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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="ami_2.html#K3">SCM-Instr-Loc</a>  ) or ( ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K4">INT</a>  ) or ( ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2 &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="scmpds_1.html#K1" target="_self">SCMPDS-Instr</a>  ) );<br/>
<div><i><font color="Green">E7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:9_1_1_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="numbers.html#K4">INT</a> </span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="scmpds_1.html#K1" target="_self">SCMPDS-Instr</a> ,<a href="ami_2.html#K3">SCM-Instr-Loc</a> </span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><a href="numbers.html#K4">INT</a> ,<a href="scmpds_1.html#K1" target="_self">SCMPDS-Instr</a> ,<a href="ami_2.html#K3">SCM-Instr-Loc</a> </span><a href="enumset1.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#T42">ENUMSET1:42</a></i>;<br/><a NAME="E2:9_1_1_1"/><b>then </b><i><font color="Green">E8</font></i>: 
(  <a href="numbers.html#K4">INT</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="numbers.html#K4">INT</a> </span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="scmpds_1.html#K1" target="_self">SCMPDS-Instr</a> ,<a href="ami_2.html#K3">SCM-Instr-Loc</a> </span><a href="tarski.html#K2">}</a></span> &amp;  <a href="scmpds_1.html#K1" target="_self">SCMPDS-Instr</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="numbers.html#K4">INT</a> </span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="scmpds_1.html#K1" target="_self">SCMPDS-Instr</a> ,<a href="ami_2.html#K3">SCM-Instr-Loc</a> </span><a href="tarski.html#K2">}</a></span> &amp;  <a href="ami_2.html#K3">SCM-Instr-Loc</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="numbers.html#K4">INT</a> </span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="scmpds_1.html#K1" target="_self">SCMPDS-Instr</a> ,<a href="ami_2.html#K3">SCM-Instr-Loc</a> </span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i><a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><a NAME="E3:9_1_1_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>, <a href="ami_2.html#K3">SCM-Instr-Loc</a> ] or <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K4">INT</a> ] or <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>, <a href="scmpds_1.html#K1" target="_self">SCMPDS-Instr</a> ] )
 <b>by </b><i><a class="ref" href="scmpds_1.html#T14" target="_self">Th14</a></i>;<br/><b>hence </b><a NAME="E4:9_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="numbers.html#K4">INT</a> </span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="scmpds_1.html#K1" target="_self">SCMPDS-Instr</a> ,<a href="ami_2.html#K3">SCM-Instr-Loc</a> </span><a href="tarski.html#K2">}</a></span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 <b>by </b><i><a class="txt" href="scmpds_1.html#E2:9_1_1_1"><i><font color="Green">E8</font></i></a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>1</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="numbers.html#K4">INT</a> </span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="scmpds_1.html#K1" target="_self">SCMPDS-Instr</a> ,<a href="ami_2.html#K3">SCM-Instr-Loc</a> </span><a href="tarski.html#K2">}</a></span><b> such that </b><br/><a NAME="E3:9_1_1"/><i><font color="Green">E8</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="scmpds_1.html#E1:9_1_1"><i><font color="Green">E7</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon">c<sub>1</sub></font>
;<br/>

<a NAME="E4:9_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[0,<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> 0]
 
<b>by </b><i><a class="txt" href="scmpds_1.html#E3:9_1_1"><i><font color="Green">E8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:9_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="ami_2.html#K3">SCM-Instr-Loc</a> 
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E6:9_1_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1,<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>] &amp; <font color="Maroon">S<sub>1</sub></font>[<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2,<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span>] )
 
<b>by </b><i><a class="txt" href="scmpds_1.html#E3:9_1_1"><i><font color="Green">E8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:9_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K4">INT</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a>  <a href="scmpds_1.html#K1" target="_self">SCMPDS-Instr</a>  )
 <b>by </b><i><a class="ref" href="scmpds_1.html#T16" target="_self">Th16</a></i>;<br/>


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