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

<b>let </b><font color="Maroon">a</font> be    <a href="scmpds_2.html#M1">Int_position</a> ;<br/><b>let </b><font color="Maroon">k1</font> be   <a href="int_1.html#NM1">Integer</a>;<br/>



<b>set </b><font color="Maroon">k2</font> = 5;<br/>
<b>set </b><font color="Maroon">i</font> = <font color="Maroon">a</font>,<font color="Maroon">k1</font> <a href="scmpds_2.html#K8">&lt;&gt;0_goto</a> 5;<br/>
<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">k1</font> <a href="scmpds_2.html#K8">&lt;&gt;0_goto</a> 5</span>)</span>,<font color="Olive">l</font></span>)</span> where <font color="Olive">l</font> is   <a href="ami_1.html#NM2">Instruction-Location</a> of <a href="scmpds_2.html#K1">SCMPDS</a>  : verum </span>}</span> ;<br/>
<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:49_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="amistd_1.html#K2">JUMP</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">k1</font> <a href="scmpds_2.html#K8">&lt;&gt;0_goto</a> 5</span>)</span>
 ;<br/><b>set </b><font color="Maroon">nl1</font> = 5;<br/><b>set </b><font color="Maroon">nl2</font> = 8;<br/><b>set </b><font color="Maroon">l1</font> =  <a href="scmpds_3.html#K2">inspos</a> 5;<br/><a NAME="E2:49_1"/><i><font color="Green">E31</font></i>: 
 <a href="amistd_1.html#K1">NIC</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">k1</font> <a href="scmpds_2.html#K8">&lt;&gt;0_goto</a> 5</span>)</span>,<span class="p1">(<span class="default"><a href="scmpds_3.html#K2">inspos</a> 5</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">k1</font> <a href="scmpds_2.html#K8">&lt;&gt;0_goto</a> 5</span>)</span>,<font color="Olive">l</font></span>)</span> where <font color="Olive">l</font> is   <a href="ami_1.html#NM2">Instruction-Location</a> of <a href="scmpds_2.html#K1">SCMPDS</a>  : verum </span>}</span> 
 ;<br/><a NAME="E3:49_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="amistd_1.html#K1">NIC</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">k1</font> <a href="scmpds_2.html#K8">&lt;&gt;0_goto</a> 5</span>)</span>,<span class="p1">(<span class="default"><a href="scmpds_3.html#K2">inspos</a> 5</span>)</span>
 <b>by </b><i><a class="txt" href="scmpds_9.html#E6:49_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="scmpds_9.html#E7:49_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/><a NAME="E4:49_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ami_1.html#K6">IC</a> <span class="p3">(<span class="default"><a href="ami_1.html#K9">Following</a> <font color="Olive">s1</font></span>)</span></span>)</span> where <font color="Olive">s1</font> is   <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a>  : (  <a href="ami_1.html#K6">IC</a> <font color="Olive">s1</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_3.html#K2">inspos</a> 5 &amp; <font color="Olive">s1</font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K2">inspos</a> 5</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>,<font color="Maroon">k1</font> <a href="scmpds_2.html#K8">&lt;&gt;0_goto</a> 5 ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">s1</font> being   <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a> <b> such that </b><br/><a NAME="E6:49_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K9">Following</a> <font color="Maroon">s1</font></span>)</span>
 <b>and </b><br/><a NAME="E7:49_1"/><i><font color="Green">E38</font></i>: 
 <a href="ami_1.html#K6">IC</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_3.html#K2">inspos</a> 5
 <b>and </b><br/><a NAME="E8:49_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">s1</font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K2">inspos</a> 5</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>,<font color="Maroon">k1</font> <a href="scmpds_2.html#K8">&lt;&gt;0_goto</a> 5
 ;<br/><b>consider </b><font color="Maroon">m1</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="E10:49_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">m1</font> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K6">IC</a> <font color="Maroon">s1</font>
 <b>and </b><br/><a NAME="E11:49_1"/><i><font color="Green">E79</font></i>: 
 <a href="scmpds_2.html#K19">ICplusConst</a> <font color="Maroon">s1</font>,5 <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">m1</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="nat_1.html#K2">*</a> 5</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="scmpds_2.html#D20">SCMPDS_2:def 20</a></i>;<br/><i><font color="Green">E80</font></i>: <font color="Maroon">m1</font> = 
 <a href="ami_3.html#K16">il.</a> 5
<b>by </b><i>, , <a class="ref" href="scmpds_3.html#D2">SCMPDS_3:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 5</span>)</span> <a href="nat_1.html#K1">+</a> 2
<b>by </b><i><a class="ref" href="ami_3.html#D20">AMI_3:def 20</a></i>
;<br/><b>set </b><font color="Maroon">l2</font> =  <a href="scmpds_3.html#K2">inspos</a> 8;<br/><a NAME="E13:49_1"/>
 <a href="amistd_1.html#K1">NIC</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">k1</font> <a href="scmpds_2.html#K8">&lt;&gt;0_goto</a> 5</span>)</span>,<span class="p1">(<span class="default"><a href="scmpds_3.html#K2">inspos</a> 8</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="amistd_1.html#K1">NIC</a> <span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">k1</font> <a href="scmpds_2.html#K8">&lt;&gt;0_goto</a> 5</span>)</span>,<font color="Olive">l</font></span>)</span> where <font color="Olive">l</font> is   <a href="ami_1.html#NM2">Instruction-Location</a> of <a href="scmpds_2.html#K1">SCMPDS</a>  : verum </span>}</span> 
 ;<br/><a NAME="E14:49_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="amistd_1.html#K1">NIC</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">k1</font> <a href="scmpds_2.html#K8">&lt;&gt;0_goto</a> 5</span>)</span>,<span class="p1">(<span class="default"><a href="scmpds_3.html#K2">inspos</a> 8</span>)</span>
 <b>by </b><i><a class="txt" href="scmpds_9.html#E6:49_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/><a NAME="E15:49_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ami_1.html#K6">IC</a> <span class="p3">(<span class="default"><a href="ami_1.html#K9">Following</a> <font color="Olive">s2</font></span>)</span></span>)</span> where <font color="Olive">s2</font> is   <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a>  : (  <a href="ami_1.html#K6">IC</a> <font color="Olive">s2</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_3.html#K2">inspos</a> 8 &amp; <font color="Olive">s2</font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K2">inspos</a> 8</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>,<font color="Maroon">k1</font> <a href="scmpds_2.html#K8">&lt;&gt;0_goto</a> 5 ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">s2</font> being   <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a> <b> such that </b><br/><a NAME="E17:49_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K9">Following</a> <font color="Maroon">s2</font></span>)</span>
 <b>and </b><br/><a NAME="E18:49_1"/><i><font color="Green">E82</font></i>: 
 <a href="ami_1.html#K6">IC</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_3.html#K2">inspos</a> 8
 <b>and </b><br/><a NAME="E19:49_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">s2</font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K2">inspos</a> 8</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>,<font color="Maroon">k1</font> <a href="scmpds_2.html#K8">&lt;&gt;0_goto</a> 5
 ;<br/><b>consider </b><font color="Maroon">m2</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="E21:49_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">m2</font> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K6">IC</a> <font color="Maroon">s2</font>
 <b>and </b><br/><a NAME="E22:49_1"/><i><font color="Green">E85</font></i>: 
 <a href="scmpds_2.html#K19">ICplusConst</a> <font color="Maroon">s2</font>,5 <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">m2</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="nat_1.html#K2">*</a> 5</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="scmpds_2.html#D20">SCMPDS_2:def 20</a></i>;<br/><i><font color="Green">E86</font></i>: <font color="Maroon">m2</font> = 
 <a href="ami_3.html#K16">il.</a> 8
<b>by </b><i>, , <a class="ref" href="scmpds_3.html#D2">SCMPDS_3:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 8</span>)</span> <a href="nat_1.html#K1">+</a> 2
<b>by </b><i><a class="ref" href="ami_3.html#D20">AMI_3:def 20</a></i>
;<br/><a NAME="E24:49_1"/><i><font color="Green">E92</font></i>: 
 <a href="scmpds_3.html#K2">inspos</a> 5 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="scmpds_3.html#K2">inspos</a> 8
 <b>by </b><i><a class="ref" href="scmpds_3.html#T31">SCMPDS_3:31</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1_3"/>
( ( <font color="Maroon">s1</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">k1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">s2</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">k1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) or ( <font color="Maroon">s1</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">k1</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">s2</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">k1</font></span>)</span> <a href="hidden.html#R1">=</a> 0 ) or ( <font color="Maroon">s1</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">k1</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">s2</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">k1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) or ( <font color="Maroon">s1</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">k1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">s2</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">k1</font></span>)</span> <a href="hidden.html#R1">=</a> 0 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_3_1"><b>suppose </b></a><b>that </b><a NAME="E1:49_1_3_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">s1</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">k1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E2:49_1_3_1"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">s2</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">k1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><i><font color="Green">E95</font></i>: <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K4">Exec</a> <span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">k1</font> <a href="scmpds_2.html#K8">&lt;&gt;0_goto</a> 5</span>)</span>,<font color="Maroon">s1</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">SCMPDS</a> </span>)</span>
<b>by </b><i><a class="ref" href="scmpds_9.html#T5" target="_self">Th5</a>, , , </i>
<br/>.= 
<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">m1</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="nat_1.html#K2">*</a> 5</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
<b>by </b><i>, , <a class="ref" href="scmpds_2.html#T67">SCMPDS_2:67</a></i>
;<br/><font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K4">Exec</a> <span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">k1</font> <a href="scmpds_2.html#K8">&lt;&gt;0_goto</a> 5</span>)</span>,<font color="Maroon">s2</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">SCMPDS</a> </span>)</span>
<b>by </b><i><a class="ref" href="scmpds_9.html#T6" target="_self">Th6</a>, , , </i>
<br/>.= 
<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">m2</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="nat_1.html#K2">*</a> 5</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
<b>by </b><i>, , <a class="ref" href="scmpds_2.html#T67">SCMPDS_2:67</a></i>
;<br/><a NAME="E5:49_1_3_1"/><b>then </b>
 <a href="int_2.html#K1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m1</font> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 5</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="int_2.html#K1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m2</font> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 5</span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E6:49_1_3_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 5</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 5</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 8</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 5</span>)</span> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 5</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 5</span>)</span> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> 8</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 5</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="scmpds_9.html#E8:49_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="scmpds_9.html#T7" target="_self">Th7</a>, </i>;<br/><b>hence </b><a NAME="E7:49_1_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_3_2"><b>suppose </b></a><b>that </b><a NAME="E1:49_1_3_2"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">s1</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">k1</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>and </b><br/><a NAME="E2:49_1_3_2"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">s2</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">k1</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><i><font color="Green">E110</font></i>: <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K4">Exec</a> <span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">k1</font> <a href="scmpds_2.html#K8">&lt;&gt;0_goto</a> 5</span>)</span>,<font color="Maroon">s1</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">SCMPDS</a> </span>)</span>
<b>by </b><i><a class="ref" href="scmpds_9.html#T5" target="_self">Th5</a>, , , </i>
<br/>.= 
 <a href="scmpds_2.html#K18">Next</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K2">inspos</a> 5</span>)</span>
<b>by </b><i>, , <a class="ref" href="scmpds_2.html#T67">SCMPDS_2:67</a></i>
;<br/><font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K4">Exec</a> <span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">k1</font> <a href="scmpds_2.html#K8">&lt;&gt;0_goto</a> 5</span>)</span>,<font color="Maroon">s2</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">SCMPDS</a> </span>)</span>
<b>by </b><i><a class="ref" href="scmpds_9.html#T6" target="_self">Th6</a>, , , </i>
<br/>.= 
 <a href="scmpds_2.html#K18">Next</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K2">inspos</a> 8</span>)</span>
<b>by </b><i>, , <a class="ref" href="scmpds_2.html#T67">SCMPDS_2:67</a></i>
;<br/><b>hence </b><a NAME="E5:49_1_3_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_3_3"><b>suppose </b></a><b>that </b><a NAME="E1:49_1_3_3"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">s1</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">k1</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>and </b><br/><a NAME="E2:49_1_3_3"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">s2</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">k1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>consider </b><font color="Maroon">n1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K3">SCM-Instr-Loc</a> <b> such that </b><br/><a NAME="E4:49_1_3_3"/><i><font color="Green">E113</font></i>: 
( <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_3.html#K2">inspos</a> 5 &amp;  <a href="scmpds_2.html#K18">Next</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K2">inspos</a> 5</span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_2.html#K15">Next</a> <font color="Maroon">n1</font> )
 <b>by </b><i><a class="ref" href="scmpds_2.html#D19">SCMPDS_2:def 19</a></i>;<br/><b>consider </b><font color="Maroon">w1</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="E6:49_1_3_3"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">w1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="scmpds_2.html#T1">SCMPDS_2:1</a></i>;<br/><i><font color="Green">E115</font></i>: <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K4">Exec</a> <span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">k1</font> <a href="scmpds_2.html#K8">&lt;&gt;0_goto</a> 5</span>)</span>,<font color="Maroon">s1</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">SCMPDS</a> </span>)</span>
<b>by </b><i><a class="ref" href="scmpds_9.html#T5" target="_self">Th5</a>, , , </i>
<br/>.= 
 <a href="ami_2.html#K15">Next</a> <font color="Maroon">n1</font>
<b>by </b><i>, , , <a class="ref" href="scmpds_2.html#T67">SCMPDS_2:67</a></i>
<br/>.= 
<font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 2
<b>by </b><i><a class="ref" href="ami_2.html#D15">AMI_2:def 15</a></i>
;<br/><font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K4">Exec</a> <span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">k1</font> <a href="scmpds_2.html#K8">&lt;&gt;0_goto</a> 5</span>)</span>,<font color="Maroon">s2</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">SCMPDS</a> </span>)</span>
<b>by </b><i><a class="ref" href="scmpds_9.html#T6" target="_self">Th6</a>, , , </i>
<br/>.= 
<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">m2</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="nat_1.html#K2">*</a> 5</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
<b>by </b><i>, , <a class="ref" href="scmpds_2.html#T67">SCMPDS_2:67</a></i>
;<br/><a NAME="E9:49_1_3_3"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">w1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a>  <a href="int_2.html#K1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m2</font> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 5</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scmpds_9.html#E8"><i><font color="Green">Lemma55</font></i></a>, </i>;<br/><a NAME="E10:49_1_3_3"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">w1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m2</font> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 5</span>)</span> or <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">w1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m2</font> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 5</span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="absvalue.html#T1">ABSVALUE:1</a></i>;<br/><b>hence </b><a NAME="E11:49_1_3_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, , <a class="txt" href="scmpds_9.html#E8:49_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="scmpds_9.html#T7" target="_self">Th7</a>, , <a class="txt" href="scmpds_9.html#E8"><i><font color="Green">Lemma55</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_3_4"><b>suppose </b></a><b>that </b><a NAME="E1:49_1_3_4"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">s1</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">k1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E2:49_1_3_4"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">s2</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">k1</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>consider </b><font color="Maroon">n2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K3">SCM-Instr-Loc</a> <b> such that </b><br/><a NAME="E4:49_1_3_4"/><i><font color="Green">E137</font></i>: 
( <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_3.html#K2">inspos</a> 8 &amp;  <a href="scmpds_2.html#K18">Next</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K2">inspos</a> 8</span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_2.html#K15">Next</a> <font color="Maroon">n2</font> )
 <b>by </b><i><a class="ref" href="scmpds_2.html#D19">SCMPDS_2:def 19</a></i>;<br/><b>consider </b><font color="Maroon">w2</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="E6:49_1_3_4"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">w2</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="scmpds_2.html#T1">SCMPDS_2:1</a></i>;<br/><i><font color="Green">E140</font></i>: <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K4">Exec</a> <span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">k1</font> <a href="scmpds_2.html#K8">&lt;&gt;0_goto</a> 5</span>)</span>,<font color="Maroon">s1</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">SCMPDS</a> </span>)</span>
<b>by </b><i><a class="ref" href="scmpds_9.html#T5" target="_self">Th5</a>, , , </i>
<br/>.= 
<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">m1</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="nat_1.html#K2">*</a> 5</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
<b>by </b><i>, , <a class="ref" href="scmpds_2.html#T67">SCMPDS_2:67</a></i>
;<br/><font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K4">Exec</a> <span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">k1</font> <a href="scmpds_2.html#K8">&lt;&gt;0_goto</a> 5</span>)</span>,<font color="Maroon">s2</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">SCMPDS</a> </span>)</span>
<b>by </b><i><a class="ref" href="scmpds_9.html#T6" target="_self">Th6</a>, , , </i>
<br/>.= 
 <a href="ami_2.html#K15">Next</a> <font color="Maroon">n2</font>
<b>by </b><i>, , , <a class="ref" href="scmpds_2.html#T67">SCMPDS_2:67</a></i>
<br/>.= 
<font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 2
<b>by </b><i><a class="ref" href="ami_2.html#D15">AMI_2:def 15</a></i>
;<br/><a NAME="E9:49_1_3_4"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">w2</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a>  <a href="int_2.html#K1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m1</font> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 5</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scmpds_9.html#E8"><i><font color="Green">Lemma55</font></i></a>, </i>;<br/><a NAME="E10:49_1_3_4"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">w2</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 5</span>)</span> or <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">w2</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m1</font> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 5</span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="absvalue.html#T1">ABSVALUE:1</a></i>;<br/><b>hence </b><a NAME="E11:49_1_3_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="scmpds_9.html#E8:49_1"><i><font color="Green">E44</font></i></a>, , , <a class="ref" href="scmpds_9.html#T7" target="_self">Th7</a>, , <a class="txt" href="scmpds_9.html#E8"><i><font color="Green">Lemma55</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>thus </b><a NAME="E2:49"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R1">c=</a>  <a href="amistd_1.html#K2">JUMP</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">k1</font> <a href="scmpds_2.html#K8">&lt;&gt;0_goto</a> 5</span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>


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