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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">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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>,<font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b4">s0</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>  ex <font color="Olive" title="b5">f</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 <br/>( <font color="Olive" title="b5">f</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> &amp; ( 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><br/> for <font color="Olive" title="b7">C</font>, <font color="Olive" title="b8">I</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">A</font>  st not <font color="Olive" title="b5">f</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <font color="Olive" title="b8">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Olive" title="b7">C</font>,<font color="Olive" title="b5">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b6">s</font>,<font color="Olive" title="b7">C</font> holds <br/><font color="Olive" title="b5">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b6">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Olive" title="b7">C</font>,<font color="Olive" title="b8">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">s0</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">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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>,<font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b3">s0</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>  ex <font color="Olive" title="b4">f</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 <br/>( <font color="Olive" title="b4">f</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> &amp; ( 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><br/> for <font color="Olive" title="b6">C</font>, <font color="Olive" title="b7">I</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">A</font>  st not <font color="Olive" title="b4">f</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <font color="Olive" title="b7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Olive" title="b6">C</font>,<font color="Olive" title="b4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b5">s</font>,<font color="Olive" title="b6">C</font> holds <br/><font color="Olive" title="b4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b5">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Olive" title="b6">C</font>,<font color="Olive" title="b7">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">s0</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">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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>,<font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">s0</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>  ex <font color="Olive" title="b3">f</font> being   <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> st <br/>( <font color="Olive" title="b3">f</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> &amp; ( 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><br/> for <font color="Olive" title="b5">C</font>, <font color="Olive" title="b6">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 not <font color="Olive" title="b3">f</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <font color="Olive" title="b6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Olive" title="b5">C</font>,<font color="Olive" title="b3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b4">s</font>,<font color="Olive" title="b5">C</font> holds <br/><font color="Olive" title="b3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b4">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Olive" title="b5">C</font>,<font color="Olive" title="b6">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">s0</font> ) )</span><br/><b>let </b><font color="Maroon" title="c4">g</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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>,<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">s0</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>  ex <font color="Olive" title="b2">f</font> being   <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> st <br/>( <font color="Olive" title="b2">f</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> &amp; ( for <font color="Olive" title="b3">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b4">C</font>, <font color="Olive" title="b5">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 not <font color="Olive" title="b2">f</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <font color="Olive" title="b5">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Olive" title="b4">C</font>,<font color="Olive" title="b2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b3">s</font>,<font color="Olive" title="b4">C</font> holds <br/><font color="Olive" title="b2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b3">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Olive" title="b4">C</font>,<font color="Olive" title="b5">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s0</font> ) )</span><br/><b>let </b><font color="Maroon" title="c5">s0</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">  ex <font color="Olive" title="b1">f</font> being   <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> st <br/>( <font color="Olive" title="b1">f</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> &amp; ( 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><br/> for <font color="Olive" title="b3">C</font>, <font color="Olive" title="b4">I</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">A</font>  st not <font color="Olive" title="b1">f</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <font color="Olive" title="b4">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Olive" title="b3">C</font>,<font color="Olive" title="b1">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b2">s</font>,<font color="Olive" title="b3">C</font> holds <br/><font color="Olive" title="b1">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b2">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Olive" title="b3">C</font>,<font color="Olive" title="b4">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s0</font> ) )</span><br/>









<b>reconsider </b><font color="Maroon" title="c6">Ss0</font> = <font color="Maroon" title="c1">S</font> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <font color="Maroon" title="c5">s0</font> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font> ;<br/>
<b>set </b><font color="Maroon" title="c7">Z</font> =  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font>;<br/>
<b>deffunc </b><font color="Maroon">H<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>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">(<span class="default"><a href="funct_5.html#K3" title="FUNCT_5:func.3">curry'</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<a NAME="E2:152"/><i><font color="Green" title="E95">A1</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">H<sub>1</sub></font>(<font color="Olive" title="b1">I</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="152_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">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="c8">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">H<sub>1</sub></font>(<font color="Maroon" title="c8">I</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font> )</span><br/>

<b>assume </b><a NAME="E1:152_1"/><i><font color="Green" title="E96">A2</font></i>: 
<font color="Maroon" title="c8">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">H<sub>1</sub></font>(<font color="Maroon" title="c8">I</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c9">B</font> =  <a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon" title="c3">A</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c10">I</font> = <font color="Maroon" title="c8">I</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">B</font> <b>by </b><i><a class="txt" href="aofa_000.html#E1:152_1"><i><font color="Green" title="E96">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">g</font> = <font color="Maroon" title="c4">g</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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>,<font color="Maroon" title="c9">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c1">S</font> ;<br/>
<a NAME="E5:152_1"/>
<span class="p1">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c11">g</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c10">I</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font>
 
;<br/>
<b>hence </b><a NAME="E6:152_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">I</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">S</font>,<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"> verum</span><br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c8">idS</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">S</font> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font> <b>by </b><i><a class="ref" href="altcat_1.html#T2" title="ALTCAT_1:th.2">ALTCAT_1:2</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font>,   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font>) <b>-&gt; </b>   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font> = $2 <a href="aofa_000.html#K4" title="AOFA_000:func.4">*</a> $1;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font>,   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font>,   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font>) <b>-&gt; </b>   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font> = <span class="p1">(<span class="default">$2,<font color="Maroon" title="c2">T</font> <a href="aofa_000.html#K3" title="AOFA_000:func.3">+*</a> $3</span>)</span> <a href="aofa_000.html#K4" title="AOFA_000:func.4">*</a> $1;<br/>
<b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font>,   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font>) <b>-&gt; </b>   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">Ss0</font> <a href="aofa_000.html#K8" title="AOFA_000:func.8">iter</a> <span class="p2">(<span class="default">$1 <a href="aofa_000.html#K4" title="AOFA_000:func.4">*</a> $2</span>)</span></span>)</span> <a href="aofa_000.html#K4" title="AOFA_000:func.4">*</a> $1;<br/>
<b>consider </b><font color="Maroon" title="c9">h</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A</font>, <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E5:152"/><i><font color="Green" title="E96">A3</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" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">I</font>)
 <b>and </b><br/><a NAME="E6:152"/><i><font color="Green" title="E97">A4</font></i>: 
<font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <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="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">idS</font>
 <b>and </b><br/><a NAME="E7:152"/><i><font color="Green" title="E98">A5</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> holds  <font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <span class="p1">(<span class="default"><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></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Olive" title="b1">I1</font>,<font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Olive" title="b2">I2</font>)
 <b>and </b><br/><a NAME="E8:152"/><i><font color="Green" title="E99">A6</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> holds  <font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <span class="p1">(<span class="default"><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></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Olive" title="b1">C</font>,<font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Olive" title="b2">I1</font>,<font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Olive" title="b3">I2</font>)
 <b>and </b><br/><a NAME="E9:152"/><i><font color="Green" title="E100">A7</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> holds  <font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <span class="p1">(<span class="default"><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></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Olive" title="b1">C</font>,<font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Olive" title="b2">I</font>)
 <b>from </b><i><a class="ref" href="aofa_000.html#S7" target="_self" title="AOFA_000:sch.7">AOFA_000:sch 7</a>(<a class="txt" href="aofa_000.html#E2:152"><i><font color="Green" title="E95">A1</font></i></a>);<br/></i>
<a NAME="E10:152"/>
<font color="Maroon" title="c9">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A</font>,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<a NAME="E11:152"/><b>then </b>
 <a href="funct_5.html#K4" title="FUNCT_5:func.4">uncurry'</a> <font color="Maroon" title="c9">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</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>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c1">S</font>
 
<b>by </b><i><a class="ref" href="funct_6.html#T20" title="FUNCT_6:th.20">FUNCT_6:20</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">f</font> =  <a href="funct_5.html#K4" title="FUNCT_5:func.4">uncurry'</a> <font color="Maroon" title="c9">h</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c1">S</font> <b>by </b><i><a class="ref" href="funct_2.html#T121" title="FUNCT_2:th.121">FUNCT_2:121</a></i>;<br/>
<a NAME="E13:152"/><i><font color="Green" title="E101">A8</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">idS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E14:152"/><i><font color="Green" title="E102">A9</font></i>: 
<font color="Maroon" title="c10">f</font> is <a href="aofa_000.html#V13" title="AOFA_000:attr.13">complying_with_empty-instruction</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="152_2"><b>proof </b></a><div class="add">

<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>; <i><font color="Red">:: according to </font></i><a class="ref" href="aofa_000.html#D28" target="_self" title="AOFA_000:def.28">AOFA_000:def 28</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">f</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"><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="c11">s</font></span><br/>

<a NAME="E1:152_2"/>
<font color="Maroon" title="c8">idS</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/>
<b>hence </b><a NAME="E2:152_2"/>
<font color="Maroon" title="c10">f</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"><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="c11">s</font>
 <b>by </b><i><a class="txt" href="aofa_000.html#E6:152"><i><font color="Green" title="E97">A4</font></i></a>, <a class="txt" href="aofa_000.html#E13:152"><i><font color="Green" title="E101">A8</font></i></a>, <a class="ref" href="funct_5.html#T46" title="FUNCT_5:th.46">FUNCT_5:46</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="E15:152"/><i><font color="Green" title="E103">A10</font></i>: 
<font color="Maroon" title="c10">f</font> is <a href="aofa_000.html#V14" title="AOFA_000:attr.14">complying_with_catenation</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="152_3"><b>proof </b></a><div class="add">

<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>; <i><font color="Red">:: according to </font></i><a class="ref" href="aofa_000.html#D29" target="_self" title="AOFA_000:def.29">AOFA_000:def 29</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">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> holds  <font color="Maroon" title="c10">f</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="Olive" title="b1">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Olive" title="b2">I2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Olive" title="b1">I1</font></span>)</span>,<font color="Olive" title="b2">I2</font></span><br/><b>let </b><font color="Maroon" title="c12">I1</font>, <font color="Maroon" title="c13">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" title="c10">f</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="c12">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c13">I2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">I1</font></span>)</span>,<font color="Maroon" title="c13">I2</font></span><br/>



<a NAME="E1:152_3"/><i><font color="Green" title="E104">A11</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c13">I2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c12">I1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c13">I2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c10">f</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="c12">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c13">I2</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c13">I2</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font>
<b>by </b><i><a class="txt" href="aofa_000.html#E13:152"><i><font color="Green" title="E101">A8</font></i></a>, <a class="ref" href="funct_5.html#T46" title="FUNCT_5:th.46">FUNCT_5:46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c13">I2</font></span>)</span> <a href="aofa_000.html#K4" title="AOFA_000:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c12">I1</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font>
<b>by </b><i><a class="txt" href="aofa_000.html#E7:152"><i><font color="Green" title="E98">A5</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c13">I2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c12">I1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c13">I2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">I1</font></span>)</span>
<b>by </b><i><a class="txt" href="aofa_000.html#E13:152"><i><font color="Green" title="E101">A8</font></i></a>, <a class="txt" href="aofa_000.html#E1:152_3"><i><font color="Green" title="E104">A11</font></i></a>, <a class="ref" href="funct_5.html#T46" title="FUNCT_5:th.46">FUNCT_5:46</a></i>
<br/>.= 
<font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">I1</font></span>)</span>,<font color="Maroon" title="c13">I2</font>
<b>by </b><i><a class="txt" href="aofa_000.html#E13:152"><i><font color="Green" title="E101">A8</font></i></a>, <a class="txt" href="aofa_000.html#E1:152_3"><i><font color="Green" title="E104">A11</font></i></a>, <a class="ref" href="funct_5.html#T46" title="FUNCT_5:th.46">FUNCT_5:46</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="E16:152"/><i><font color="Green" title="E104">A12</font></i>: 
<font color="Maroon" title="c10">f</font> <a href="aofa_000.html#R2" title="AOFA_000:pred.2">complies_with_if_wrt</a> <font color="Maroon" title="c2">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="152_4"><b>proof </b></a><div class="add">

<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>; <i><font color="Red">:: according to </font></i><a class="ref" href="aofa_000.html#D30" target="_self" title="AOFA_000:def.30">AOFA_000:def 30</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">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> holds <br/> ( ( <font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font> implies <font color="Maroon" title="c10">f</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"><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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Olive" title="b1">C</font></span>)</span>,<font color="Olive" title="b2">I1</font> ) &amp; ( <font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Olive" title="b1">C</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c2">T</font> implies <font color="Maroon" title="c10">f</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"><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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Olive" title="b1">C</font></span>)</span>,<font color="Olive" title="b3">I2</font> ) )</span><br/><b>let </b><font color="Maroon" title="c12">C</font>, <font color="Maroon" title="c13">I1</font>, <font color="Maroon" title="c14">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" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font> implies <font color="Maroon" title="c10">f</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"><a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Maroon" title="c12">C</font>,<font color="Maroon" title="c13">I1</font>,<font color="Maroon" title="c14">I2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">C</font></span>)</span>,<font color="Maroon" title="c13">I1</font> ) &amp; ( <font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">C</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c2">T</font> implies <font color="Maroon" title="c10">f</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"><a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Maroon" title="c12">C</font>,<font color="Maroon" title="c13">I1</font>,<font color="Maroon" title="c14">I2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">C</font></span>)</span>,<font color="Maroon" title="c14">I2</font> ) )</span><br/>



<a NAME="E1:152_4"/><i><font color="Green" title="E105">A13</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <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="c12">C</font>,<font color="Maroon" title="c13">I1</font>,<font color="Maroon" title="c14">I2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c12">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c13">I1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c14">I2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><i><font color="Green" title="E106">A14</font></i>: <font color="Maroon" title="c10">f</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"><a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Maroon" title="c12">C</font>,<font color="Maroon" title="c13">I1</font>,<font color="Maroon" title="c14">I2</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <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="c12">C</font>,<font color="Maroon" title="c13">I1</font>,<font color="Maroon" title="c14">I2</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font>
<b>by </b><i><a class="txt" href="aofa_000.html#E13:152"><i><font color="Green" title="E101">A8</font></i></a>, <a class="ref" href="funct_5.html#T46" title="FUNCT_5:th.46">FUNCT_5:46</a></i>
<br/>.= 
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c12">C</font>,<font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c13">I1</font>,<font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c14">I2</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font>
<b>by </b><i><a class="txt" href="aofa_000.html#E8:152"><i><font color="Green" title="E99">A6</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c13">I1</font></span>)</span>,<font color="Maroon" title="c2">T</font> <a href="aofa_000.html#K3" title="AOFA_000:func.3">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c14">I2</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c12">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
;<br/>
<a NAME="E3:152_4"/><i><font color="Green" title="E107">A15</font></i>: 
<font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c12">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font>
 
<b>by </b><i><a class="txt" href="aofa_000.html#E13:152"><i><font color="Green" title="E101">A8</font></i></a>, <a class="txt" href="aofa_000.html#E1:152_4"><i><font color="Green" title="E105">A13</font></i></a>, <a class="ref" href="funct_5.html#T46" title="FUNCT_5:th.46">FUNCT_5:46</a></i>;<br/>
<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">C</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c2">T</font> implies <font color="Maroon" title="c10">f</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"><a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Maroon" title="c12">C</font>,<font color="Maroon" title="c13">I1</font>,<font color="Maroon" title="c14">I2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">C</font></span>)</span>,<font color="Maroon" title="c14">I2</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:152_4_2"/>
<font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="c10">f</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"><a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Maroon" title="c12">C</font>,<font color="Maroon" title="c13">I1</font>,<font color="Maroon" title="c14">I2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">C</font></span>)</span>,<font color="Maroon" title="c13">I1</font></span><br/><b>hence </b><font color="Maroon" title="c10">f</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"><a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Maroon" title="c12">C</font>,<font color="Maroon" title="c13">I1</font>,<font color="Maroon" title="c14">I2</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c13">I1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">C</font></span>)</span>
<b>by </b><i><a class="txt" href="aofa_000.html#E1:152_4"><i><font color="Green" title="E105">A13</font></i></a>, <a class="txt" href="aofa_000.html#E2:152_4"><i><font color="Green" title="E106">A14</font></i></a>, <a class="txt" href="aofa_000.html#E3:152_4"><i><font color="Green" title="E107">A15</font></i></a>, <a class="ref" href="aofa_000.html#T4" target="_self" title="AOFA_000:th.4">Th4</a></i>
<br/>.= 
<font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">C</font></span>)</span>,<font color="Maroon" title="c13">I1</font>
<b>by </b><i><a class="txt" href="aofa_000.html#E13:152"><i><font color="Green" title="E101">A8</font></i></a>, <a class="txt" href="aofa_000.html#E1:152_4"><i><font color="Green" title="E105">A13</font></i></a>, <a class="ref" href="funct_5.html#T46" title="FUNCT_5:th.46">FUNCT_5:46</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>

<b>assume </b><a NAME="E5:152_4"/>
<font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">C</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <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="c10">f</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"><a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Maroon" title="c12">C</font>,<font color="Maroon" title="c13">I1</font>,<font color="Maroon" title="c14">I2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">C</font></span>)</span>,<font color="Maroon" title="c14">I2</font></span><br/>

<b>hence </b><font color="Maroon" title="c10">f</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"><a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Maroon" title="c12">C</font>,<font color="Maroon" title="c13">I1</font>,<font color="Maroon" title="c14">I2</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c14">I2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">C</font></span>)</span>
<b>by </b><i><a class="txt" href="aofa_000.html#E1:152_4"><i><font color="Green" title="E105">A13</font></i></a>, <a class="txt" href="aofa_000.html#E2:152_4"><i><font color="Green" title="E106">A14</font></i></a>, <a class="txt" href="aofa_000.html#E3:152_4"><i><font color="Green" title="E107">A15</font></i></a>, <a class="ref" href="aofa_000.html#T5" target="_self" title="AOFA_000:th.5">Th5</a></i>
<br/>.= 
<font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">C</font></span>)</span>,<font color="Maroon" title="c14">I2</font>
<b>by </b><i><a class="txt" href="aofa_000.html#E13:152"><i><font color="Green" title="E101">A8</font></i></a>, <a class="txt" href="aofa_000.html#E1:152_4"><i><font color="Green" title="E105">A13</font></i></a>, <a class="ref" href="funct_5.html#T46" title="FUNCT_5:th.46">FUNCT_5:46</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="E17:152"/>
<font color="Maroon" title="c10">f</font> <a href="aofa_000.html#R3" title="AOFA_000:pred.3">complies_with_while_wrt</a> <font color="Maroon" title="c2">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="152_5"><b>proof </b></a><div class="add">

<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>; <i><font color="Red">:: according to </font></i><a class="ref" href="aofa_000.html#D31" target="_self" title="AOFA_000:def.31">AOFA_000:def 31</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">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> holds <br/> ( ( <font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font> implies <font color="Maroon" title="c10">f</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"><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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Olive" title="b1">C</font></span>)</span>,<font color="Olive" title="b2">I</font></span>)</span>,<span class="p1">(<span class="default"><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></span>)</span> ) &amp; ( <font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Olive" title="b1">C</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c2">T</font> implies <font color="Maroon" title="c10">f</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"><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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Olive" title="b1">C</font> ) )</span><br/><b>let </b><font color="Maroon" title="c12">C</font>, <font color="Maroon" title="c13">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="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font> implies <font color="Maroon" title="c10">f</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"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Maroon" title="c12">C</font>,<font color="Maroon" title="c13">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">C</font></span>)</span>,<font color="Maroon" title="c13">I</font></span>)</span>,<span class="p1">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Maroon" title="c12">C</font>,<font color="Maroon" title="c13">I</font></span>)</span> ) &amp; ( <font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">C</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c2">T</font> implies <font color="Maroon" title="c10">f</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"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Maroon" title="c12">C</font>,<font color="Maroon" title="c13">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">C</font> ) )</span><br/>



<a NAME="E1:152_5"/><i><font color="Green" title="E105">A16</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <span class="p2">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Maroon" title="c12">C</font>,<font color="Maroon" title="c13">I</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c12">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c13">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c12">C</font></span>)</span> <a href="aofa_000.html#K4" title="AOFA_000:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c13">I</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><i><font color="Green" title="E106">A17</font></i>: <font color="Maroon" title="c10">f</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"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Maroon" title="c12">C</font>,<font color="Maroon" title="c13">I</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <span class="p2">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Maroon" title="c12">C</font>,<font color="Maroon" title="c13">I</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font>
<b>by </b><i><a class="txt" href="aofa_000.html#E13:152"><i><font color="Green" title="E101">A8</font></i></a>, <a class="ref" href="funct_5.html#T46" title="FUNCT_5:th.46">FUNCT_5:46</a></i>
<br/>.= 
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c12">C</font>,<font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c13">I</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font>
<b>by </b><i><a class="txt" href="aofa_000.html#E9:152"><i><font color="Green" title="E100">A7</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">Ss0</font> <a href="aofa_000.html#K8" title="AOFA_000:func.8">iter</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c12">C</font></span>)</span> <a href="aofa_000.html#K4" title="AOFA_000:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c13">I</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c12">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
;<br/>
<a NAME="E3:152_5"/><i><font color="Green" title="E107">A18</font></i>: 
<font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c12">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font>
 
<b>by </b><i><a class="txt" href="aofa_000.html#E13:152"><i><font color="Green" title="E101">A8</font></i></a>, <a class="txt" href="aofa_000.html#E1:152_5"><i><font color="Green" title="E105">A16</font></i></a>, <a class="ref" href="funct_5.html#T46" title="FUNCT_5:th.46">FUNCT_5:46</a></i>;<br/>
<a NAME="E4:152_5"/><i><font color="Green" title="E108">A19</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c12">C</font></span>)</span> <a href="aofa_000.html#K4" title="AOFA_000:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c13">I</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">S</font>
 
;<br/>
<div><i><font color="Green" title="E109">B20</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="152_5_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">z</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">(<span class="default"><font color="Maroon" title="c6">Ss0</font> <a href="aofa_000.html#K4" title="AOFA_000:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c12">C</font></span>)</span> <a href="aofa_000.html#K4" title="AOFA_000:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c13">I</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Ss0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">z</font></span><br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c6">Ss0</font> <a href="aofa_000.html#K4" title="AOFA_000:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c12">C</font></span>)</span> <a href="aofa_000.html#K4" title="AOFA_000:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c13">I</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">z</font> = 
<font color="Maroon" title="c6">Ss0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c12">C</font></span>)</span> <a href="aofa_000.html#K4" title="AOFA_000:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c13">I</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">z</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon" title="c5">s0</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<font color="Maroon" title="c6">Ss0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">z</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</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>
<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">C</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c2">T</font> implies <font color="Maroon" title="c10">f</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"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Maroon" title="c12">C</font>,<font color="Maroon" title="c13">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">C</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:152_5_3"/>
<font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="c10">f</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"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Maroon" title="c12">C</font>,<font color="Maroon" title="c13">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">C</font></span>)</span>,<font color="Maroon" title="c13">I</font></span>)</span>,<span class="p1">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Maroon" title="c12">C</font>,<font color="Maroon" title="c13">I</font></span>)</span></span><br/><b>hence </b><font color="Maroon" title="c10">f</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"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Maroon" title="c12">C</font>,<font color="Maroon" title="c13">I</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">Ss0</font> <a href="aofa_000.html#K8" title="AOFA_000:func.8">iter</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c12">C</font></span>)</span> <a href="aofa_000.html#K4" title="AOFA_000:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c13">I</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c12">C</font></span>)</span> <a href="aofa_000.html#K4" title="AOFA_000:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c13">I</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">C</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="aofa_000.html#E1:152_5"><i><font color="Green" title="E105">A16</font></i></a>, <a class="txt" href="aofa_000.html#E2:152_5"><i><font color="Green" title="E106">A17</font></i></a>, <a class="txt" href="aofa_000.html#E3:152_5"><i><font color="Green" title="E107">A18</font></i></a>, <a class="txt" href="aofa_000.html#E4:152_5"><i><font color="Green" title="E108">A19</font></i></a>, <a class="txt" href="aofa_000.html#E5:152_5"><i><font color="Green" title="E109">B20</font></i></a>, <a class="ref" href="aofa_000.html#T11" target="_self" title="AOFA_000:th.11">Th11</a>, <a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">Ss0</font> <a href="aofa_000.html#K8" title="AOFA_000:func.8">iter</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c12">C</font></span>)</span> <a href="aofa_000.html#K4" title="AOFA_000:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c13">I</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c12">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c13">I</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">C</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">Ss0</font> <a href="aofa_000.html#K8" title="AOFA_000:func.8">iter</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c12">C</font></span>)</span> <a href="aofa_000.html#K4" title="AOFA_000:func.4">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c13">I</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K4" title="AOFA_000:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c12">C</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c13">I</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">C</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">Ss0</font> <a href="aofa_000.html#K8" title="AOFA_000:func.8">iter</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c12">C</font></span>)</span> <a href="aofa_000.html#K4" title="AOFA_000:func.4">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c13">I</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K4" title="AOFA_000:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c12">C</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">C</font></span>)</span>,<font color="Maroon" title="c13">I</font></span>)</span>
<b>by </b><i><a class="txt" href="aofa_000.html#E13:152"><i><font color="Green" title="E101">A8</font></i></a>, <a class="txt" href="aofa_000.html#E1:152_5"><i><font color="Green" title="E105">A16</font></i></a>, <a class="ref" href="funct_5.html#T46" title="FUNCT_5:th.46">FUNCT_5:46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <span class="p2">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Maroon" title="c12">C</font>,<font color="Maroon" title="c13">I</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">C</font></span>)</span>,<font color="Maroon" title="c13">I</font></span>)</span>
<b>by </b><i><a class="txt" href="aofa_000.html#E9:152"><i><font color="Green" title="E100">A7</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">C</font></span>)</span>,<font color="Maroon" title="c13">I</font></span>)</span>,<span class="p1">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Maroon" title="c12">C</font>,<font color="Maroon" title="c13">I</font></span>)</span>
<b>by </b><i><a class="txt" href="aofa_000.html#E13:152"><i><font color="Green" title="E101">A8</font></i></a>, <a class="txt" href="aofa_000.html#E1:152_5"><i><font color="Green" title="E105">A16</font></i></a>, <a class="ref" href="funct_5.html#T46" title="FUNCT_5:th.46">FUNCT_5:46</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>

<b>thus </b><a NAME="E7:152_5"/>
( <font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">C</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c2">T</font> implies <font color="Maroon" title="c10">f</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"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Maroon" title="c12">C</font>,<font color="Maroon" title="c13">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">C</font> )
 <b>by </b><i><a class="txt" href="aofa_000.html#E1:152_5"><i><font color="Green" title="E105">A16</font></i></a>, <a class="txt" href="aofa_000.html#E2:152_5"><i><font color="Green" title="E106">A17</font></i></a>, <a class="txt" href="aofa_000.html#E3:152_5"><i><font color="Green" title="E107">A18</font></i></a>, <a class="txt" href="aofa_000.html#E4:152_5"><i><font color="Green" title="E108">A19</font></i></a>, <a class="ref" href="aofa_000.html#T12" target="_self" title="AOFA_000:th.12">Th12</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>then reconsider </b><font color="Maroon" title="c11">f</font> = <font color="Maroon" title="c10">f</font> as    <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> <b>by </b><i><a class="txt" href="aofa_000.html#E14:152"><i><font color="Green" title="E102">A9</font></i></a>, <a class="txt" href="aofa_000.html#E15:152"><i><font color="Green" title="E103">A10</font></i></a>, <a class="txt" href="aofa_000.html#E16:152"><i><font color="Green" title="E104">A12</font></i></a>, <a class="ref" href="aofa_000.html#D32" target="_self" title="AOFA_000:def.32">Def32</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c11">f</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">f</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> &amp; ( 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><br/> for <font color="Olive" title="b2">C</font>, <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>  st not <font color="Maroon" title="c11">f</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <font color="Olive" title="b3">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Olive" title="b2">C</font>,<font color="Maroon" title="c11">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">C</font> holds <br/><font color="Maroon" title="c11">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Olive" title="b2">C</font>,<font color="Olive" title="b3">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s0</font> ) )</span><br/>

<a NAME="E19:152"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E20:152"/><b>then </b>
<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="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">f</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T118" title="ZFMISC_1:th.118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E21:152"/><b>then </b><i><font color="Green" title="E105">A21</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> &amp; <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</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="hidden.html#R1" title="HIDDEN:pred.1">=</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="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="152_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">a</font> be    <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"> ( <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> implies <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">a</font> )</span><br/><b>assume </b><a NAME="E1:152_6"/><i><font color="Green" title="E106">A22</font></i>: 
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</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">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">a</font></span><br/><b>then consider </b><font color="Maroon" title="c13">s</font>, <font color="Maroon" title="c14">I</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:152_6"/><i><font color="Green" title="E107">A23</font></i>: 
( <font color="Maroon" title="c13">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon" title="c14">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> &amp; <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c14">I</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">s</font> = <font color="Maroon" title="c13">s</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="txt" href="aofa_000.html#E3:152_6"><i><font color="Green" title="E107">A23</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">I</font> = <font color="Maroon" title="c14">I</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">A</font> <b>by </b><i><a class="txt" href="aofa_000.html#E3:152_6"><i><font color="Green" title="E107">A23</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">EI</font> = <span class="p1">(<span class="default"><a href="funct_5.html#K3" title="FUNCT_5:func.3">curry'</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">I</font> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font> <b>by </b><i><a class="txt" href="aofa_000.html#E3:152_6"><i><font color="Green" title="E107">A23</font></i></a>, <a class="txt" href="aofa_000.html#E2:152"><i><font color="Green" title="E95">A1</font></i></a></i>;<br/><a NAME="E7:152_6"/>
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</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>,<font color="Maroon" title="c1">S</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/><a NAME="E8:152_6"/><b>then </b>
 <a href="funct_5.html#K3" title="FUNCT_5:func.3">curry'</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p1">(<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 class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font></span>)</span>
 <b>by </b><i><a class="txt" href="aofa_000.html#E1:152_6"><i><font color="Green" title="E106">A22</font></i></a>, <a class="ref" href="funct_6.html#T19" title="FUNCT_6:th.19">FUNCT_6:19</a></i>;<br/><a NAME="E9:152_6"/><b>then </b><i><font color="Green" title="E108">A24</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_5.html#K3" title="FUNCT_5:func.3">curry'</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon" title="c3">A</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">EI</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c16">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T169" title="FUNCT_2:th.169">FUNCT_2:169</a></i>;<br/><b>thus </b><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">a</font> = 
<font color="Maroon" title="c4">g</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c15">s</font>,<font color="Maroon" title="c16">I</font>
<b>by </b><i><a class="txt" href="aofa_000.html#E3:152_6"><i><font color="Green" title="E107">A23</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c17">EI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">s</font>
<b>by </b><i><a class="txt" href="aofa_000.html#E3:152_6"><i><font color="Green" title="E107">A23</font></i></a>, <a class="txt" href="aofa_000.html#E9:152_6"><i><font color="Green" title="E108">A24</font></i></a>, <a class="ref" href="funct_5.html#T41" title="FUNCT_5:th.41">FUNCT_5:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c16">I</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">s</font>
<b>by </b><i><a class="txt" href="aofa_000.html#E3:152_6"><i><font color="Green" title="E107">A23</font></i></a>, <a class="txt" href="aofa_000.html#E5:152"><i><font color="Green" title="E96">A3</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c11">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c15">s</font>,<font color="Maroon" title="c16">I</font>
<b>by </b><i><a class="txt" href="aofa_000.html#E13:152"><i><font color="Green" title="E101">A8</font></i></a>, <a class="txt" href="aofa_000.html#E9:152_6"><i><font color="Green" title="E108">A24</font></i></a>, <a class="ref" href="funct_5.html#T46" title="FUNCT_5:th.46">FUNCT_5:46</a></i>
<br/>.= 
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">a</font>
<b>by </b><i><a class="txt" href="aofa_000.html#E3:152_6"><i><font color="Green" title="E107">A23</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E23:152"/>
<font color="Maroon" title="c11">f</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font>
 <b>by </b><i><a class="txt" href="aofa_000.html#E21:152"><i><font color="Green" title="E105">A21</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>; <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><br/> for <font color="Olive" title="b2">C</font>, <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>  st not <font color="Maroon" title="c11">f</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <font color="Olive" title="b3">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Olive" title="b2">C</font>,<font color="Maroon" title="c11">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">C</font> holds <br/><font color="Maroon" title="c11">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Olive" title="b2">C</font>,<font color="Olive" title="b3">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s0</font></span><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"> 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 not <font color="Maroon" title="c11">f</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <font color="Olive" title="b2">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Olive" title="b1">C</font>,<font color="Maroon" title="c11">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c12">s</font>,<font color="Olive" title="b1">C</font> holds <br/><font color="Maroon" title="c11">f</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="Olive" title="b1">C</font>,<font color="Olive" title="b2">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s0</font></span><br/><b>let </b><font color="Maroon" title="c13">C</font>, <font color="Maroon" title="c14">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"> ( not <font color="Maroon" title="c11">f</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <font color="Maroon" title="c14">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c13">C</font>,<font color="Maroon" title="c11">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c12">s</font>,<font color="Maroon" title="c13">C</font> implies <font color="Maroon" title="c11">f</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="c13">C</font>,<font color="Maroon" title="c14">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s0</font> )</span><br/>



<b>assume </b><a NAME="E24:152"/><i><font color="Green" title="E106">A25</font></i>: 
not <font color="Maroon" title="c11">f</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <font color="Maroon" title="c14">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c13">C</font>,<font color="Maroon" title="c11">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c12">s</font>,<font color="Maroon" title="c13">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">f</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="c13">C</font>,<font color="Maroon" title="c14">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s0</font></span><br/>

<a NAME="E25:152"/><i><font color="Green" title="E107">A26</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <span class="p2">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Maroon" title="c13">C</font>,<font color="Maroon" title="c14">I</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c13">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c14">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c13">C</font></span>)</span> <a href="aofa_000.html#K4" title="AOFA_000:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c14">I</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><i><font color="Green" title="E108">A27</font></i>: <font color="Maroon" title="c11">f</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="c13">C</font>,<font color="Maroon" title="c14">I</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <span class="p2">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Maroon" title="c13">C</font>,<font color="Maroon" title="c14">I</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">s</font>
<b>by </b><i><a class="txt" href="aofa_000.html#E13:152"><i><font color="Green" title="E101">A8</font></i></a>, <a class="ref" href="funct_5.html#T46" title="FUNCT_5:th.46">FUNCT_5:46</a></i>
<br/>.= 
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c13">C</font>,<font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c14">I</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">s</font>
<b>by </b><i><a class="txt" href="aofa_000.html#E9:152"><i><font color="Green" title="E100">A7</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">Ss0</font> <a href="aofa_000.html#K8" title="AOFA_000:func.8">iter</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c13">C</font></span>)</span> <a href="aofa_000.html#K4" title="AOFA_000:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c14">I</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c13">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">s</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
;<br/>
<a NAME="E27:152"/><i><font color="Green" title="E109">A28</font></i>: 
<font color="Maroon" title="c11">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c12">s</font>,<font color="Maroon" title="c13">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c13">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">s</font>
 
<b>by </b><i><a class="txt" href="aofa_000.html#E13:152"><i><font color="Green" title="E101">A8</font></i></a>, <a class="txt" href="aofa_000.html#E25:152"><i><font color="Green" title="E107">A26</font></i></a>, <a class="ref" href="funct_5.html#T46" title="FUNCT_5:th.46">FUNCT_5:46</a></i>;<br/>
<a NAME="E28:152"/><i><font color="Green" title="E110">A29</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c13">C</font></span>)</span> <a href="aofa_000.html#K4" title="AOFA_000:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c14">I</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">S</font>
 
;<br/>
<a NAME="E29:152"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font>
 
;<br/>
<a NAME="E30:152"/><b>then </b>
<font color="Maroon" title="c9">h</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c11">f</font>
 
<b>by </b><i><a class="ref" href="funct_5.html#T55" title="FUNCT_5:th.55">FUNCT_5:55</a></i>;<br/>
<a NAME="E31:152"/><b>then </b>
<span class="p1">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c13">C</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c13">C</font></span>)</span> <a href="aofa_000.html#K4" title="AOFA_000:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c14">I</font></span>)</span>
 
<b>by </b><i><a class="txt" href="aofa_000.html#E7:152"><i><font color="Green" title="E98">A5</font></i></a></i>;<br/>
<a NAME="E32:152"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c13">C</font></span>)</span> <a href="aofa_000.html#K4" title="AOFA_000:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c14">I</font></span>)</span></span>)</span> <a href="aofa_000.html#K5" title="AOFA_000:func.5">orbit</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c12">s</font>,<font color="Maroon" title="c13">C</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">T</font>
 
<b>by </b><i><a class="txt" href="aofa_000.html#E24:152"><i><font color="Green" title="E106">A25</font></i></a>, <a class="ref" href="aofa_000.html#T87" target="_self" title="AOFA_000:th.87">Th87</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c11">f</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="c13">C</font>,<font color="Maroon" title="c14">I</font></span>)</span> = 
<font color="Maroon" title="c6">Ss0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c12">s</font>,<font color="Maroon" title="c13">C</font></span>)</span>
<b>by </b><i><a class="txt" href="aofa_000.html#E25:152"><i><font color="Green" title="E107">A26</font></i></a>, <a class="txt" href="aofa_000.html#E26:152"><i><font color="Green" title="E108">A27</font></i></a>, <a class="txt" href="aofa_000.html#E27:152"><i><font color="Green" title="E109">A28</font></i></a>, <a class="txt" href="aofa_000.html#E28:152"><i><font color="Green" title="E110">A29</font></i></a>, <a class="ref" href="aofa_000.html#D7" target="_self" title="AOFA_000:def.7">Def7</a></i>
<br/>.= 
<font color="Maroon" title="c5">s0</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</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>
