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<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">T</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c3">A</font> be  <a href="freealg.html#V3" title="FREEALG:attr.3">free</a> <a href="aofa_000.html#V12" title="AOFA_000:attr.12">ECIW-strict</a> <a href="aofa_000.html#NM1" title="AOFA_000:NM.1">preIfWhileAlgebra</a>;<br/><b>let </b><font color="Maroon" title="c4">f1</font>, <font color="Maroon" title="c5">f2</font> be    <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>;<br/>







<b>assume </b><a NAME="E1:170"/><i><font color="Green" title="E103">A1</font></i>: 
<font color="Maroon" title="c4">f1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<span class="p2">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon" title="c3">A</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">f2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<span class="p2">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon" title="c3">A</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/>

<b>set </b><font color="Maroon" title="c6">g</font> = <font color="Maroon" title="c4">f1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<span class="p2">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon" title="c3">A</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<b>set </b><font color="Maroon" title="c7">TP1</font> =  <a href="aofa_000.html#K21" title="AOFA_000:func.21">TerminatingPrograms</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c4">f1</font>;<br/>
<b>set </b><font color="Maroon" title="c8">TP2</font> =  <a href="aofa_000.html#K21" title="AOFA_000:func.21">TerminatingPrograms</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c5">f2</font>;<br/>
<b>thus </b><a NAME="E2:170"/>
 <a href="aofa_000.html#K21" title="AOFA_000:func.21">TerminatingPrograms</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c4">f1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="aofa_000.html#K21" title="AOFA_000:func.21">TerminatingPrograms</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c5">f2</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="170_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">q</font>, <font color="Maroon" title="c10">I</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D3" title="RELAT_1:def.3">RELAT_1:def 3</a><br/>

<b>assume </b><a NAME="E1:170_1"/><i><font color="Green" title="E104">A2</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c10">I</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aofa_000.html#K21" title="AOFA_000:func.21">TerminatingPrograms</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c4">f1</font>
 ;<br/>

<a NAME="E2:170_1"/><b>then </b>
( <font color="Maroon" title="c9">q</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon" title="c10">I</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">A</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E3:170_1"/>
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c10">I</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a>  <a href="aofa_000.html#K21" title="AOFA_000:func.21">TerminatingPrograms</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c5">f2</font>
 <b>by </b><i><a class="txt" href="aofa_000.html#E1:170"><i><font color="Green" title="E103">A1</font></i></a>, <a class="txt" href="aofa_000.html#E1:170_1"><i><font color="Green" title="E104">A2</font></i></a>, <a class="txt" href="aofa_000.html#E140"><i><font color="Green" title="E102">Lm3</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c9">q</font>, <font color="Maroon" title="c10">I</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D3" title="RELAT_1:def.3">RELAT_1:def 3</a><br/>

<b>assume </b><a NAME="E3:170"/><i><font color="Green" title="E104">A3</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c10">I</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aofa_000.html#K21" title="AOFA_000:func.21">TerminatingPrograms</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c5">f2</font>
 ;<br/>

<a NAME="E4:170"/><b>then </b>
( <font color="Maroon" title="c9">q</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon" title="c10">I</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">A</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E5:170"/>
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c10">I</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a>  <a href="aofa_000.html#K21" title="AOFA_000:func.21">TerminatingPrograms</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c4">f1</font>
 <b>by </b><i><a class="txt" href="aofa_000.html#E1:170"><i><font color="Green" title="E103">A1</font></i></a>, <a class="txt" href="aofa_000.html#E3:170"><i><font color="Green" title="E104">A3</font></i></a>, <a class="txt" href="aofa_000.html#E140"><i><font color="Green" title="E102">Lm3</font></i></a></i>;<br/>


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