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<b>let </b><font color="Maroon" title="c6">k1</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/>

<b>assume </b><a NAME="E1:96"/><i><font color="Green" title="E54">A1</font></i>: 
<font color="Maroon" title="c6">k1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/>

<b>set </b><font color="Maroon" title="c7">n</font> =  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c6">k1</font>;<br/>
<a NAME="E2:96"/>
<span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c6">k1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></i>;<br/>
<a NAME="E3:96"/><b>then </b>
2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c6">k1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_2.html#K5" title="AMI_2:func.5">SCM-Instr-Loc</a> 
 
<b>by </b><i><a class="ref" href="ami_2.html#D3" title="AMI_2:def.3">AMI_2:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">loc</font> = 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c6">k1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="ami_1.html#NM2" title="AMI_1:NM.2">Instruction-Location</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>  ;<br/>
<b>consider </b><font color="Maroon" title="c9">s</font> being   <a href="ami_1.html#NM4" title="AMI_1:NM.4">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> <b> such that </b><br/><a NAME="E6:96"/><i><font color="Green" title="E55">A2</font></i>: 
( <font color="Maroon" title="c9">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">loc</font> &amp; ( for <font color="Olive" title="b1">d</font> being   <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</a>  holds  <font color="Maroon" title="c9">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) )
 <b>by </b><i><a class="ref" href="scmpds_2.html#T74" target="_self" title="SCMPDS_2:th.74">Th74</a></i>;<br/>
<a NAME="E7:96"/><i><font color="Green" title="E56">A3</font></i>: 
<font color="Maroon" title="c8">loc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c9">s</font>
 
<b>by </b><i><a class="txt" href="scmpds_2.html#E6:96"><i><font color="Green" title="E55">A2</font></i></a></i>;<br/>
<a NAME="E8:96"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c6">k1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k1</font>
 
<b>by </b><i><a class="ref" href="absvalue.html#T11" title="ABSVALUE:th.11">ABSVALUE:11</a></i>;<br/>
<a NAME="E9:96"/><b>then </b>
0 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c6">k1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k1</font>
 
;<br/>
<a NAME="E10:96"/><b>then </b><i><font color="Green" title="E57">A4</font></i>: 
<span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c6">k1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">k1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E12:96"/><i><font color="Green" title="E58">A5</font></i>: 
( <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c9">s</font> &amp;  <a href="scmpds_2.html#K19" title="SCMPDS_2:func.19">ICplusConst</a> <font color="Maroon" title="c9">s</font>,<font color="Maroon" title="c6">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">k1</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 )
 <b>by </b><i><a class="ref" href="scmpds_2.html#D20" target="_self" title="SCMPDS_2:def.20">Def20</a></i>;<br/>
<i><font color="Green" title="E59">A6</font></i>: <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K4" title="SCMPDS_2:func.4">goto</a> <font color="Maroon" title="c6">k1</font></span>)</span>,<font color="Maroon" title="c9">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span> = 
<span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c6">k1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">k1</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
<b>by </b><i><a class="txt" href="scmpds_2.html#E7:96"><i><font color="Green" title="E56">A3</font></i></a>, <a class="txt" href="scmpds_2.html#E12:96"><i><font color="Green" title="E58">A5</font></i></a>, <a class="ref" href="scmpds_2.html#T66" target="_self" title="SCMPDS_2:th.66">Th66</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c6">k1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">k1</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
<b>by </b><i><a class="ref" href="complex1.html#T151" title="COMPLEX1:th.151">COMPLEX1:151</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c6">k1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">k1</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c6">k1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">k1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2
<b>by </b><i><a class="txt" href="scmpds_2.html#E10:96"><i><font color="Green" title="E57">A4</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">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" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c6">k1</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">k1</font></span>)</span>

;<br/>
<b>assume </b><a NAME="E14:96"/>
 <a href="scmpds_2.html#K4" title="SCMPDS_2:func.4">goto</a> <font color="Maroon" title="c6">k1</font> is <a href="ami_1.html#V3" title="AMI_1:attr.3">halting</a>
 ;<br/>

<a NAME="E15:96"/><b>then </b>
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K4" title="SCMPDS_2:func.4">goto</a> <font color="Maroon" title="c6">k1</font></span>)</span>,<font color="Maroon" title="c9">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c6">k1</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="scmpds_2.html#E6:96"><i><font color="Green" title="E55">A2</font></i></a>, <a class="ref" href="ami_1.html#D8" title="AMI_1:def.8">AMI_1:def 8</a></i>;<br/>
<a NAME="E16:96"/><b>then </b>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c6">k1</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c6">k1</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="scmpds_2.html#E13:96"><i><font color="Green" title="E59">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E17:96"/>
contradiction
 <b>by </b><i><a class="txt" href="scmpds_2.html#E1:96"><i><font color="Green" title="E54">A1</font></i></a></i>;<br/>


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