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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> ; <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="c1">S</font><br/> for <font color="Olive" title="b2">A</font> being <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/> for <font color="Olive" title="b3">f1</font>, <font color="Olive" title="b4">f2</font> being   <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Olive" title="b2">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Olive" title="b1">T</font>  st <font color="Olive" title="b3">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="Olive" title="b2">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="Olive" title="b4">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="Olive" title="b2">A</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/>for <font color="Olive" title="b5">I</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">A</font><br/> for <font color="Olive" title="b6">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b6">s</font>,<font color="Olive" title="b5">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="Olive" title="b2">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Olive" title="b1">T</font>,<font color="Olive" title="b3">f1</font> holds <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b6">s</font>,<font color="Olive" title="b5">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="Olive" title="b2">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Olive" title="b1">T</font>,<font color="Olive" title="b4">f2</font> &amp; <font color="Olive" title="b3">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b6">s</font>,<font color="Olive" title="b5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b6">s</font>,<font color="Olive" title="b5">I</font> )</span><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>; <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">A</font> being <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/> for <font color="Olive" title="b2">f1</font>, <font color="Olive" title="b3">f2</font> being   <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Olive" title="b1">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>  st <font color="Olive" title="b2">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="Olive" title="b1">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="Olive" title="b3">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="Olive" title="b1">A</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <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="Olive" title="b1">A</font><br/> for <font color="Olive" title="b5">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">s</font>,<font color="Olive" title="b4">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="Olive" title="b1">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>,<font color="Olive" title="b2">f1</font> holds <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">s</font>,<font color="Olive" title="b4">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="Olive" title="b1">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>,<font color="Olive" title="b3">f2</font> &amp; <font color="Olive" title="b2">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b5">s</font>,<font color="Olive" title="b4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b5">s</font>,<font color="Olive" title="b4">I</font> )</span><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>; <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">f1</font>, <font color="Olive" title="b2">f2</font> being   <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>  st <font color="Olive" title="b1">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="Olive" title="b2">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> holds <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="c3">A</font><br/> for <font color="Olive" title="b4">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">s</font>,<font color="Olive" title="b3">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="Olive" title="b1">f1</font> holds <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">s</font>,<font color="Olive" title="b3">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="Olive" title="b2">f2</font> &amp; <font color="Olive" title="b1">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b4">s</font>,<font color="Olive" title="b3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b4">s</font>,<font color="Olive" title="b3">I</font> )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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> implies 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="c3">A</font><br/> for <font color="Olive" title="b2">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">s</font>,<font color="Olive" title="b1">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> holds <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">s</font>,<font color="Olive" title="b1">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> &amp; <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b2">s</font>,<font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b2">s</font>,<font color="Olive" title="b1">I</font> ) )</span><br/>







<b>assume </b><a NAME="E1:172"/><i><font color="Green" title="E106">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>
 ; <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="c3">A</font><br/> for <font color="Olive" title="b2">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">s</font>,<font color="Olive" title="b1">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> holds <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">s</font>,<font color="Olive" title="b1">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> &amp; <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b2">s</font>,<font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b2">s</font>,<font color="Olive" title="b1">I</font> )</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>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">A</font>] <b>means </b>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="c1">S</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">s</font>,$1</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> 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>,$1</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> &amp; <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">s</font>,$1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">s</font>,$1 );<br/>
<a NAME="E2:172"/><i><font color="Green" title="E107">A2</font></i>: 
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="c3">A</font>  st <font color="Olive" title="b1">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon" title="c3">A</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">I</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="172_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">I</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">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="c9">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon" title="c3">A</font> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">I</font>] )</span><br/>

<b>assume </b><a NAME="E1:172_1"/><i><font color="Green" title="E108">A3</font></i>: 
<font color="Maroon" title="c9">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon" title="c3">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">I</font>]</span><br/>

<b>let </b><font color="Maroon" title="c10">s</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>; <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="c10">s</font>,<font color="Maroon" title="c9">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> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c9">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> &amp; <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c9">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c9">I</font> ) )</span><br/>

<b>assume </b><a NAME="E2:172_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c9">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>
 ; <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="c10">s</font>,<font color="Maroon" title="c9">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> &amp; <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c9">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c9">I</font> )</span><br/>

<b>thus </b><a NAME="E3:172_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c9">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>
 <b>by </b><i><a class="txt" href="#E1:172_1"><i><font color="Green" title="E108">A3</font></i></a>, <a class="ref" href="aofa_000.html#T94" target="_self" title="AOFA_000:th.94">Th94</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c9">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c9">I</font></span><br/>

<a NAME="E4:172_1"/><i><font color="Green" title="E109">A4</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c9">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> <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>
 
<b>by </b><i><a class="txt" href="#E1:172_1"><i><font color="Green" title="E108">A3</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c9">I</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<span class="p3">(<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></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c9">I</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c5">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c9">I</font>
<b>by </b><i><a class="txt" href="#E1:172"><i><font color="Green" title="E106">A1</font></i></a>, <a class="txt" href="#E4:172_1"><i><font color="Green" title="E109">A4</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/> <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>
<a NAME="E3:172"/><i><font color="Green" title="E108">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="aofa_000.html#K12" title="AOFA_000:func.12">EmptyIns</a> <font color="Maroon" title="c3">A</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="172_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">s</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>; <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="c9">s</font>,<span class="p2">(<span class="default"><a href="aofa_000.html#K12" title="AOFA_000:func.12">EmptyIns</a> <font color="Maroon" title="c3">A</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="c3">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c4">f1</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">s</font>,<span class="p2">(<span class="default"><a href="aofa_000.html#K12" title="AOFA_000:func.12">EmptyIns</a> <font color="Maroon" title="c3">A</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="c3">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c5">f2</font> &amp; <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K12" title="AOFA_000:func.12">EmptyIns</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K12" title="AOFA_000:func.12">EmptyIns</a> <font color="Maroon" title="c3">A</font></span>)</span> ) )</span><br/>

<a NAME="E1:172_2"/>
<font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K12" title="AOFA_000:func.12">EmptyIns</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">s</font>
 
<b>by </b><i><a class="ref" href="aofa_000.html#D28" target="_self" title="AOFA_000:def.28">Def28</a></i>;<br/>
<b>hence </b><a NAME="E2:172_2"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">s</font>,<span class="p2">(<span class="default"><a href="aofa_000.html#K12" title="AOFA_000:func.12">EmptyIns</a> <font color="Maroon" title="c3">A</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="c3">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c4">f1</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">s</font>,<span class="p2">(<span class="default"><a href="aofa_000.html#K12" title="AOFA_000:func.12">EmptyIns</a> <font color="Maroon" title="c3">A</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="c3">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c5">f2</font> &amp; <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K12" title="AOFA_000:func.12">EmptyIns</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K12" title="AOFA_000:func.12">EmptyIns</a> <font color="Maroon" title="c3">A</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="aofa_000.html#T96" target="_self" title="AOFA_000:th.96">Th96</a>, <a class="ref" href="aofa_000.html#D28" target="_self" title="AOFA_000:def.28">Def28</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>
<a NAME="E4:172"/><i><font color="Green" title="E109">A6</font></i>: 
for <font color="Olive" title="b1">I1</font>, <font color="Olive" title="b2">I2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">A</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">I1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">I2</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Olive" title="b2">I2</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="172_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">I1</font>, <font color="Maroon" title="c10">I2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">I1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">I2</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c10">I2</font>] )</span><br/>

<b>assume </b><a NAME="E1:172_3"/><i><font color="Green" title="E110">A7</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">I1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">I2</font>] )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c10">I2</font>]</span><br/>

<b>let </b><font color="Maroon" title="c11">s</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>; <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="c11">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c10">I2</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="c3">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c4">f1</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c10">I2</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="c3">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c5">f2</font> &amp; <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c10">I2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c10">I2</font></span>)</span> ) )</span><br/>

<b>assume </b><a NAME="E2:172_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c10">I2</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="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 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="c11">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c10">I2</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="c3">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c5">f2</font> &amp; <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c10">I2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c10">I2</font></span>)</span> )</span><br/>

<a NAME="E3:172_3"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c9">I1</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> &amp; <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">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c9">I1</font></span>)</span>,<font color="Maroon" title="c10">I2</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> )
 
<b>by </b><i><a class="ref" href="aofa_000.html#T97" target="_self" title="AOFA_000:th.97">Th97</a></i>;<br/>
<a NAME="E4:172_3"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c9">I1</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> &amp; <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">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c9">I1</font></span>)</span>,<font color="Maroon" title="c10">I2</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> &amp; <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c9">I1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c9">I1</font> &amp; <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c9">I1</font></span>)</span>,<font color="Maroon" title="c10">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c9">I1</font></span>)</span>,<font color="Maroon" title="c10">I2</font> &amp; <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c9">I1</font></span>)</span>,<font color="Maroon" title="c10">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c10">I2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E1:172_3"><i><font color="Green" title="E110">A7</font></i></a>, <a class="ref" href="aofa_000.html#D29" target="_self" title="AOFA_000:def.29">Def29</a></i>;<br/>
<b>hence </b><a NAME="E5:172_3"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c10">I2</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="c3">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c5">f2</font> &amp; <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c10">I2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c10">I2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="aofa_000.html#D35" target="_self" title="AOFA_000:def.35">Def35</a>, <a class="ref" href="aofa_000.html#D29" target="_self" title="AOFA_000:def.29">Def29</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>
<a NAME="E5:172"/><i><font color="Green" title="E110">A8</font></i>: 
for <font color="Olive" title="b1">C</font>, <font color="Olive" title="b2">I1</font>, <font color="Olive" title="b3">I2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">A</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">C</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">I1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b3">I2</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Olive" title="b1">C</font>,<font color="Olive" title="b2">I1</font>,<font color="Olive" title="b3">I2</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="172_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">C</font>, <font color="Maroon" title="c10">I1</font>, <font color="Maroon" title="c11">I2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">C</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">I1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">I2</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Maroon" title="c9">C</font>,<font color="Maroon" title="c10">I1</font>,<font color="Maroon" title="c11">I2</font>] )</span><br/>

<b>assume </b><a NAME="E1:172_4"/><i><font color="Green" title="E111">A9</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">C</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">I1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">I2</font>] )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[ <a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Maroon" title="c9">C</font>,<font color="Maroon" title="c10">I1</font>,<font color="Maroon" title="c11">I2</font>]</span><br/>

<a NAME="E2:172_4"/><i><font color="Green" title="E112">A10</font></i>: 
( <font color="Maroon" title="c4">f1</font> <a href="aofa_000.html#R2" title="AOFA_000:pred.2">complies_with_if_wrt</a> <font color="Maroon" title="c2">T</font> &amp; <font color="Maroon" title="c5">f2</font> <a href="aofa_000.html#R2" title="AOFA_000:pred.2">complies_with_if_wrt</a> <font color="Maroon" title="c2">T</font> )
 
<b>by </b><i><a class="ref" href="aofa_000.html#D32" target="_self" title="AOFA_000:def.32">Def32</a></i>;<br/>
<b>set </b><font color="Maroon" title="c12">J</font> =  <a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Maroon" title="c9">C</font>,<font color="Maroon" title="c10">I1</font>,<font color="Maroon" title="c11">I2</font>;<br/>
<b>let </b><font color="Maroon" title="c13">s</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>; <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="c13">s</font>,<span class="p2">(<span class="default"><a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Maroon" title="c9">C</font>,<font color="Maroon" title="c10">I1</font>,<font color="Maroon" title="c11">I2</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="c3">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c4">f1</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">s</font>,<span class="p2">(<span class="default"><a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Maroon" title="c9">C</font>,<font color="Maroon" title="c10">I1</font>,<font color="Maroon" title="c11">I2</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="c3">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c5">f2</font> &amp; <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Maroon" title="c9">C</font>,<font color="Maroon" title="c10">I1</font>,<font color="Maroon" title="c11">I2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Maroon" title="c9">C</font>,<font color="Maroon" title="c10">I1</font>,<font color="Maroon" title="c11">I2</font></span>)</span> ) )</span><br/>

<b>assume </b><a NAME="E3:172_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">s</font>,<span class="p2">(<span class="default"><a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Maroon" title="c9">C</font>,<font color="Maroon" title="c10">I1</font>,<font color="Maroon" title="c11">I2</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="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 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="c13">s</font>,<span class="p2">(<span class="default"><a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Maroon" title="c9">C</font>,<font color="Maroon" title="c10">I1</font>,<font color="Maroon" title="c11">I2</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="c3">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c5">f2</font> &amp; <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Maroon" title="c9">C</font>,<font color="Maroon" title="c10">I1</font>,<font color="Maroon" title="c11">I2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Maroon" title="c9">C</font>,<font color="Maroon" title="c10">I1</font>,<font color="Maroon" title="c11">I2</font></span>)</span> )</span><br/>

<a NAME="E4:172_4"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">s</font>,<font color="Maroon" title="c9">C</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> &amp; ( <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">s</font>,<font color="Maroon" title="c9">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font> implies <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">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">s</font>,<font color="Maroon" title="c9">C</font></span>)</span>,<font color="Maroon" title="c10">I1</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> ) &amp; ( <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">s</font>,<font color="Maroon" title="c9">C</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c2">T</font> implies <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">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">s</font>,<font color="Maroon" title="c9">C</font></span>)</span>,<font color="Maroon" title="c11">I2</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> ) )
 
<b>by </b><i><a class="ref" href="aofa_000.html#T98" target="_self" title="AOFA_000:th.98">Th98</a></i>;<br/>
<a NAME="E5:172_4"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">s</font>,<font color="Maroon" title="c9">C</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> &amp; <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">s</font>,<font color="Maroon" title="c9">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">s</font>,<font color="Maroon" title="c9">C</font> &amp; ( <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">s</font>,<font color="Maroon" title="c9">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font> implies ( <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">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">s</font>,<font color="Maroon" title="c9">C</font></span>)</span>,<font color="Maroon" title="c10">I1</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> &amp; <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">s</font>,<font color="Maroon" title="c9">C</font></span>)</span>,<font color="Maroon" title="c10">I1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">s</font>,<font color="Maroon" title="c9">C</font></span>)</span>,<font color="Maroon" title="c10">I1</font> &amp; <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Maroon" title="c9">C</font>,<font color="Maroon" title="c10">I1</font>,<font color="Maroon" title="c11">I2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">s</font>,<font color="Maroon" title="c9">C</font></span>)</span>,<font color="Maroon" title="c10">I1</font> ) ) &amp; ( <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">s</font>,<font color="Maroon" title="c9">C</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c2">T</font> implies ( <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">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">s</font>,<font color="Maroon" title="c9">C</font></span>)</span>,<font color="Maroon" title="c11">I2</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> &amp; <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">s</font>,<font color="Maroon" title="c9">C</font></span>)</span>,<font color="Maroon" title="c11">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">s</font>,<font color="Maroon" title="c9">C</font></span>)</span>,<font color="Maroon" title="c11">I2</font> &amp; <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Maroon" title="c9">C</font>,<font color="Maroon" title="c10">I1</font>,<font color="Maroon" title="c11">I2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">s</font>,<font color="Maroon" title="c9">C</font></span>)</span>,<font color="Maroon" title="c11">I2</font> ) ) &amp; ( <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">s</font>,<font color="Maroon" title="c9">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font> or <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">s</font>,<font color="Maroon" title="c9">C</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c2">T</font> ) )
 
<b>by </b><i><a class="txt" href="#E1:172_4"><i><font color="Green" title="E111">A9</font></i></a>, <a class="txt" href="#E2:172_4"><i><font color="Green" title="E112">A10</font></i></a>, <a class="ref" href="aofa_000.html#D30" target="_self" title="AOFA_000:def.30">Def30</a></i>;<br/>
<b>hence </b><a NAME="E6:172_4"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">s</font>,<span class="p2">(<span class="default"><a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Maroon" title="c9">C</font>,<font color="Maroon" title="c10">I1</font>,<font color="Maroon" title="c11">I2</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="c3">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c5">f2</font> &amp; <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Maroon" title="c9">C</font>,<font color="Maroon" title="c10">I1</font>,<font color="Maroon" title="c11">I2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Maroon" title="c9">C</font>,<font color="Maroon" title="c10">I1</font>,<font color="Maroon" title="c11">I2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="aofa_000.html#D35" target="_self" title="AOFA_000:def.35">Def35</a>, <a class="txt" href="#E2:172_4"><i><font color="Green" title="E112">A10</font></i></a>, <a class="ref" href="aofa_000.html#D30" target="_self" title="AOFA_000:def.30">Def30</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>
<a NAME="E6:172"/><i><font color="Green" title="E111">A11</font></i>: 
for <font color="Olive" title="b1">C</font>, <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="c3">A</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">C</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">I</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Olive" title="b1">C</font>,<font color="Olive" title="b2">I</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="172_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">C</font>, <font color="Maroon" title="c10">I</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">C</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">I</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Maroon" title="c9">C</font>,<font color="Maroon" title="c10">I</font>] )</span><br/>

<b>assume </b><a NAME="E1:172_5"/><i><font color="Green" title="E112">A12</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">C</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">I</font>] )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[ <a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Maroon" title="c9">C</font>,<font color="Maroon" title="c10">I</font>]</span><br/>

<b>set </b><font color="Maroon" title="c11">J</font> =  <a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Maroon" title="c9">C</font>,<font color="Maroon" title="c10">I</font>;<br/>
<b>let </b><font color="Maroon" title="c12">s</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>; <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="c12">s</font>,<span class="p2">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Maroon" title="c9">C</font>,<font color="Maroon" title="c10">I</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="c3">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c4">f1</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">s</font>,<span class="p2">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Maroon" title="c9">C</font>,<font color="Maroon" title="c10">I</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="c3">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c5">f2</font> &amp; <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c12">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Maroon" title="c9">C</font>,<font color="Maroon" title="c10">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c12">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Maroon" title="c9">C</font>,<font color="Maroon" title="c10">I</font></span>)</span> ) )</span><br/>

<b>assume </b><a NAME="E2:172_5"/><i><font color="Green" title="E113">A13</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">s</font>,<span class="p2">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Maroon" title="c9">C</font>,<font color="Maroon" title="c10">I</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="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 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="c12">s</font>,<span class="p2">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Maroon" title="c9">C</font>,<font color="Maroon" title="c10">I</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="c3">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c5">f2</font> &amp; <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c12">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Maroon" title="c9">C</font>,<font color="Maroon" title="c10">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c12">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Maroon" title="c9">C</font>,<font color="Maroon" title="c10">I</font></span>)</span> )</span><br/>

<a NAME="E3:172_5"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">s</font>,<font color="Maroon" title="c9">C</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>
 
<b>by </b><i><a class="ref" href="aofa_000.html#T99" target="_self" title="AOFA_000:th.99">Th99</a></i>;<br/>
<a NAME="E4:172_5"/><b>then </b><i><font color="Green" title="E114">A14</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">s</font>,<font color="Maroon" title="c9">C</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> &amp; <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c12">s</font>,<font color="Maroon" title="c9">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c12">s</font>,<font color="Maroon" title="c9">C</font> )
 
<b>by </b><i><a class="txt" href="#E1:172_5"><i><font color="Green" title="E112">A12</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">r</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E6:172_5"/><i><font color="Green" title="E115">A15</font></i>: 
( <font color="Maroon" title="c13">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c12">s</font>,<font color="Maroon" title="c9">C</font> &amp; <font color="Maroon" title="c13">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c2">T</font> )
 <b>and </b><br/><a NAME="E7:172_5"/><i><font color="Green" title="E116">A16</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">r</font> holds <br/>( <font color="Maroon" title="c13">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font> &amp; <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="c13">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c9">C</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="c3">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c4">f1</font> &amp; <font color="Maroon" title="c13">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c9">C</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E2:172_5"><i><font color="Green" title="E113">A13</font></i></a>, <a class="ref" href="aofa_000.html#T99" target="_self" title="AOFA_000:th.99">Th99</a></i>;<br/>
<a NAME="E8:172_5"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">r</font> holds <br/>( <font color="Maroon" title="c13">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font> &amp; <font color="Maroon" title="c13">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c9">C</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E7:172_5"><i><font color="Green" title="E116">A16</font></i></a></i>;<br/>
<a NAME="E9:172_5"/><b>then </b><i><font color="Green" title="E117">A17</font></i>: 
<font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c12">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Maroon" title="c9">C</font>,<font color="Maroon" title="c10">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E6:172_5"><i><font color="Green" title="E115">A15</font></i></a>, <a class="ref" href="aofa_000.html#T86" target="_self" title="AOFA_000:th.86">Th86</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 &amp; $1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">r</font> implies ( <font color="Maroon" title="c13">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font> &amp; <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="c13">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c9">C</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="c3">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c5">f2</font> &amp; <font color="Maroon" title="c13">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c9">C</font></span>)</span> ) );<br/>
<a NAME="E10:172_5"/><i><font color="Green" title="E118">A18</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="172_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c14">i</font>]</span><br/>

<b>assume </b><a NAME="E1:172_5_1"/><i><font color="Green" title="E119">A19</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">r</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font> &amp; <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="c13">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c9">C</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="c3">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c5">f2</font> &amp; <font color="Maroon" title="c13">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c9">C</font></span>)</span> )</span><br/>

<b>hence </b><a NAME="E2:172_5_1"/><i><font color="Green" title="E120">A20</font></i>: 
<font color="Maroon" title="c13">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="#E7:172_5"><i><font color="Green" title="E116">A16</font></i></a></i>; <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"><span class="p2">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c9">C</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="c3">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c5">f2</font> &amp; <font color="Maroon" title="c13">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c9">C</font></span>)</span> )</span><br/>

<b>then reconsider </b><font color="Maroon" title="c15">ri</font> = <font color="Maroon" title="c13">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<a NAME="E4:172_5_1"/><i><font color="Green" title="E121">A21</font></i>: 
<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="c13">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c9">C</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="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="#E1:172_5_1"><i><font color="Green" title="E119">A19</font></i></a>, <a class="txt" href="#E7:172_5"><i><font color="Green" title="E116">A16</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:172_5_1"/>
<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="c13">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c9">C</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="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="#E1:172_5"><i><font color="Green" title="E112">A12</font></i></a>, <a class="txt" href="#E2:172_5_1"><i><font color="Green" title="E120">A20</font></i></a>, <a class="txt" href="#E4:172"><i><font color="Green" title="E109">A6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c9">C</font></span>)</span></span><br/>

<a NAME="E6:172_5_1"/><i><font color="Green" title="E122">A22</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">ri</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> &amp; <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">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c15">ri</font>,<font color="Maroon" title="c10">I</font></span>)</span>,<font color="Maroon" title="c9">C</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> )
 
<b>by </b><i><a class="txt" href="#E4:172_5_1"><i><font color="Green" title="E121">A21</font></i></a>, <a class="ref" href="aofa_000.html#T97" target="_self" title="AOFA_000:th.97">Th97</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c13">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c9">C</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:172_5_1"><i><font color="Green" title="E119">A19</font></i></a>, <a class="txt" href="#E7:172_5"><i><font color="Green" title="E116">A16</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c15">ri</font>,<font color="Maroon" title="c10">I</font></span>)</span>,<font color="Maroon" title="c9">C</font>
<b>by </b><i><a class="ref" href="aofa_000.html#D29" target="_self" title="AOFA_000:def.29">Def29</a></i>
<br/>.= 
<font color="Maroon" title="c5">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c15">ri</font>,<font color="Maroon" title="c10">I</font></span>)</span>,<font color="Maroon" title="c9">C</font>
<b>by </b><i><a class="txt" href="#E1:172_5"><i><font color="Green" title="E112">A12</font></i></a>, <a class="txt" href="#E6:172_5_1"><i><font color="Green" title="E122">A22</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c15">ri</font>,<font color="Maroon" title="c10">I</font></span>)</span>,<font color="Maroon" title="c9">C</font>
<b>by </b><i><a class="txt" href="#E1:172_5"><i><font color="Green" title="E112">A12</font></i></a>, <a class="txt" href="#E6:172_5_1"><i><font color="Green" title="E122">A22</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">f2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c9">C</font></span>)</span>
<b>by </b><i><a class="ref" href="aofa_000.html#D29" target="_self" title="AOFA_000:def.29">Def29</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>
<a NAME="E11:172_5"/><b>then </b>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">r</font> holds <br/>( <font color="Maroon" title="c13">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font> &amp; <font color="Maroon" title="c13">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c9">C</font></span>)</span> )
 
;<br/>
<b>hence </b><a NAME="E12:172_5"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">s</font>,<span class="p2">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Maroon" title="c9">C</font>,<font color="Maroon" title="c10">I</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="c3">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c5">f2</font> &amp; <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c12">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Maroon" title="c9">C</font>,<font color="Maroon" title="c10">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c12">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Maroon" title="c9">C</font>,<font color="Maroon" title="c10">I</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E4:172_5"><i><font color="Green" title="E114">A14</font></i></a>, <a class="txt" href="#E6:172_5"><i><font color="Green" title="E115">A15</font></i></a>, <a class="txt" href="#E9:172_5"><i><font color="Green" title="E117">A17</font></i></a>, <a class="txt" href="#E10:172_5"><i><font color="Green" title="E118">A18</font></i></a>, <a class="ref" href="aofa_000.html#T86" target="_self" title="AOFA_000:th.86">Th86</a>, <a class="ref" href="aofa_000.html#D35" target="_self" title="AOFA_000:def.35">Def35</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>let </b><font color="Maroon" title="c9">I</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">A</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">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <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="c9">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> 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="c9">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> &amp; <font color="Maroon" title="c4">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">s</font>,<font color="Maroon" title="c9">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">s</font>,<font color="Maroon" title="c9">I</font> )</span><br/>

<b>thus </b><a NAME="E7:172"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">I</font>]
 <b>from </b><i><a class="ref" href="aofa_000.html#S2" target="_self" title="AOFA_000:sch.2">AOFA_000:sch 2</a>(<a class="txt" href="#E2:172"><i><font color="Green" title="E107">A2</font></i></a>, <a class="txt" href="#E3:172"><i><font color="Green" title="E108">A5</font></i></a>, <a class="txt" href="#E4:172"><i><font color="Green" title="E109">A6</font></i></a>, <a class="txt" href="#E5:172"><i><font color="Green" title="E110">A8</font></i></a>, <a class="txt" href="#E6:172"><i><font color="Green" title="E111">A11</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>
