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

<b>let </b><font color="Maroon">s1</font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a> , <font color="Maroon">s2</font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a> ;<br/><b>let </b><font color="Maroon">k1</font> be   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">k2</font> be   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">m</font> be   <a href="int_1.html#NM1">Integer</a>;<br/>



<b>assume </b><a NAME="E1:18"/><i><font color="Green">E33</font></i>: 
(  <a href="ami_1.html#K6">IC</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K6">IC</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">k1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k2</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K6">IC</a> <font color="Maroon">s1</font> &amp; <span class="p1">(<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="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k1</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<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="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k2</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 ;<br/>

<b>assume </b><a NAME="E2:18"/><i><font color="Green">E35</font></i>: 
 <a href="scmpds_2.html#K19">ICplusConst</a> <font color="Maroon">s1</font>,<font color="Maroon">k1</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K19">ICplusConst</a> <font color="Maroon">s2</font>,<font color="Maroon">k2</font>
 ;<br/>

<b>consider </b><font color="Maroon">i</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="E4:18"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K6">IC</a> <font color="Maroon">s1</font> &amp;  <a href="scmpds_2.html#K19">ICplusConst</a> <font color="Maroon">s1</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">i</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><a class="ref" href="scmpds_2.html#D20">SCMPDS_2:def 20</a></i>;<br/>
<i><font color="Green">E38</font></i>:  <a href="scmpds_2.html#K19">ICplusConst</a> <font color="Maroon">s1</font>,<font color="Maroon">k1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</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/>.= 
<font color="Maroon">m</font> <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>consider </b><font color="Maroon">j</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="E7:18"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K6">IC</a> <font color="Maroon">s2</font> &amp;  <a href="scmpds_2.html#K19">ICplusConst</a> <font color="Maroon">s2</font>,<font color="Maroon">k2</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">j</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">k2</font></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/>
 <a href="scmpds_2.html#K19">ICplusConst</a> <font color="Maroon">s2</font>,<font color="Maroon">k2</font> = 
<span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k2</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/>.= 
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k2</font></span>)</span>

;<br/>
<a NAME="E9:18"/><b>then </b>
2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> 2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k2</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E10:18"/>
contradiction
 <b>by </b><i/>;<br/>


</div>
