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<div class="add">

<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="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> ;<br/><b>let </b><font color="Maroon" title="c2">I</font> be  <a href="scmfsa_4.html#V1" title="SCMFSA_4:attr.1">initial</a> <a href="ami_1.html#NM5" title="AMI_1:NM.5">FinPartState</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> ;<br/>



<b>assume </b><a NAME="E1:45"/><i><font color="Green" title="E38">A1</font></i>: 
<font color="Maroon" title="c2">I</font> <a href="scmfsa8a.html#R1" title="SCMFSA8A:pred.1">is_pseudo-closed_on</a> <font color="Maroon" title="c1">s</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c3">s2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ami_1.html#K16" title="AMI_1:func.16">+*</a> <span class="p3">(<span class="default"><a href="ami_1.html#K26" title="AMI_1:func.26">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ami_1.html#K16" title="AMI_1:func.16">+*</a> <span class="p2">(<span class="default"><a href="ami_1.html#K26" title="AMI_1:func.26">Start-At</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span>;<br/>
<i><font color="Green" title="E39">A2</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ami_1.html#K16" title="AMI_1:func.16">+*</a> <span class="p3">(<span class="default"><a href="ami_1.html#K26" title="AMI_1:func.26">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ami_1.html#K16" title="AMI_1:func.16">+*</a> <span class="p2">(<span class="default"><a href="ami_1.html#K26" title="AMI_1:func.26">Start-At</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span> = 
<font color="Maroon" title="c1">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ami_1.html#K16" title="AMI_1:func.16">+*</a> <span class="p3">(<span class="default"><a href="ami_1.html#K26" title="AMI_1:func.26">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K16" title="AMI_1:func.16">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ami_1.html#K16" title="AMI_1:func.16">+*</a> <span class="p3">(<span class="default"><a href="ami_1.html#K26" title="AMI_1:func.26">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#T15" title="FUNCT_4:th.15">FUNCT_4:15</a></i>
<br/>.= 
<font color="Maroon" title="c1">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ami_1.html#K16" title="AMI_1:func.16">+*</a> <span class="p2">(<span class="default"><a href="ami_1.html#K26" title="AMI_1:func.26">Start-At</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span>

;<br/>
<a NAME="E3:45"/><b>then </b><i><font color="Green" title="E40">A3</font></i>: 
(  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ami_1.html#K16" title="AMI_1:func.16">+*</a> <span class="p0">(<span class="default"><a href="ami_1.html#K26" title="AMI_1:func.26">Start-At</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ami_1.html#K16" title="AMI_1:func.16">+*</a> <span class="p5">(<span class="default"><a href="ami_1.html#K26" title="AMI_1:func.26">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <span class="p2">(<span class="default"><a href="scmfsa8a.html#K3" title="SCMFSA8A:func.3">pseudo-LifeSpan</a> <font color="Maroon" title="c1">s</font>,<font color="Maroon" title="c2">I</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="ami_1.html#K31" title="AMI_1:func.31">ProgramPart</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span> &amp; ( for <font color="Olive" title="b1">n</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>   st not  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ami_1.html#K16" title="AMI_1:func.16">+*</a> <span class="p0">(<span class="default"><a href="ami_1.html#K26" title="AMI_1:func.26">Start-At</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ami_1.html#K16" title="AMI_1:func.16">+*</a> <span class="p5">(<span class="default"><a href="ami_1.html#K26" title="AMI_1:func.26">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">I</font> holds <br/> <a href="scmfsa8a.html#K3" title="SCMFSA8A:func.3">pseudo-LifeSpan</a> <font color="Maroon" title="c1">s</font>,<font color="Maroon" title="c2">I</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 
<b>by </b><i><a class="txt" href="scmfsa8c.html#E1:45"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="scmfsa8a.html#D5" title="SCMFSA8A:def.5">SCMFSA8A:def 5</a></i>;<br/>
<a NAME="E4:45"/>
(  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ami_1.html#K16" title="AMI_1:func.16">+*</a> <span class="p0">(<span class="default"><a href="ami_1.html#K26" title="AMI_1:func.26">Start-At</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ami_1.html#K16" title="AMI_1:func.16">+*</a> <span class="p5">(<span class="default"><a href="ami_1.html#K26" title="AMI_1:func.26">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <span class="p2">(<span class="default"><a href="scmfsa8a.html#K3" title="SCMFSA8A:func.3">pseudo-LifeSpan</a> <font color="Maroon" title="c1">s</font>,<font color="Maroon" title="c2">I</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="ami_1.html#K31" title="AMI_1:func.31">ProgramPart</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span> &amp; ( for <font color="Olive" title="b1">n</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>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="scmfsa8a.html#K3" title="SCMFSA8A:func.3">pseudo-LifeSpan</a> <font color="Maroon" title="c1">s</font>,<font color="Maroon" title="c2">I</font> holds <br/> <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ami_1.html#K16" title="AMI_1:func.16">+*</a> <span class="p0">(<span class="default"><a href="ami_1.html#K26" title="AMI_1:func.26">Start-At</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ami_1.html#K16" title="AMI_1:func.16">+*</a> <span class="p5">(<span class="default"><a href="ami_1.html#K26" title="AMI_1:func.26">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">I</font> ) )
 
<b>by </b><i><a class="txt" href="scmfsa8c.html#E1:45"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="scmfsa8c.html#E2:45"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="scmfsa8a.html#D5" title="SCMFSA8A:def.5">SCMFSA8A:def 5</a></i>;<br/>
<b>hence </b><a NAME="E5:45"/>
<font color="Maroon" title="c2">I</font> <a href="scmfsa8a.html#R1" title="SCMFSA8A:pred.1">is_pseudo-closed_on</a> <font color="Maroon" title="c1">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ami_1.html#K16" title="AMI_1:func.16">+*</a> <span class="p2">(<span class="default"><a href="ami_1.html#K26" title="AMI_1:func.26">Start-At</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="scmfsa8a.html#D3" title="SCMFSA8A:def.3">SCMFSA8A:def 3</a></i>;<br/>

<b>hence </b><a NAME="E6:45"/>
 <a href="scmfsa8a.html#K3" title="SCMFSA8A:func.3">pseudo-LifeSpan</a> <font color="Maroon" title="c1">s</font>,<font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa8a.html#K3" title="SCMFSA8A:func.3">pseudo-LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ami_1.html#K16" title="AMI_1:func.16">+*</a> <span class="p3">(<span class="default"><a href="ami_1.html#K26" title="AMI_1:func.26">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">I</font>
 <b>by </b><i><a class="txt" href="scmfsa8c.html#E3:45"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="scmfsa8a.html#D5" title="SCMFSA8A:def.5">SCMFSA8A:def 5</a></i>;<br/>


</div>
