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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S</font> being 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/> for <font color="Olive" title="b2">T</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Olive" title="b1">S</font><br/> for <font color="Olive" title="b3">f</font> being   <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">T</font><br/> for <font color="Olive" title="b4">I</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font> holds <br/> ( <font color="Olive" title="b4">I</font> <a href="aofa_000.html#R5" title="AOFA_000:pred.5">is_terminating_wrt</a> <font color="Olive" title="b3">f</font> iff <font color="Olive" title="b4">I</font> <a href="aofa_000.html#R6" title="AOFA_000:pred.6">is_terminating_wrt</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b1">S</font> )</span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">T</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font><br/> for <font color="Olive" title="b2">f</font> being   <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="Olive" title="b1">T</font><br/> for <font color="Olive" title="b3">I</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font> holds <br/> ( <font color="Olive" title="b3">I</font> <a href="aofa_000.html#R5" title="AOFA_000:pred.5">is_terminating_wrt</a> <font color="Olive" title="b2">f</font> iff <font color="Olive" title="b3">I</font> <a href="aofa_000.html#R6" title="AOFA_000:pred.6">is_terminating_wrt</a> <font color="Olive" title="b2">f</font>,<font color="Maroon" title="c2">S</font> )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being   <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/> for <font color="Olive" title="b2">I</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font> holds <br/> ( <font color="Olive" title="b2">I</font> <a href="aofa_000.html#R5" title="AOFA_000:pred.5">is_terminating_wrt</a> <font color="Olive" title="b1">f</font> iff <font color="Olive" title="b2">I</font> <a href="aofa_000.html#R6" title="AOFA_000:pred.6">is_terminating_wrt</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c2">S</font> )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">I</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font> holds <br/> ( <font color="Olive" title="b1">I</font> <a href="aofa_000.html#R5" title="AOFA_000:pred.5">is_terminating_wrt</a> <font color="Maroon" title="c4">f</font> iff <font color="Olive" title="b1">I</font> <a href="aofa_000.html#R6" title="AOFA_000:pred.6">is_terminating_wrt</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">S</font> )</span><br/><b>let </b><font color="Maroon" title="c5">I</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">I</font> <a href="aofa_000.html#R5" title="AOFA_000:pred.5">is_terminating_wrt</a> <font color="Maroon" title="c4">f</font> iff <font color="Maroon" title="c5">I</font> <a href="aofa_000.html#R6" title="AOFA_000:pred.6">is_terminating_wrt</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">S</font> )</span><br/>









<b>thus </b><a NAME="E1:175"/>
( <font color="Maroon" title="c5">I</font> <a href="aofa_000.html#R5" title="AOFA_000:pred.5">is_terminating_wrt</a> <font color="Maroon" title="c4">f</font> implies <font color="Maroon" title="c5">I</font> <a href="aofa_000.html#R6" title="AOFA_000:pred.6">is_terminating_wrt</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">S</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">I</font> <a href="aofa_000.html#R6" title="AOFA_000:pred.6">is_terminating_wrt</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">S</font> implies <font color="Maroon" title="c5">I</font> <a href="aofa_000.html#R5" title="AOFA_000:pred.5">is_terminating_wrt</a> <font color="Maroon" title="c4">f</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="175_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:175_1"/><i><font color="Green" title="E107">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> holds  <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="c5">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>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="aofa_000.html#D37" target="_self" title="AOFA_000:def.37">AOFA_000:def 37</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">I</font> <a href="aofa_000.html#R6" title="AOFA_000:pred.6">is_terminating_wrt</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">S</font></span><br/>

<b>let </b><font color="Maroon" title="c6">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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c5">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> )</span><br/>

<b>thus </b><a NAME="E2:175_1"/>
( <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c5">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>by </b><i><a class="txt" href="aofa_000.html#E1:175_1"><i><font color="Green" title="E107">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E2:175"/><i><font color="Green" title="E107">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="c2">S</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="c5">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>
 ; <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="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">I</font> <a href="aofa_000.html#R5" title="AOFA_000:pred.5">is_terminating_wrt</a> <font color="Maroon" title="c4">f</font></span><br/>

<b>let </b><font color="Maroon" title="c6">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#D37" target="_self" title="AOFA_000:def.37">AOFA_000:def 37</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c5">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></span><br/>

<b>thus </b><a NAME="E3:175"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c5">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>by </b><i><a class="txt" href="aofa_000.html#E2:175"><i><font color="Green" title="E107">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
