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<b>let </b><font color="Maroon" title="c1">s</font> be   <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> ;<br/><b>let </b><font color="Maroon" title="c2">m1</font>, <font color="Maroon" title="c3">m2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:5"/><i><font color="Green" title="E7">A1</font></i>: 
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_3.html#K2" title="SCMPDS_3:func.2">inspos</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c3">m2</font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon" title="c4">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:5"/><i><font color="Green" title="E8">A2</font></i>: 
( <font color="Maroon" title="c4">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="c1">s</font> &amp;  <a href="scmpds_2.html#K19" title="SCMPDS_2:func.19">ICplusConst</a> <font color="Maroon" title="c1">s</font>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">m2</font></span>)</span> <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="c4">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> <span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">m2</font></span>)</span></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" title="SCMPDS_2:def.20">SCMPDS_2:def 20</a></i>;<br/>
<i><font color="Green" title="E9">A3</font></i>: <font color="Maroon" title="c4">m</font> = 
 <a href="ami_3.html#K16" title="AMI_3:func.16">il.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c3">m2</font></span>)</span>
<b>by </b><i><a class="txt" href="scmp_gcd.html#E1:5"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="scmp_gcd.html#E3:5"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="scmpds_3.html#D2" title="SCMPDS_3:def.2">SCMPDS_3:def 2</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"><font color="Maroon" title="c2">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c3">m2</font></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="ami_3.html#D20" title="AMI_3:def.20">AMI_3:def 20</a></i>
;<br/>
<a NAME="E5:5"/>
2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">m1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/>
<b>hence </b> <a href="scmpds_2.html#K19" title="SCMPDS_2:func.19">ICplusConst</a> <font color="Maroon" title="c1">s</font>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">m2</font></span>)</span> = 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">m1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
<b>by </b><i><a class="txt" href="scmp_gcd.html#E3:5"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="scmp_gcd.html#E4:5"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
<br/>.= 
 <a href="ami_3.html#K16" title="AMI_3:func.16">il.</a> <font color="Maroon" title="c2">m1</font>
<b>by </b><i><a class="ref" href="ami_3.html#D20" title="AMI_3:def.20">AMI_3:def 20</a></i>
<br/>.= 
 <a href="scmpds_3.html#K2" title="SCMPDS_3:func.2">inspos</a> <font color="Maroon" title="c2">m1</font>
<b>by </b><i><a class="ref" href="scmpds_3.html#D2" title="SCMPDS_3:def.2">SCMPDS_3:def 2</a></i>
;<br/><br/>


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