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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">I</font> be  <a href="ami_3.html#V1">programmed</a>  <a href="ami_1.html#M1">FinPartState</a> of  <a href="scmfsa_2.html#K1">SCM+FSA</a> , <font color="Maroon">J</font> be  <a href="ami_3.html#V1">programmed</a>  <a href="ami_1.html#M1">FinPartState</a> of  <a href="scmfsa_2.html#K1">SCM+FSA</a> ;<br/>



<a NAME="E1:37"/><i><font color="Green">E27</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_4.html#K7" target="_self">IncAddr</a> <font color="Maroon">J</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">J</font>
 
<b>by </b><i><a class="ref" href="scmfsa_4.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E2:37"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_4.html#K7" target="_self">IncAddr</a> <font color="Maroon">I</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="scmfsa_4.html#T1" target="_self">Th1</a></i>;<br/>
<b>then </b><i><font color="Green">E30</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_4.html#K7" target="_self">IncAddr</a> <font color="Maroon">I</font>,<font color="Maroon">n</font></span>)</span> <a href="scmfsa_4.html#K1" target="_self">+*</a> <span class="p2">(<span class="default"><a href="scmfsa_4.html#K7" target="_self">IncAddr</a> <font color="Maroon">J</font>,<font color="Maroon">n</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">I</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">J</font></span>)</span>
<b>by </b><i><a class="txt" href="scmfsa_4.html#E1:37"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="scmfsa_4.html#K1" target="_self">+*</a> <font color="Maroon">J</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
;<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">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:37_2"/><i><font color="Green">E31</font></i>: 
 <a href="scmfsa_2.html#K5">insloc</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="scmfsa_4.html#K1" target="_self">+*</a> <font color="Maroon">J</font></span>)</span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_2_1"/>
(  <a href="scmfsa_2.html#K5">insloc</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">J</font> or  not  <a href="scmfsa_2.html#K5">insloc</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">J</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1"><b>suppose </b></a><a NAME="E1:37_2_1_1"/><i><font color="Green">E32</font></i>: 
 <a href="scmfsa_2.html#K5">insloc</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">J</font>
 ;<br/><div class="add"><i><font color="Green">E68</font></i>:  <a href="ami_5.html#K5">pi</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="scmfsa_4.html#K1" target="_self">+*</a> <font color="Maroon">J</font></span>)</span>,<span class="p1">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <font color="Maroon">m</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="scmfsa_4.html#K1" target="_self">+*</a> <font color="Maroon">J</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i>, <a class="ref" href="ami_5.html#D5">AMI_5:def 5</a></i>
<br/>.= 
<font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="txt" href="scmfsa_4.html#E3:37"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
 <a href="ami_5.html#K5">pi</a> <font color="Maroon">J</font>,<span class="p1">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="txt" href="scmfsa_4.html#E3:37"><i><font color="Green">E30</font></i></a>, <a class="ref" href="ami_5.html#D5">AMI_5:def 5</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_4.html#K7" target="_self">IncAddr</a> <font color="Maroon">I</font>,<font color="Maroon">n</font></span>)</span> <a href="scmfsa_4.html#K1" target="_self">+*</a> <span class="p2">(<span class="default"><a href="scmfsa_4.html#K7" target="_self">IncAddr</a> <font color="Maroon">J</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <font color="Maroon">m</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="scmfsa_4.html#K7" target="_self">IncAddr</a> <font color="Maroon">J</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="txt" href="scmfsa_4.html#E1:37"><i><font color="Green">E27</font></i></a>, <a class="txt" href="scmfsa_4.html#E3:37"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
 <a href="scmfsa_4.html#K4" target="_self">IncAddr</a> <span class="p1">(<span class="default"><a href="ami_5.html#K5">pi</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="scmfsa_4.html#K1" target="_self">+*</a> <font color="Maroon">J</font></span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <font color="Maroon">m</font></span>)</span></span>)</span>,<font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="scmfsa_4.html#E3:37"><i><font color="Green">E30</font></i></a>, <a class="txt" href="scmfsa_4.html#E1:37_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="scmfsa_4.html#T1" target="_self">Th1</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_2"><b>suppose </b></a><a NAME="E1:37_2_1_2"/><i><font color="Green">E78</font></i>: 
not  <a href="scmfsa_2.html#K5">insloc</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">J</font>
 ;<br/><div class="add"><a NAME="E2:37_2_1_2"/>
 <a href="scmfsa_2.html#K5">insloc</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">I</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">J</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><a NAME="E3:37_2_1_2"/><b>then </b><i><font color="Green">E79</font></i>: 
 <a href="scmfsa_2.html#K5">insloc</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="txt" href="scmfsa_4.html#E3:37"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><i><font color="Green">E86</font></i>:  <a href="ami_5.html#K5">pi</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="scmfsa_4.html#K1" target="_self">+*</a> <font color="Maroon">J</font></span>)</span>,<span class="p1">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <font color="Maroon">m</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="scmfsa_4.html#K1" target="_self">+*</a> <font color="Maroon">J</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i>, <a class="ref" href="ami_5.html#D5">AMI_5:def 5</a></i>
<br/>.= 
<font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="txt" href="scmfsa_4.html#E3:37"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
 <a href="ami_5.html#K5">pi</a> <font color="Maroon">I</font>,<span class="p1">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="txt" href="scmfsa_4.html#E1:37_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="ami_5.html#D5">AMI_5:def 5</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_4.html#K7" target="_self">IncAddr</a> <font color="Maroon">I</font>,<font color="Maroon">n</font></span>)</span> <a href="scmfsa_4.html#K1" target="_self">+*</a> <span class="p2">(<span class="default"><a href="scmfsa_4.html#K7" target="_self">IncAddr</a> <font color="Maroon">J</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <font color="Maroon">m</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="scmfsa_4.html#K7" target="_self">IncAddr</a> <font color="Maroon">I</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="txt" href="scmfsa_4.html#E1:37"><i><font color="Green">E27</font></i></a>, <a class="txt" href="scmfsa_4.html#E3:37"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
 <a href="scmfsa_4.html#K4" target="_self">IncAddr</a> <span class="p1">(<span class="default"><a href="ami_5.html#K5">pi</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="scmfsa_4.html#K1" target="_self">+*</a> <font color="Maroon">J</font></span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <font color="Maroon">m</font></span>)</span></span>)</span>,<font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="scmfsa_4.html#E1:37_2"><i><font color="Green">E31</font></i></a>, , <a class="ref" href="scmfsa_4.html#T1" target="_self">Th1</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:37"/>
 <a href="scmfsa_4.html#K7" target="_self">IncAddr</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="scmfsa_4.html#K1" target="_self">+*</a> <font color="Maroon">J</font></span>)</span>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_4.html#K7" target="_self">IncAddr</a> <font color="Maroon">I</font>,<font color="Maroon">n</font></span>)</span> <a href="scmfsa_4.html#K1" target="_self">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_4.html#K7" target="_self">IncAddr</a> <font color="Maroon">J</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="scmfsa_4.html#T1" target="_self">Th1</a></i>;<br/>


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