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<b>let </b><font color="Maroon" title="c1">s1</font>, <font color="Maroon" title="c2">s2</font> be   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">k1</font>, <font color="Olive" title="b2">k2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c2">s2</font> &amp; <font color="Olive" title="b1">k1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">k2</font> holds <br/> <a href="scmpds_2.html#K18" title="SCMPDS_2:func.18">ICplusConst</a> <font color="Maroon" title="c1">s1</font>,<font color="Olive" title="b1">k1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="scmpds_2.html#K18" title="SCMPDS_2:func.18">ICplusConst</a> <font color="Maroon" title="c2">s2</font>,<font color="Olive" title="b2">k2</font></span><br/><b>let </b><font color="Maroon" title="c3">k1</font>, <font color="Maroon" title="c4">k2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c2">s2</font> &amp; <font color="Maroon" title="c3">k1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">k2</font> implies  <a href="scmpds_2.html#K18" title="SCMPDS_2:func.18">ICplusConst</a> <font color="Maroon" title="c1">s1</font>,<font color="Maroon" title="c3">k1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="scmpds_2.html#K18" title="SCMPDS_2:func.18">ICplusConst</a> <font color="Maroon" title="c2">s2</font>,<font color="Maroon" title="c4">k2</font> )</span><br/>



<b>assume </b><a NAME="E1:15"/><i><font color="Green" title="E10">A1</font></i>: 
(  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c2">s2</font> &amp; <font color="Maroon" title="c3">k1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">k2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="scmpds_2.html#K18" title="SCMPDS_2:func.18">ICplusConst</a> <font color="Maroon" title="c1">s1</font>,<font color="Maroon" title="c3">k1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="scmpds_2.html#K18" title="SCMPDS_2:func.18">ICplusConst</a> <font color="Maroon" title="c2">s2</font>,<font color="Maroon" title="c4">k2</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">m</font> =  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c1">s1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ami_1.html#D4" title="AMI_1:def.4">AMI_1:def 4</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">mm</font> = <font color="Maroon" title="c5">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2;<br/>
<a NAME="E3:15"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <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> <font color="Maroon" title="c3">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c3">k1</font>
 
;<br/>
<a NAME="E4:15"/><b>then </b><i><font color="Green" title="E11">A2</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <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> <font color="Maroon" title="c3">k1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/>
<a NAME="E5:15"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <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> <font color="Maroon" title="c4">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c4">k2</font>
 
;<br/>
<a NAME="E6:15"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <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> <font color="Maroon" title="c4">k2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<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 NAME="E7:15"/>
 <a href="scmpds_2.html#K18" title="SCMPDS_2:func.18">ICplusConst</a> <font color="Maroon" title="c1">s1</font>,<font color="Maroon" title="c3">k1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="scmpds_2.html#K18" title="SCMPDS_2:func.18">ICplusConst</a> <font color="Maroon" title="c2">s2</font>,<font color="Maroon" title="c4">k2</font>
 <b>by </b><i><a class="txt" href="scmpds_3.html#E1:15"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="scmpds_3.html#E4:15"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="scmpds_3.html#T18" target="_self" title="SCMPDS_3:th.18">Th18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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