<?xml version="1.0"?>
<div class="add">

<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">I</font> being  <a href="ami_1.html#NM3" title="AMI_1:NM.3">Instruction</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">I</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sf_mastr.html#K1" title="SF_MASTR:func.1">UsedIntLoc</a> <font color="Olive" title="b1">I</font> );<br/>
<a NAME="E1:22_1_1"/><i><font color="Green" title="E19">A1</font></i>: 
for <font color="Olive" title="b1">e</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="ami_1.html#U3" title="AMI_1:sel.3">Instructions</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>   ex <font color="Olive" title="b2">u</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <a href="scmfsa_2.html#K2" title="SCMFSA_2:func.2">Int-Locations</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="22_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">e</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="ami_1.html#U3" title="AMI_1:sel.3">Instructions</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> ;<br/>

<b>reconsider </b><font color="Maroon" title="c3">f</font> = <font color="Maroon" title="c2">e</font> as   <a href="ami_1.html#NM3" title="AMI_1:NM.3">Instruction</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c4">u</font> =  <a href="sf_mastr.html#K1" title="SF_MASTR:func.1">UsedIntLoc</a> <font color="Maroon" title="c3">f</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <a href="scmfsa_2.html#K2" title="SCMFSA_2:func.2">Int-Locations</a>  ;<br/>
<b>take </b>
<font color="Maroon" title="c4">u</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c3">f</font>
;<br/>

<b>thus </b><a NAME="E3:22_1_1_1"/>
( <font color="Maroon" title="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c4">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sf_mastr.html#K1" title="SF_MASTR:func.1">UsedIntLoc</a> <font color="Maroon" title="c3">f</font> )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c2">UIL</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="ami_1.html#U3" title="AMI_1:sel.3">Instructions</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> , <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <a href="scmfsa_2.html#K2" title="SCMFSA_2:func.2">Int-Locations</a> <b> such that </b><br/><a NAME="E3:22_1_1"/><i><font color="Green" title="E20">A2</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="ami_1.html#U3" title="AMI_1:sel.3">Instructions</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c2">UIL</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="sf_mastr.html#E1:22_1_1"><i><font color="Green" title="E19">A1</font></i></a>);<br/></i>
<b>set </b><font color="Maroon" title="c3">IT</font> =  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">UIL</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">p</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">dp</font> =  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">p</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<a NAME="E5:22_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">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#T118" title="AMI_1:th.118">AMI_1:118</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">p'</font> = <font color="Maroon" title="c1">p</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">dp</font>,the <a href="ami_1.html#U3" title="AMI_1:sel.3">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="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">Up</font> = <font color="Maroon" title="c2">UIL</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">p'</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">dp</font>, <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <a href="scmfsa_2.html#K2" title="SCMFSA_2:func.2">Int-Locations</a>  ;<br/>
<a NAME="E8:22_1_1"/><i><font color="Green" title="E21">A3</font></i>: 
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">UIL</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">Up</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a></i>;<br/>
<a NAME="E9:22_1_1"/>
 <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <a href="scmfsa_2.html#K2" title="SCMFSA_2:func.2">Int-Locations</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="scmfsa_2.html#K2" title="SCMFSA_2:func.2">Int-Locations</a> 
 
<b>by </b><i><a class="ref" href="finsub_1.html#T26" title="FINSUB_1:th.26">FINSUB_1:26</a></i>;<br/>
<a NAME="E10:22_1_1"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">Up</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="scmfsa_2.html#K2" title="SCMFSA_2:func.2">Int-Locations</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:22_1_1"/><b>then </b><i><font color="Green" title="E22">A4</font></i>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">Up</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="scmfsa_2.html#K2" title="SCMFSA_2:func.2">Int-Locations</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T95" title="ZFMISC_1:th.95">ZFMISC_1:95</a></i>;<br/>
<b>take </b>
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">UIL</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">p</font></span>)</span>
;<br/>

<b>thus </b><a NAME="E12:22_1_1"/>
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">UIL</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">p</font></span>)</span> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="scmfsa_2.html#K2" title="SCMFSA_2:func.2">Int-Locations</a> 
 <b>by </b><i><a class="txt" href="sf_mastr.html#E8:22_1_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="sf_mastr.html#E11:22_1_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#T99" title="ZFMISC_1:th.99">ZFMISC_1:99</a></i>;<br/>

<b>take </b>
<font color="Maroon" title="c2">UIL</font>
;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c7">i</font> be   <a href="ami_1.html#NM3" title="AMI_1:NM.3">Instruction</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> ;<br/><a NAME="E1:22_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c2">UIL</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">i</font>]
 <b>by </b><i><a class="txt" href="sf_mastr.html#E3:22_1_1"><i><font color="Green" title="E20">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E2:22_1_1_2"/>
<font color="Maroon" title="c2">UIL</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sf_mastr.html#K1" title="SF_MASTR:func.1">UsedIntLoc</a> <font color="Maroon" title="c7">i</font>
 ;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E14:22_1_1"/>
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">UIL</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">UIL</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">p</font></span>)</span>
 ;<br/>


</div>
