec</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K4" target="_self">goto</a> <font color="Maroon">k1</font></span>)</span>,<font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmpds_2.html#K1" target="_self">SCMPDS</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">k1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="ami_1.html#D8">AMI_1:def 8</a></i>;<br/>
<a NAME="E16:96"/><b>then </b>
2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">k1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">k1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E17:96"/>
contradiction
 <b>by </b><i/>;<br/>


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