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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>() ) or ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Olive" title="b1">m</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">m</font>) ) or ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <font color="Olive" title="b1">m</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">m</font>) ) or ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <font color="Olive" title="b1">m</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">m</font>) ) );<br/>
<a NAME="E1:36_1"/><i><font color="Green" title="E20">A1</font></i>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  holds <br/> ex <font color="Olive" title="b2">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e</font>,<font color="Olive" title="b2">u</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:36_1_1"/>
<font color="Maroon" title="c1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> 
 ;<br/>

<a NAME="E2:36_1_1"/><b>then </b>
( <font color="Maroon" title="c1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2" title="SCMFSA_2:func.2">Int-Locations</a>  <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="scmfsa_2.html#K3" title="SCMFSA_2:func.3">FinSeq-Locations</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> </span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  )
 
<b>by </b><i><a class="ref" href="scmfsa_2.html#T8" title="SCMFSA_2:th.8">SCMFSA_2:8</a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E3:36_1_1"/><b>then </b><i><font color="Green" title="E21">A2</font></i>: 
( <font color="Maroon" title="c1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="scmfsa_2.html#K2" title="SCMFSA_2:func.2">Int-Locations</a>  <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="scmfsa_2.html#K3" title="SCMFSA_2:func.3">FinSeq-Locations</a>  or <font color="Maroon" title="c1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> </span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_1"/>
( <font color="Maroon" title="c1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> </span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="scmfsa_2.html#K2" title="SCMFSA_2:func.2">Int-Locations</a>  or <font color="Maroon" title="c1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="scmfsa_2.html#K3" title="SCMFSA_2:func.3">FinSeq-Locations</a>  or <font color="Maroon" title="c1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  )
 </span><b>by </b><i><a class="txt" href="scmfsa6a.html#E3:36_1_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_1_1"/>
<font color="Maroon" title="c1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> </span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_1_1_1_1_1"/>
<font color="Maroon" title="c1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_1_2"/>
<font color="Maroon" title="c1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="scmfsa_2.html#K2" title="SCMFSA_2:func.2">Int-Locations</a> 
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c1">e</font> is    <a href="scmfsa_2.html#M1" title="SCMFSA_2:mode.1">Int-Location</a> 
 <b>by </b><i><a class="ref" href="scmfsa_2.html#T11" title="SCMFSA_2:th.11">SCMFSA_2:11</a></i>;<br/><b>hence </b><a NAME="E3:36_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <font color="Olive" title="b1">m</font>
 <b>by </b><i><a class="ref" href="scmfsa_2.html#T19" title="SCMFSA_2:th.19">SCMFSA_2:19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_1_3"><b>case </b></a><a NAME="E1:36_1_1_1_1_3"/>
<font color="Maroon" title="c1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="scmfsa_2.html#K3" title="SCMFSA_2:func.3">FinSeq-Locations</a> 
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c1">e</font> is    <a href="scmfsa_2.html#M2" title="SCMFSA_2:mode.2">FinSeq-Location</a> 
 <b>by </b><i><a class="ref" href="scmfsa_2.html#T12" title="SCMFSA_2:th.12">SCMFSA_2:12</a></i>;<br/><b>hence </b><a NAME="E3:36_1_1_1_1_3"/>
 ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <font color="Olive" title="b1">m</font>
 <b>by </b><i><a class="ref" href="scmfsa_2.html#T20" title="SCMFSA_2:th.20">SCMFSA_2:20</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_1_4"><b>case </b></a><a NAME="E1:36_1_1_1_1_4"/>
<font color="Maroon" title="c1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_1_1_1_1_4"/>
 ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Olive" title="b1">m</font>
 <b>by </b><i><a class="ref" href="scmfsa_2.html#T21" title="SCMFSA_2:th.21">SCMFSA_2:21</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c2">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:36_1_1"/><i><font color="Green" title="E22">A3</font></i>: 
( <font color="Maroon" title="c1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  or <font color="Maroon" title="c1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Maroon" title="c2">m</font> or <font color="Maroon" title="c1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <font color="Maroon" title="c2">m</font> or <font color="Maroon" title="c1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <font color="Maroon" title="c2">m</font> )
 ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_2"/>
( <font color="Maroon" title="c1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  or <font color="Maroon" title="c1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Maroon" title="c2">m</font> or <font color="Maroon" title="c1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <font color="Maroon" title="c2">m</font> or <font color="Maroon" title="c1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <font color="Maroon" title="c2">m</font> )
 </span><b>by </b><i><a class="txt" href="scmfsa6a.html#E6:36_1_1"><i><font color="Green" title="E22">A3</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2_1"><b>suppose </b></a><a NAME="E1:36_1_1_2_1"/><i><font color="Green" title="E23">A4</font></i>: 
<font color="Maroon" title="c1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> 
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c3">u</font> = <font color="Maroon">F<sub>4</sub></font>();<br/><b>thus </b><a NAME="E2:36_1_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">e</font>,<font color="Maroon" title="c3">u</font>]
 <b>by </b><i><a class="txt" href="scmfsa6a.html#E1:36_1_1_2_1"><i><font color="Green" title="E23">A4</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2_2"><b>suppose </b></a><a NAME="E1:36_1_1_2_2"/><i><font color="Green" title="E23">A5</font></i>: 
<font color="Maroon" title="c1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Maroon" title="c2">m</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c3">u</font> = <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c2">m</font>);<br/><b>thus </b><a NAME="E2:36_1_1_2_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">e</font>,<font color="Maroon" title="c3">u</font>]
 <b>by </b><i><a class="txt" href="scmfsa6a.html#E1:36_1_1_2_2"><i><font color="Green" title="E23">A5</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2_3"><b>suppose </b></a><a NAME="E1:36_1_1_2_3"/><i><font color="Green" title="E23">A6</font></i>: 
<font color="Maroon" title="c1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <font color="Maroon" title="c2">m</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c3">u</font> = <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c2">m</font>);<br/><b>thus </b><a NAME="E2:36_1_1_2_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">e</font>,<font color="Maroon" title="c3">u</font>]
 <b>by </b><i><a class="txt" href="scmfsa6a.html#E1:36_1_1_2_3"><i><font color="Green" title="E23">A6</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2_4"><b>suppose </b></a><a NAME="E1:36_1_1_2_4"/><i><font color="Green" title="E23">A7</font></i>: 
<font color="Maroon" title="c1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <font color="Maroon" title="c2">m</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c3">u</font> = <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c2">m</font>);<br/><b>thus </b><a NAME="E2:36_1_1_2_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">e</font>,<font color="Maroon" title="c3">u</font>]
 <b>by </b><i><a class="txt" href="scmfsa6a.html#E1:36_1_1_2_4"><i><font color="Green" title="E23">A7</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c1">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:36_1"/><i><font color="Green" title="E21">A8</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> 
 <b>and </b><br/><a NAME="E4:36_1"/><i><font color="Green" title="E22">A9</font></i>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font>]
 <b>from </b><i><a class="ref" href="zfrefle1.html#S1" title="ZFREFLE1:sch.1">ZFREFLE1:sch 1</a>(<a class="txt" href="scmfsa6a.html#E1:36_1"><i><font color="Green" title="E20">A1</font></i></a>);<br/></i>
<a NAME="E5:36_1"/><i><font color="Green" title="E23">A10</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="ami_1.html#U5" title="AMI_1:sel.5">Object-Kind</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:36_1_2"/><i><font color="Green" title="E24">A11</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="ami_1.html#U5" title="AMI_1:sel.5">Object-Kind</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> 
 ;<br/><a NAME="E2:36_1_2"/><b>then </b>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2" title="SCMFSA_2:func.2">Int-Locations</a>  <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="scmfsa_2.html#K3" title="SCMFSA_2:func.3">FinSeq-Locations</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> </span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  )
 <b>by </b><i><a class="txt" href="scmfsa6a.html#E5:36_1"><i><font color="Green" title="E23">A10</font></i></a>, <a class="ref" href="scmfsa_2.html#T8" title="SCMFSA_2:th.8">SCMFSA_2:8</a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:36_1_2"/><b>then </b><i><font color="Green" title="E25">A12</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="scmfsa_2.html#K2" title="SCMFSA_2:func.2">Int-Locations</a>  <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="scmfsa_2.html#K3" title="SCMFSA_2:func.3">FinSeq-Locations</a>  or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> </span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><b>consider </b><font color="Maroon" title="c3">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:36_1_2"/><i><font color="Green" title="E26">A13</font></i>: 
( ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  &amp; <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>() ) or ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">m</font>) ) or ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">m</font>) ) or ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c3">m</font>) ) )
 <b>by </b><i><a class="txt" href="scmfsa6a.html#E4:36_1"><i><font color="Green" title="E22">A9</font></i></a>, <a class="txt" href="scmfsa6a.html#E5:36_1"><i><font color="Green" title="E23">A10</font></i></a>, <a class="txt" href="scmfsa6a.html#E1:36_1_2"><i><font color="Green" title="E24">A11</font></i></a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_2_1"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="scmfsa_2.html#K2" title="SCMFSA_2:func.2">Int-Locations</a>  or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="scmfsa_2.html#K3" title="SCMFSA_2:func.3">FinSeq-Locations</a>  or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> </span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  )
 </span><b>by </b><i><a class="txt" href="scmfsa6a.html#E3:36_1_2"><i><font color="Green" title="E25">A12</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_2_1_1"><b>suppose </b></a><a NAME="E1:36_1_2_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="scmfsa_2.html#K2" title="SCMFSA_2:func.2">Int-Locations</a> 
 ;<br/><div class="add"><a NAME="E2:36_1_2_1_1"/><b>then </b><i><font color="Green" title="E27">A14</font></i>: 
<font color="Maroon" title="c2">x</font> is    <a href="scmfsa_2.html#M1" title="SCMFSA_2:mode.1">Int-Location</a> 
 <b>by </b><i><a class="ref" href="scmfsa_2.html#T11" title="SCMFSA_2:th.11">SCMFSA_2:11</a></i>;<br/><b>then </b>the <a href="ami_1.html#U5" title="AMI_1:sel.5">Object-Kind</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> = 
 <a href="ami_1.html#K3" title="AMI_1:func.3">ObjectKind</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <font color="Maroon" title="c3">m</font></span>)</span>
<b>by </b><i><a class="txt" href="scmfsa6a.html#E5:36_1_2"><i><font color="Green" title="E26">A13</font></i></a>, <a class="ref" href="ami_1.html#D6" title="AMI_1:def.6">AMI_1:def 6</a>, <a class="ref" href="scmfsa_2.html#T81" title="SCMFSA_2:th.81">SCMFSA_2:81</a>, <a class="ref" href="scmfsa_2.html#T83" title="SCMFSA_2:th.83">SCMFSA_2:83</a>, <a class="ref" href="scmfsa_2.html#T84" title="SCMFSA_2:th.84">SCMFSA_2:84</a></i>
<br/>.= 
 <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
<b>by </b><i><a class="ref" href="scmfsa_2.html#T26" title="SCMFSA_2:th.26">SCMFSA_2:26</a></i>
;<br/><b>hence </b><a NAME="E4:36_1_2_1_1"/>
<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="ami_1.html#U5" title="AMI_1:sel.5">Object-Kind</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="scmfsa6a.html#E5:36_1_2"><i><font color="Green" title="E26">A13</font></i></a>, <a class="txt" href="scmfsa6a.html#E2:36_1_2_1_1"><i><font color="Green" title="E27">A14</font></i></a>, <a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a>, <a class="ref" href="scmfsa_2.html#T81" title="SCMFSA_2:th.81">SCMFSA_2:81</a>, <a class="ref" href="scmfsa_2.html#T83" title="SCMFSA_2:th.83">SCMFSA_2:83</a>, <a class="ref" href="scmfsa_2.html#T84" title="SCMFSA_2:th.84">SCMFSA_2:84</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_2_1_2"><b>suppose </b></a><a NAME="E1:36_1_2_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="scmfsa_2.html#K3" title="SCMFSA_2:func.3">FinSeq-Locations</a> 
 ;<br/><div class="add"><a NAME="E2:36_1_2_1_2"/><b>then </b><i><font color="Green" title="E27">A15</font></i>: 
<font color="Maroon" title="c2">x</font> is    <a href="scmfsa_2.html#M2" title="SCMFSA_2:mode.2">FinSeq-Location</a> 
 <b>by </b><i><a class="ref" href="scmfsa_2.html#T12" title="SCMFSA_2:th.12">SCMFSA_2:12</a></i>;<br/><b>then </b>the <a href="ami_1.html#U5" title="AMI_1:sel.5">Object-Kind</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> = 
 <a href="ami_1.html#K3" title="AMI_1:func.3">ObjectKind</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <font color="Maroon" title="c3">m</font></span>)</span>
<b>by </b><i><a class="txt" href="scmfsa6a.html#E5:36_1_2"><i><font color="Green" title="E26">A13</font></i></a>, <a class="ref" href="ami_1.html#D6" title="AMI_1:def.6">AMI_1:def 6</a>, <a class="ref" href="scmfsa_2.html#T82" title="SCMFSA_2:th.82">SCMFSA_2:82</a>, <a class="ref" href="scmfsa_2.html#T83" title="SCMFSA_2:th.83">SCMFSA_2:83</a>, <a class="ref" href="scmfsa_2.html#T85" title="SCMFSA_2:th.85">SCMFSA_2:85</a></i>
<br/>.= 
<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
<b>by </b><i><a class="ref" href="scmfsa_2.html#T27" title="SCMFSA_2:th.27">SCMFSA_2:27</a></i>
;<br/><b>hence </b><a NAME="E4:36_1_2_1_2"/>
<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="ami_1.html#U5" title="AMI_1:sel.5">Object-Kind</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="scmfsa6a.html#E5:36_1_2"><i><font color="Green" title="E26">A13</font></i></a>, <a class="txt" href="scmfsa6a.html#E2:36_1_2_1_2"><i><font color="Green" title="E27">A15</font></i></a>, <a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a>, <a class="ref" href="scmfsa_2.html#T82" title="SCMFSA_2:th.82">SCMFSA_2:82</a>, <a class="ref" href="scmfsa_2.html#T83" title="SCMFSA_2:th.83">SCMFSA_2:83</a>, <a class="ref" href="scmfsa_2.html#T85" title="SCMFSA_2:th.85">SCMFSA_2:85</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_2_1_3"><b>suppose </b></a><a NAME="E1:36_1_2_1_3"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> </span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:36_1_2_1_3"/><b>then </b><i><font color="Green" title="E27">A16</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then </b>the <a href="ami_1.html#U5" title="AMI_1:sel.5">Object-Kind</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> = 
 <a href="ami_1.html#K3" title="AMI_1:func.3">ObjectKind</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> </span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#D6" title="AMI_1:def.6">AMI_1:def 6</a></i>
<br/>.= 
the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> 
<b>by </b><i><a class="ref" href="ami_1.html#D11" title="AMI_1:def.11">AMI_1:def 11</a></i>
;<br/><b>hence </b><a NAME="E4:36_1_2_1_3"/>
<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="ami_1.html#U5" title="AMI_1:sel.5">Object-Kind</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="scmfsa6a.html#E5:36_1_2"><i><font color="Green" title="E26">A13</font></i></a>, <a class="txt" href="scmfsa6a.html#E2:36_1_2_1_3"><i><font color="Green" title="E27">A16</font></i></a>, <a class="ref" href="ami_1.html#T48" title="AMI_1:th.48">AMI_1:48</a>, <a class="ref" href="scmfsa_2.html#T81" title="SCMFSA_2:th.81">SCMFSA_2:81</a>, <a class="ref" href="scmfsa_2.html#T82" title="SCMFSA_2:th.82">SCMFSA_2:82</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_2_1_4"><b>suppose </b></a><a NAME="E1:36_1_2_1_4"/><i><font color="Green" title="E27">A17</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> 
 ;<br/><div class="add"><b>then </b>the <a href="ami_1.html#U5" title="AMI_1:sel.5">Object-Kind</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> = 
 <a href="ami_1.html#K3" title="AMI_1:func.3">ObjectKind</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Maroon" title="c3">m</font></span>)</span>
<b>by </b><i><a class="txt" href="scmfsa6a.html#E5:36_1_2"><i><font color="Green" title="E26">A13</font></i></a>, <a class="ref" href="ami_1.html#T48" title="AMI_1:th.48">AMI_1:48</a>, <a class="ref" href="ami_1.html#D6" title="AMI_1:def.6">AMI_1:def 6</a>, <a class="ref" href="scmfsa_2.html#T84" title="SCMFSA_2:th.84">SCMFSA_2:84</a>, <a class="ref" href="scmfsa_2.html#T85" title="SCMFSA_2:th.85">SCMFSA_2:85</a></i>
<br/>.= 
the <a href="ami_1.html#U4" title="AMI_1:sel.4">Instructions</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> 
<b>by </b><i><a class="ref" href="ami_1.html#D14" title="AMI_1:def.14">AMI_1:def 14</a></i>
;<br/><b>hence </b><a NAME="E3:36_1_2_1_4"/>
<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="ami_1.html#U5" title="AMI_1:sel.5">Object-Kind</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="scmfsa6a.html#E5:36_1_2"><i><font color="Green" title="E26">A13</font></i></a>, <a class="txt" href="scmfsa6a.html#E1:36_1_2_1_4"><i><font color="Green" title="E27">A17</font></i></a>, <a class="ref" href="ami_1.html#T48" title="AMI_1:th.48">AMI_1:48</a>, <a class="ref" href="scmfsa_2.html#T84" title="SCMFSA_2:th.84">SCMFSA_2:84</a>, <a class="ref" href="scmfsa_2.html#T85" title="SCMFSA_2:th.85">SCMFSA_2:85</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" title="c2">f</font> = <font color="Maroon" title="c1">f</font> as   <a href="ami_1.html#NM4" title="AMI_1:NM.4">State</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  <b>by </b><i><a class="txt" href="scmfsa6a.html#E3:36_1"><i><font color="Green" title="E21">A8</font></i></a>, <a class="txt" href="scmfsa6a.html#E5:36_1"><i><font color="Green" title="E23">A10</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c2">f</font>
;<br/>

<b>consider </b><font color="Maroon" title="c3">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E9:36_1"/><i><font color="Green" title="E24">A18</font></i>: 
( (  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  &amp; <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>() ) or (  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">m</font>) ) or (  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">m</font>) ) or (  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c3">m</font>) ) )
 <b>by </b><i><a class="txt" href="scmfsa6a.html#E4:36_1"><i><font color="Green" title="E22">A9</font></i></a></i>;<br/>
<b>thus </b><a NAME="E10:36_1"/>
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>()
 <b>by </b><i><a class="txt" href="scmfsa6a.html#E9:36_1"><i><font color="Green" title="E24">A18</font></i></a>, <a class="ref" href="ami_1.html#T48" title="AMI_1:th.48">AMI_1:48</a>, <a class="ref" href="ami_1.html#D15" title="AMI_1:def.15">AMI_1:def 15</a>, <a class="ref" href="scmfsa_2.html#T81" title="SCMFSA_2:th.81">SCMFSA_2:81</a>, <a class="ref" href="scmfsa_2.html#T82" title="SCMFSA_2:th.82">SCMFSA_2:82</a></i>;<br/>

<b>let </b><font color="Maroon" title="c4">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>consider </b><font color="Maroon" title="c5">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E12:36_1"/><i><font color="Green" title="E25">A19</font></i>: 
( (  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  &amp; <font color="Maroon" title="c2">f</font> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>() ) or (  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Maroon" title="c5">m</font> &amp; <font color="Maroon" title="c2">f</font> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c5">m</font>) ) or (  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <font color="Maroon" title="c5">m</font> &amp; <font color="Maroon" title="c2">f</font> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">m</font>) ) or (  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <font color="Maroon" title="c5">m</font> &amp; <font color="Maroon" title="c2">f</font> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c5">m</font>) ) )
 <b>by </b><i><a class="txt" href="scmfsa6a.html#E4:36_1"><i><font color="Green" title="E22">A9</font></i></a></i>;<br/>
<b>thus </b><a NAME="E13:36_1"/>
<font color="Maroon" title="c2">f</font> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c4">i</font>)
 <b>by </b><i><a class="txt" href="scmfsa6a.html#E12:36_1"><i><font color="Green" title="E25">A19</font></i></a>, <a class="ref" href="ami_1.html#T48" title="AMI_1:th.48">AMI_1:48</a>, <a class="ref" href="scmfsa_2.html#T84" title="SCMFSA_2:th.84">SCMFSA_2:84</a>, <a class="ref" href="scmfsa_2.html#T85" title="SCMFSA_2:th.85">SCMFSA_2:85</a></i>;<br/>

<b>consider </b><font color="Maroon" title="c6">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E15:36_1"/><i><font color="Green" title="E26">A20</font></i>: 
( (  <a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  &amp; <font color="Maroon" title="c2">f</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>() ) or (  <a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Maroon" title="c6">m</font> &amp; <font color="Maroon" title="c2">f</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c6">m</font>) ) or (  <a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <font color="Maroon" title="c6">m</font> &amp; <font color="Maroon" title="c2">f</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c6">m</font>) ) or (  <a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <font color="Maroon" title="c6">m</font> &amp; <font color="Maroon" title="c2">f</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c6">m</font>) ) )
 <b>by </b><i><a class="txt" href="scmfsa6a.html#E4:36_1"><i><font color="Green" title="E22">A9</font></i></a></i>;<br/>
<b>thus </b><a NAME="E16:36_1"/>
<font color="Maroon" title="c2">f</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c4">i</font>)
 <b>by </b><i><a class="txt" href="scmfsa6a.html#E15:36_1"><i><font color="Green" title="E26">A20</font></i></a>, <a class="ref" href="scmfsa_2.html#T81" title="SCMFSA_2:th.81">SCMFSA_2:81</a>, <a class="ref" href="scmfsa_2.html#T83" title="SCMFSA_2:th.83">SCMFSA_2:83</a>, <a class="ref" href="scmfsa_2.html#T84" title="SCMFSA_2:th.84">SCMFSA_2:84</a></i>;<br/>

<b>consider </b><font color="Maroon" title="c7">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E18:36_1"/><i><font color="Green" title="E27">A21</font></i>: 
( (  <a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  &amp; <font color="Maroon" title="c2">f</font> <a href="scmfsa_2.html#K21" title="SCMFSA_2:func.21">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>() ) or (  <a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Maroon" title="c7">m</font> &amp; <font color="Maroon" title="c2">f</font> <a href="scmfsa_2.html#K21" title="SCMFSA_2:func.21">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c7">m</font>) ) or (  <a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <font color="Maroon" title="c7">m</font> &amp; <font color="Maroon" title="c2">f</font> <a href="scmfsa_2.html#K21" title="SCMFSA_2:func.21">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c7">m</font>) ) or (  <a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <font color="Maroon" title="c7">m</font> &amp; <font color="Maroon" title="c2">f</font> <a href="scmfsa_2.html#K21" title="SCMFSA_2:func.21">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c7">m</font>) ) )
 <b>by </b><i><a class="txt" href="scmfsa6a.html#E4:36_1"><i><font color="Green" title="E22">A9</font></i></a></i>;<br/>
<b>thus </b><a NAME="E19:36_1"/>
<font color="Maroon" title="c2">f</font> <a href="scmfsa_2.html#K21" title="SCMFSA_2:func.21">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c4">i</font>)
 <b>by </b><i><a class="txt" href="scmfsa6a.html#E18:36_1"><i><font color="Green" title="E27">A21</font></i></a>, <a class="ref" href="scmfsa_2.html#T82" title="SCMFSA_2:th.82">SCMFSA_2:82</a>, <a class="ref" href="scmfsa_2.html#T83" title="SCMFSA_2:th.83">SCMFSA_2:83</a>, <a class="ref" href="scmfsa_2.html#T85" title="SCMFSA_2:th.85">SCMFSA_2:85</a></i>;<br/>


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