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<b>let </b><font color="Maroon" title="c1">A</font> be   <a href="aofa_000.html#NM1" title="AOFA_000:NM.1">preIfWhileAlgebra</a>;<br/><b>let </b><font color="Maroon" title="c2">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="c3">T</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font>;<br/><b>let </b><font color="Maroon" title="c4">f</font> be    <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font>;<br/><b>let </b><font color="Maroon" title="c5">P</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c6">I</font>, <font color="Maroon" title="c7">J</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:180"/><i><font color="Green" title="E108">A1</font></i>: 
for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">s</font>,<font color="Maroon" title="c6">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="c1">A</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">f</font>
 <b>and </b><br/><a NAME="E2:180"/><i><font color="Green" title="E109">A2</font></i>: 
for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">s</font>,<font color="Maroon" title="c7">J</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="c1">A</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">f</font>
 <b>and </b><br/><a NAME="E3:180"/><i><font color="Green" title="E110">A3</font></i>: 
for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">s</font>,<font color="Maroon" title="c6">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="aofa_000.html#D38" target="_self" title="AOFA_000:def.38">AOFA_000:def 38</a>,<a class="ref" href="aofa_000.html#D39" target="_self" title="AOFA_000:def.39">AOFA_000:def 39</a><br/>

<b>let </b><font color="Maroon" title="c8">s</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="aofa_000.html#D38" target="_self" title="AOFA_000:def.38">AOFA_000:def 38</a><br/>

<b>assume </b><a NAME="E4:180"/>
<font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 ;<br/>

<a NAME="E5:180"/><b>then </b><i><font color="Green" title="E111">A4</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">s</font>,<font color="Maroon" title="c6">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="c1">A</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c8">s</font>,<font color="Maroon" title="c6">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> )
 
<b>by </b><i><a class="txt" href="aofa_000.html#E1:180"><i><font color="Green" title="E108">A1</font></i></a>, <a class="txt" href="aofa_000.html#E3:180"><i><font color="Green" title="E110">A3</font></i></a></i>;<br/>
<a NAME="E6:180"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c8">s</font>,<font color="Maroon" title="c6">I</font></span>)</span>,<font color="Maroon" title="c7">J</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="c1">A</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="txt" href="aofa_000.html#E2:180"><i><font color="Green" title="E109">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:180"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c7">J</font></span>)</span></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="c1">A</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="txt" href="aofa_000.html#E5:180"><i><font color="Green" title="E111">A4</font></i></a>, <a class="ref" href="aofa_000.html#D35" target="_self" title="AOFA_000:def.35">Def35</a></i>;<br/>


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