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<b>let </b><font color="Maroon">k</font> be   <a href="int_1.html#NM1">Integer</a>;<br/>

<b>consider </b><font color="Maroon">g</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a> , <a href="numbers.html#K4">INT</a> <b> such that </b><br/><a NAME="E2:84"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a>  holds  <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="scmpds_2.html#T13" target="_self">Th13</a></i>;<br/>
<b>consider </b><font color="Maroon">S</font> being   <a href="scmpds_1.html#NM1">SCMPDS-State</a>;<br/>
<b>set </b><font color="Maroon">t</font> = <font color="Maroon">S</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font>;<br/>
<b>set </b><font color="Maroon">f</font> = the <a href="ami_1.html#U5">Object-Kind</a> of <a href="scmpds_2.html#K1" target="_self">SCMPDS</a> ;<br/>
<a NAME="E4:84"/><i><font color="Green">E23</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <a href="scmpds_1.html#K2">SCMPDS-OK</a> 
 
<b>by </b><i><a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<i><font color="Green">E25</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">S</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<a href="numbers.html#K5">NAT</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<a href="numbers.html#K5">NAT</a>  <a href="xboole_0.html#K2">\/</a> <a href="ami_2.html#K2">SCM-Data-Loc</a> 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="numbers.html#K5">NAT</a> 
<b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
;<br/>
<a NAME="E6:84"/><i><font color="Green">E26</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="ami_1.html#U5">Object-Kind</a> of <a href="scmpds_2.html#K1" target="_self">SCMPDS</a>  <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:84"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="ami_1.html#U5">Object-Kind</a> of <a href="scmpds_2.html#K1" target="_self">SCMPDS</a>  holds <br/><span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="ami_1.html#U5">Object-Kind</a> of <a href="scmpds_2.html#K1" target="_self">SCMPDS</a>  <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:84_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="ami_1.html#U5">Object-Kind</a> of <a href="scmpds_2.html#K1" target="_self">SCMPDS</a> 
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:84_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> or  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_1_1"><b>suppose </b></a><a NAME="E1:84_2_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 ;<br/><div class="add"><a NAME="E2:84_2_1_1"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ami_2.html#K2">SCM-Data-Loc</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><i><font color="Green">E64</font></i>: <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">k</font>
<b>by </b><i>, </i>
;<br/><a NAME="E4:84_2_1_1"/>
the <a href="ami_1.html#U5">Object-Kind</a> of <a href="scmpds_2.html#K1" target="_self">SCMPDS</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K4">INT</a> 
 <b>by </b><i>, <a class="ref" href="scmpds_1.html#T21">SCMPDS_1:21</a></i>;<br/><b>hence </b><a NAME="E5:84_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="ami_1.html#U5">Object-Kind</a> of <a href="scmpds_2.html#K1" target="_self">SCMPDS</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_1_2"><b>suppose </b></a><a NAME="E1:84_2_1_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 ;<br/><div class="add"><a NAME="E2:84_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><b>hence </b><a NAME="E3:84_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="ami_1.html#U5">Object-Kind</a> of <a href="scmpds_2.html#K1" target="_self">SCMPDS</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">S</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font> as   <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1" target="_self">SCMPDS</a>  <b>by </b><i>, , <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<b>take </b>
<font color="Maroon">s</font>
;<br/>

<b>let </b><font color="Maroon">d</font> be    <a href="scmpds_2.html#M1" target="_self">Int_position</a> ;<br/>

<b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">d</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a>  <b>by </b><i/>;<br/>
<a NAME="E10:84"/>
<font color="Maroon">D</font> <a href="hidden.html#R2">in</a>  <a href="ami_2.html#K2">SCM-Data-Loc</a> 
 
;<br/>
<a NAME="E11:84"/><b>then </b>
<font color="Maroon">D</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><font color="Maroon">s</font> <a href="scmpds_2.html#K2" target="_self">.</a> <font color="Maroon">d</font> = 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">D</font>
<b>by </b><i><a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">k</font>
<b>by </b><i/>
;<br/><br/>


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