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

<b>let </b><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">r</font> be  <a href="ami_3.html#V1" title="AMI_3:attr.1">programmed</a>  <a href="ami_1.html#M1" title="AMI_1:mode.1">FinPartState</a> of  <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> ;<br/>



<b>consider </b><font color="Maroon" title="c3">UIL</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of 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> , <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <a href="scmfsa_2.html#K3" title="SCMFSA_2:func.3">FinSeq-Locations</a> <b> such that </b><br/><a NAME="E2:48"/><i><font color="Green" title="E37">A1</font></i>: 
( ( for <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>  holds  <font color="Maroon" title="c3">UIL</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sf_mastr.html#K3" title="SF_MASTR:func.3">UsedInt*Loc</a> <font color="Olive" title="b1">i</font> ) &amp;  <a href="sf_mastr.html#K4" title="SF_MASTR:func.4">UsedInt*Loc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="scmfsa_4.html#K1" title="SCMFSA_4:func.1">+*</a> <font color="Maroon" title="c2">r</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="c3">UIL</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="scmfsa_4.html#K1" title="SCMFSA_4:func.1">+*</a> <font color="Maroon" title="c2">r</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="sf_mastr.html#D4" target="_self" title="SF_MASTR:def.4">Def4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">UIL1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of 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> , <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <a href="scmfsa_2.html#K3" title="SCMFSA_2:func.3">FinSeq-Locations</a> <b> such that </b><br/><a NAME="E4:48"/><i><font color="Green" title="E38">A2</font></i>: 
( ( for <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>  holds  <font color="Maroon" title="c4">UIL1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sf_mastr.html#K3" title="SF_MASTR:func.3">UsedInt*Loc</a> <font color="Olive" title="b1">i</font> ) &amp;  <a href="sf_mastr.html#K4" title="SF_MASTR:func.4">UsedInt*Loc</a> <font color="Maroon" title="c1">p</font> <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="c4">UIL1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">p</font></span>)</span> )
 <b>by </b><i><a class="ref" href="sf_mastr.html#D4" target="_self" title="SF_MASTR:def.4">Def4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">UIL2</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of 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> , <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <a href="scmfsa_2.html#K3" title="SCMFSA_2:func.3">FinSeq-Locations</a> <b> such that </b><br/><a NAME="E6:48"/><i><font color="Green" title="E39">A3</font></i>: 
( ( for <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>  holds  <font color="Maroon" title="c5">UIL2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sf_mastr.html#K3" title="SF_MASTR:func.3">UsedInt*Loc</a> <font color="Olive" title="b1">i</font> ) &amp;  <a href="sf_mastr.html#K4" title="SF_MASTR:func.4">UsedInt*Loc</a> <font color="Maroon" title="c2">r</font> <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="c5">UIL2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c2">r</font></span>)</span> )
 <b>by </b><i><a class="ref" href="sf_mastr.html#D4" target="_self" title="SF_MASTR:def.4">Def4</a></i>;<br/>
<a NAME="E7:48"/>
for <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 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>  holds  <font color="Maroon" title="c3">UIL</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">UIL1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 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> ;<br/>

<b>reconsider </b><font color="Maroon" title="c7">d</font> = <font color="Maroon" title="c6">c</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>thus </b><font color="Maroon" title="c3">UIL</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">c</font> = 
 <a href="sf_mastr.html#K3" title="SF_MASTR:func.3">UsedInt*Loc</a> <font color="Maroon" title="c7">d</font>
<b>by </b><i><a class="txt" href="sf_mastr.html#E2:48"><i><font color="Green" title="E37">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">UIL1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">c</font>
<b>by </b><i><a class="txt" href="sf_mastr.html#E4:48"><i><font color="Green" title="E38">A2</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E8:48"/><b>then </b><i><font color="Green" title="E40">A4</font></i>: 
<font color="Maroon" title="c3">UIL</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">UIL1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>
<a NAME="E9:48"/>
for <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 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>  holds  <font color="Maroon" title="c3">UIL</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">UIL2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 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> ;<br/>

<b>reconsider </b><font color="Maroon" title="c7">d</font> = <font color="Maroon" title="c6">c</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>thus </b><font color="Maroon" title="c3">UIL</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">c</font> = 
 <a href="sf_mastr.html#K3" title="SF_MASTR:func.3">UsedInt*Loc</a> <font color="Maroon" title="c7">d</font>
<b>by </b><i><a class="txt" href="sf_mastr.html#E2:48"><i><font color="Green" title="E37">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">UIL2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">c</font>
<b>by </b><i><a class="txt" href="sf_mastr.html#E6:48"><i><font color="Green" title="E39">A3</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E10:48"/><b>then </b><i><font color="Green" title="E41">A5</font></i>: 
<font color="Maroon" title="c3">UIL</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">UIL2</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>
<a NAME="E11:48"/><i><font color="Green" title="E42">A6</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="c3">UIL</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="scmfsa_4.html#K1" title="SCMFSA_4:func.1">+*</a> <font color="Maroon" title="c2">r</font></span>)</span></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="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">UIL</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="scmfsa_4.html#K1" title="SCMFSA_4:func.1">+*</a> <font color="Maroon" title="c2">r</font></span>)</span></span>)</span></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="E12:48"/><i><font color="Green" title="E43">A7</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="c3">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="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">UIL</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">p</font></span>)</span></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="E13:48"/><i><font color="Green" title="E44">A8</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="c3">UIL</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c2">r</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="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">UIL</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c2">r</font></span>)</span></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="E14:48"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">UIL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 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="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E15:48"/><b>then </b>
(  <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>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">UIL</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">UIL</font> )
 
<b>by </b><i><a class="ref" href="scmfsa_4.html#T1" title="SCMFSA_4:th.1">SCMFSA_4:1</a></i>;<br/>
<a NAME="E16:48"/><b>then </b>
<font color="Maroon" title="c3">UIL</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="scmfsa_4.html#K1" title="SCMFSA_4:func.1">+*</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">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="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">UIL</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c2">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_7.html#T10" title="FUNCT_7:th.10">FUNCT_7:10</a></i>;<br/>
<a NAME="E17:48"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">UIL</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="scmfsa_4.html#K1" title="SCMFSA_4:func.1">+*</a> <font color="Maroon" title="c2">r</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">UIL</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">UIL</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c2">r</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T18" title="FUNCT_4:th.18">FUNCT_4:18</a></i>;<br/>
<a NAME="E18:48"/><b>then </b>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">UIL</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="scmfsa_4.html#K1" title="SCMFSA_4:func.1">+*</a> <font color="Maroon" title="c2">r</font></span>)</span></span>)</span></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"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">UIL</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">UIL</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c2">r</font></span>)</span></span>)</span></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>hence </b><a NAME="E19:48"/>
 <a href="sf_mastr.html#K4" title="SF_MASTR:func.4">UsedInt*Loc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="scmfsa_4.html#K1" title="SCMFSA_4:func.1">+*</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="sf_mastr.html#K4" title="SF_MASTR:func.4">UsedInt*Loc</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><a href="sf_mastr.html#K4" title="SF_MASTR:func.4">UsedInt*Loc</a> <font color="Maroon" title="c2">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="sf_mastr.html#E2:48"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="sf_mastr.html#E4:48"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="sf_mastr.html#E6:48"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="sf_mastr.html#E8:48"><i><font color="Green" title="E40">A4</font></i></a>, <a class="txt" href="sf_mastr.html#E10:48"><i><font color="Green" title="E41">A5</font></i></a>, <a class="txt" href="sf_mastr.html#E11:48"><i><font color="Green" title="E42">A6</font></i></a>, <a class="txt" href="sf_mastr.html#E12:48"><i><font color="Green" title="E43">A7</font></i></a>, <a class="txt" href="sf_mastr.html#E13:48"><i><font color="Green" title="E44">A8</font></i></a>, <a class="ref" href="zfmisc_1.html#T96" title="ZFMISC_1:th.96">ZFMISC_1:96</a></i>;<br/>


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