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<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="scmpds_4.html#NM1">Program-block</a>;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="scmpds_4.html#K2">Initialized</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:37_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="scmpds_4.html#V1">paraclosed</a>
 ;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a> ;<br/><a NAME="E2:37_1"/>
 <a href="scmpds_4.html#K2">Initialized</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#T26">FUNCT_4:26</a></i>;<br/><a NAME="E3:37_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds  <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="scmpds_6.html#E1:37_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="scmpds_4.html#D9">SCMPDS_4:def 9</a></i>;<br/><b>hence </b><a NAME="E4:37_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="scmpds_6.html#R1" target="_self">is_closed_on</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="scmpds_6.html#D2" target="_self">Def2</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:37"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a>  holds <font color="Maroon">c<sub>3</sub></font> <a href="scmpds_6.html#R1" target="_self">is_closed_on</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</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>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:37_2"/>
 <a href="scmpds_4.html#K2">Initialized</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:37_2"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="scmpds_6.html#R1" target="_self">is_closed_on</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="scmpds_6.html#E2:37"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_4.html#T79">FUNCT_4:79</a></i>;<br/><b>hence </b><a NAME="E3:37_2"/>
 <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8">stop</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="scmpds_6.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:37"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="scmpds_4.html#V1">paraclosed</a>
 <b>by </b><i><a class="ref" href="scmpds_4.html#D9">SCMPDS_4:def 9</a></i>;<br/>


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