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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="int_1.html#NM1">Integer</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:106"/><i><font color="Green">E56</font></i>: 
 <a href="ami_1.html#K6">IC</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="scmpds_3.html#K2">inspos</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E2:106"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>and </b><br/><a NAME="E3:106"/><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K6">IC</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="scmpds_3.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K6">IC</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:106"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K6">IC</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="scmpds_2.html#K19">ICplusConst</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></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">c<sub>6</sub></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">c<sub>5</sub></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/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:106"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K6">IC</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="scmpds_2.html#K19">ICplusConst</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></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">c<sub>7</sub></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">c<sub>5</sub></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/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>5</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="scmpds_4.html#E2:106"><i><font color="Green">E57</font></i></a>, <a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<i><font color="Green">E61</font></i>: <font color="Maroon">c<sub>7</sub></font> = 
 <a href="scmpds_3.html#K2">inspos</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="scmpds_4.html#E1:106"><i><font color="Green">E56</font></i></a>, <a class="txt" href="scmpds_4.html#E3:106"><i><font color="Green">E58</font></i></a>, <a class="txt" href="scmpds_4.html#E7:106"><i><font color="Green">E60</font></i></a>, <a class="ref" href="scmpds_3.html#D3">SCMPDS_3:def 3</a></i>
<br/>.= 
 <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2

;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E11:106"/><i><font color="Green">E62</font></i>: 
 <a href="scmpds_3.html#K2">inspos</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K19">ICplusConst</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="scmpds_3.html#T32">SCMPDS_3:32</a></i>;<br/>
<i><font color="Green">E63</font></i>: <font color="Maroon">c<sub>6</sub></font> = 
 <a href="ami_3.html#K16">il.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="scmpds_4.html#E1:106"><i><font color="Green">E56</font></i></a>, <a class="txt" href="scmpds_4.html#E5:106"><i><font color="Green">E59</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2

;<br/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2 = 
 <a href="ami_3.html#K16">il.</a> <font color="Maroon">c<sub>9</sub></font>

<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">c<sub>6</sub></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">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
<b>by </b><i><a class="txt" href="scmpds_4.html#E5:106"><i><font color="Green">E59</font></i></a>, <a class="txt" href="scmpds_4.html#E11:106"><i><font color="Green">E62</font></i></a></i>
;<br/>
<b>then </b><i><font color="Green">E64</font></i>: 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font> = 
 <a href="int_2.html#K1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>4</sub></font></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">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scmpds_4.html#E12:106"><i><font color="Green">E63</font></i></a></i>
<br/>.= 
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="scmpds_2.html#K19">ICplusConst</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="scmpds_3.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font> = 
 <a href="scmpds_3.html#K2">inspos</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="scmpds_4.html#E11:106"><i><font color="Green">E62</font></i></a>, <a class="ref" href="scmpds_3.html#D3">SCMPDS_3:def 3</a></i>
<br/>.= 
 <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
<b>by </b><i><a class="txt" href="scmpds_4.html#E14:106"><i><font color="Green">E64</font></i></a></i>
<br/>.= 
 <a href="scmpds_2.html#K19">ICplusConst</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="scmpds_4.html#E7:106"><i><font color="Green">E60</font></i></a>, <a class="txt" href="scmpds_4.html#E9:106"><i><font color="Green">E61</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/>


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