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<span class="kw">let </span><font color="Maroon" title="c1">s1</font>, <font color="Maroon" title="c2">s2</font> be   <a href="memstr_0.html#NM2" title="MEMSTR_0:NM.2">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <font color="Maroon" title="c1">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">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/><span class="kw">let </span><font color="Maroon" title="c3">k1</font>, <font color="Maroon" title="c4">k2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <font color="Maroon" title="c1">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">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/>



<span class="kw">reconsider </span><font color="Maroon" title="c5">m</font> =  <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <font color="Maroon" title="c1">s1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<span class="kw">set </span><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="E2:10"/>
<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="E3:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E5">A1</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></span>;<br/>
<a NAME="E4:10"/>
<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="E5:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E6">A2</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:10"/>
(  <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <font color="Maroon" title="c1">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">hence </span><a NAME="E7:10"/>
 <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E5:10"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="scmpds_3.html#T8" target="_self" title="SCMPDS_3:th.8">Th18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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