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<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="scmpds_2.html#M1">Int_position</a> ;<br/>

<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 ex <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>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> ) );<br/>
<div><i><font color="Green">E7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a> ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0 or <font color="Maroon">c<sub>4</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1"><b>suppose </b></a><a NAME="E1:6_1_1_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = 0 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><b>take </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font>;<br/><b>thus </b><a NAME="E3:6_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>]
 <b>by </b><i><a class="txt" href="scmpds_8.html#E1:6_1_1_1"><i><font color="Green">E8</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_2"><b>suppose </b></a><a NAME="E1:6_1_1_2"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><b>take </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font>;<br/><b>thus </b><a NAME="E3:6_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>]
 <b>by </b><i><a class="txt" href="scmpds_8.html#E1:6_1_1_2"><i><font color="Green">E8</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E2:6"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="card_3.html#K4">product</a> the <a href="ami_1.html#U5">Object-Kind</a> of <a href="scmpds_2.html#K1">SCMPDS</a> , <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a>  holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font>]
 
<b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="scmpds_8.html#E1:6"><i><font color="Green">E7</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="card_3.html#K4">product</a> the <a href="ami_1.html#U5">Object-Kind</a> of <a href="scmpds_2.html#K1">SCMPDS</a> , <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:6"/><i><font color="Green">E8</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">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 ;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<div><b>hereby </b>
<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/><a NAME="E1:6_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>]
 <b>by </b><i><a class="txt" href="scmpds_8.html#E4:6"><i><font color="Green">E8</font></i></a></i>;<br/><b>hence </b><a NAME="E2:6_2"/>
( ( <font color="Maroon">c<sub>5</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0 implies <font color="Maroon">c<sub>4</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <font color="Maroon">c<sub>5</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 implies <font color="Maroon">c<sub>4</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> ) )
 ;<br/></div>
<b>end;</b></div>


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