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<b>let </b><font color="Maroon" title="c1">A</font> be  <a href="aofa_i00.html#V2" title="AOFA_I00:attr.2">Euclidean</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">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">X</font><br/> for <font color="Olive" title="b3">g</font> being <a href="aofa_i00.html#V1" title="AOFA_I00:attr.1">Euclidean</a>  <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>, <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Olive" title="b1">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Olive" title="b1">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Olive" title="b2">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <br/> for <font color="Olive" title="b4">x</font>, <font color="Olive" title="b5">y</font>, <font color="Olive" title="b6">m</font> being   <a href="aofa_i00.html#M1" title="AOFA_I00:mode.1">Variable</a> of <font color="Olive" title="b3">g</font>  st  ex <font color="Olive" title="b7">d</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b7">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b7">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b7">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Olive" title="b7">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b5">y</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <span class="p2">(<span class="default"><font color="Olive" title="b6">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Olive" title="b6">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Olive" title="b5">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Olive" title="b4">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Olive" title="b6">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Olive" title="b4">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Olive" title="b4">x</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#R6" title="AOFA_000:pred.6">is_terminating_wrt</a> <font color="Olive" title="b3">g</font>,<span class="p1">{<span class="default"> <font color="Olive" title="b7">s</font> where <font color="Olive" title="b7">s</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="Olive" title="b1">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  : <font color="Olive" title="b7">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b6">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> </span><br/><b>let </b><font color="Maroon" title="c2">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="card_4.html#V1" title="CARD_4:attr.1">countable</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">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font><br/> for <font color="Olive" title="b2">g</font> being <a href="aofa_i00.html#V1" title="AOFA_I00:attr.1">Euclidean</a>  <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>, <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Olive" title="b1">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <br/> for <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font>, <font color="Olive" title="b5">m</font> being   <a href="aofa_i00.html#M1" title="AOFA_I00:mode.1">Variable</a> of <font color="Olive" title="b2">g</font>  st  ex <font color="Olive" title="b6">d</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b6">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b6">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b6">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Olive" title="b6">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b4">y</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <span class="p2">(<span class="default"><font color="Olive" title="b5">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Olive" title="b5">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Olive" title="b4">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Olive" title="b3">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Olive" title="b5">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Olive" title="b3">x</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#R6" title="AOFA_000:pred.6">is_terminating_wrt</a> <font color="Olive" title="b2">g</font>,<span class="p1">{<span class="default"> <font color="Olive" title="b6">s</font> where <font color="Olive" title="b6">s</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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  : <font color="Olive" title="b6">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b5">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> </span><br/><b>let </b><font color="Maroon" title="c3">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</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">g</font> being <a href="aofa_i00.html#V1" title="AOFA_I00:attr.1">Euclidean</a>  <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>, <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <br/> for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font>, <font color="Olive" title="b4">m</font> being   <a href="aofa_i00.html#M1" title="AOFA_I00:mode.1">Variable</a> of <font color="Olive" title="b1">g</font>  st  ex <font color="Olive" title="b5">d</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b5">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b5">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b5">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Olive" title="b5">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b3">y</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Olive" title="b4">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Olive" title="b3">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Olive" title="b2">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Olive" title="b4">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Olive" title="b2">x</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#R6" title="AOFA_000:pred.6">is_terminating_wrt</a> <font color="Olive" title="b1">g</font>,<span class="p1">{<span class="default"> <font color="Olive" title="b5">s</font> where <font color="Olive" title="b5">s</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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  : <font color="Olive" title="b5">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b4">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> </span><br/><b>let </b><font color="Maroon" title="c4">g</font> be  <a href="aofa_i00.html#V1" title="AOFA_I00:attr.1">Euclidean</a>  <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>, <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</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">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">m</font> being   <a href="aofa_i00.html#M1" title="AOFA_I00:mode.1">Variable</a> of <font color="Maroon" title="c4">g</font>  st  ex <font color="Olive" title="b4">d</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b4">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b4">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b4">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Olive" title="b4">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Olive" title="b3">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Olive" title="b2">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Olive" title="b3">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#R6" title="AOFA_000:pred.6">is_terminating_wrt</a> <font color="Maroon" title="c4">g</font>,<span class="p1">{<span class="default"> <font color="Olive" title="b4">s</font> where <font color="Olive" title="b4">s</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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  : <font color="Olive" title="b4">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> </span><br/>







<b>set </b><font color="Maroon" title="c5">S</font> =  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ;<br/>
<b>set </b><font color="Maroon" title="c6">T</font> = <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<b>let </b><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font>, <font color="Maroon" title="c9">m</font> be    <a href="aofa_i00.html#M1" title="AOFA_I00:mode.1">Variable</a> of <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"> (  ex <font color="Olive" title="b1">d</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b1">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Olive" title="b1">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) implies <span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#R6" title="AOFA_000:pred.6">is_terminating_wrt</a> <font color="Maroon" title="c4">g</font>,<span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  : <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span>  )</span><br/>

<b>set </b><font color="Maroon" title="c10">P</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  : <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> ;<br/>
<b>given </b><font color="Maroon" title="c11">d</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><a NAME="E2:137"/><i><font color="Green" title="E69">A0</font></i>: 
( <font color="Maroon" title="c11">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c11">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c11">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c11">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 ; <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="c8">y</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#R6" title="AOFA_000:pred.6">is_terminating_wrt</a> <font color="Maroon" title="c4">g</font>,<span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  : <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> </span><br/>

<a NAME="E3:137"/><i><font color="Green" title="E70">D0</font></i>: 
( <font color="Maroon" title="c9">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c9">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">y</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">y</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">y</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">m</font> )
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E2:137"><i><font color="Green" title="E69">A0</font></i></a></i>;<br/>
<a NAME="E4:137"/><i><font color="Green" title="E71">A1</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1 <a href="aofa_000.html#R6" title="AOFA_000:pred.6">is_terminating_wrt</a> <font color="Maroon" title="c4">g</font>,<span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  : <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> 
 
<b>by </b><i><a class="ref" href="aofa_000.html#T107" title="AOFA_000:th.107">AOFA_000:107</a></i>;<br/>
<a NAME="E5:137"/><i><font color="Green" title="E72">A2</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  : <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span>  <a href="aofa_000.html#R7" title="AOFA_000:pred.7">is_invariant_wrt</a> <font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1,<font color="Maroon" title="c4">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="137_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">s</font> be    <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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="aofa_000.html#D39" title="AOFA_000:def.39">AOFA_000:def 39</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c12">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  : <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span>  or <font color="Maroon" title="c4">g</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"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  : <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E1:137_1"/>
<font color="Maroon" title="c12">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  : <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> 
 ; <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="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c12">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  : <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> </span><br/>

<a NAME="E2:137_1"/><b>then </b><i><font color="Green" title="E73">B1</font></i>: 
 ex <font color="Olive" title="b1">s'</font> being   <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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  st <br/>( <font color="Maroon" title="c12">s</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">s'</font> &amp; <font color="Olive" title="b1">s'</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
;<br/>
<a NAME="E3:137_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c12">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font>
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E3:137"><i><font color="Green" title="E70">D0</font></i></a>, <a class="ref" href="aofa_i00.html#T25" target="_self" title="AOFA_I00:th.25">Th011</a></i>;<br/>
<b>hence </b><a NAME="E4:137_1"/>
<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c12">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  : <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> 
 <b>by </b><i><a class="txt" href="aofa_i00.html#E2:137_1"><i><font color="Green" title="E73">B1</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>set </b><font color="Maroon" title="c12">C</font> = <font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<b>set </b><font color="Maroon" title="c13">I</font> =  <a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c14">J</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span>;<br/>
<a NAME="E6:137"/><i><font color="Green" title="E73">A4</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_000.html#R6" title="AOFA_000:pred.6">is_terminating_wrt</a> <font color="Maroon" title="c4">g</font>,<span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  : <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> 
 
<b>by </b><i><a class="ref" href="aofa_000.html#T107" title="AOFA_000:th.107">AOFA_000:107</a></i>;<br/>
<a NAME="E7:137"/><i><font color="Green" title="E74">A5</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  : <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span>  <a href="aofa_000.html#R7" title="AOFA_000:pred.7">is_invariant_wrt</a> <font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c4">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="137_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">s</font> be    <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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="aofa_000.html#D39" title="AOFA_000:def.39">AOFA_000:def 39</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c15">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  : <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span>  or <font color="Maroon" title="c4">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c15">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  : <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E1:137_2"/>
<font color="Maroon" title="c15">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  : <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> 
 ; <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="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c15">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  : <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> </span><br/>

<a NAME="E2:137_2"/><b>then </b><i><font color="Green" title="E75">B1</font></i>: 
 ex <font color="Olive" title="b1">s'</font> being   <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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  st <br/>( <font color="Maroon" title="c15">s</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">s'</font> &amp; <font color="Olive" title="b1">s'</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
;<br/>
<a NAME="E3:137_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font>
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E2:137"><i><font color="Green" title="E69">A0</font></i></a>, <a class="ref" href="aofa_i00.html#T38" target="_self" title="AOFA_I00:th.38">Th015</a></i>;<br/>
<b>hence </b><a NAME="E4:137_2"/>
<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  : <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> 
 <b>by </b><i><a class="txt" href="aofa_i00.html#E2:137_2"><i><font color="Green" title="E75">B1</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>
<a NAME="E8:137"/><i><font color="Green" title="E75">A6</font></i>: 
for <font color="Olive" title="b1">s</font> being   <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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>   st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">s</font> where <font color="Olive" title="b2">s</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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  : <font color="Olive" title="b2">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span>  &amp; <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b1">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">s</font> where <font color="Olive" title="b2">s</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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  : <font color="Olive" title="b2">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="137_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">s</font> be    <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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c15">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  : <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span>  &amp; <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  : <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E1:137_3"/>
<font color="Maroon" title="c15">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  : <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  : <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E2:137_3"/>
<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <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="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  : <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> </span><br/>

<a NAME="E3:137_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="aofa_i00.html#T40" target="_self" title="AOFA_I00:th.40">Th015'</a></i>;<br/>
<b>hence </b><a NAME="E4:137_3"/>
<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  : <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> 
 ; <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>defpred </b><font color="Maroon">S<sub>1</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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ] <b>means </b>$1 <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  =  <a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <span class="p1">(<span class="default">$1 <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>
<a NAME="E9:137"/><i><font color="Green" title="E76">Z0</font></i>: 
( <font color="Maroon" title="c4">g</font> <a href="aofa_000.html#R3" title="AOFA_000:pred.3">complies_with_while_wrt</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">g</font> <a href="aofa_000.html#R2" title="AOFA_000:pred.2">complies_with_if_wrt</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="aofa_000.html#D32" title="AOFA_000:def.32">AOFA_000:def 32</a></i>;<br/>
<a NAME="E10:137"/>
for <font color="Olive" title="b1">s</font> being   <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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>   st <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">s</font> where <font color="Olive" title="b2">s</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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  : <font color="Olive" title="b2">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span>  holds <br/><font color="Maroon" title="c4">g</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="137_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">s0</font> be    <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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ; <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="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s0</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  : <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span>  implies <font color="Maroon" title="c4">g</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s0</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:137_4"/>
<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s0</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  : <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> 
 ; <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="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s0</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><br/>

<b>set </b><font color="Maroon" title="c16">s1</font> = <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s0</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>;<br/>
<a NAME="E2:137_4"/><i><font color="Green" title="E77">A4</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">s0</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">s0</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">s0</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font> &amp; ( <font color="Maroon" title="c15">s0</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; ( <font color="Maroon" title="c15">s0</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E2:137"><i><font color="Green" title="E69">A0</font></i></a>, <a class="ref" href="aofa_i00.html#T38" target="_self" title="AOFA_I00:th.38">Th015</a></i>;<br/>
<a NAME="E3:137_4"/><i><font color="Green" title="E78">B0</font></i>: 
( <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s0</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s0</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>] )
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E2:137_4"><i><font color="Green" title="E77">A4</font></i></a>, <a class="ref" href="aofa_i00.html#T2" target="_self" title="AOFA_I00:th.2">LemTS</a></i>;<br/>
<a NAME="E4:137_4"/><i><font color="Green" title="E79">B1</font></i>: 
for <font color="Olive" title="b1">s</font> being   <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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">s</font>] holds <br/>( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>] implies <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>] ) &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">s</font>) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="137_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">s</font> be    <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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c17">s</font>] implies ( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>] implies <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>] ) &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">s</font>) ) )</span><br/>

<b>assume </b><a NAME="E1:137_4_1"/><i><font color="Green" title="E80">00</font></i>: 
<font color="Maroon" title="c17">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>] implies <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>] ) &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">s</font>) )</span><br/>

<b>set </b><font color="Maroon" title="c18">sJ</font> = <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c19">sC</font> = <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>;<br/>
<a NAME="E2:137_4_1"/><i><font color="Green" title="E81">01</font></i>: 
( ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> &amp; <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E2:137"><i><font color="Green" title="E69">A0</font></i></a>, <a class="ref" href="aofa_i00.html#T38" target="_self" title="AOFA_I00:th.38">Th015</a>, <a class="ref" href="aofa_000.html#D29" title="AOFA_000:def.29">AOFA_000:def 29</a></i>;<br/>
<b>hence </b><a NAME="E3:137_4_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>] iff <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="ref" href="aofa_i00.html#T2" target="_self" title="AOFA_I00:th.2">LemTS</a></i>; <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="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">s</font>)</span><br/>

<b>set </b><font color="Maroon" title="c20">q0</font> = <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c21">q1</font> = <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c22">q2</font> = <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p2">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c23">q3</font> = <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p3">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span>;<br/>
<a NAME="E4:137_4_1"/>
( <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
;<br/>
<a NAME="E5:137_4_1"/><b>then </b><i><font color="Green" title="E82">03</font></i>: 
( <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span> or <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="aofa_000.html#K12" title="AOFA_000:func.12">EmptyIns</a> <font color="Maroon" title="c1">A</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E9:137"><i><font color="Green" title="E76">Z0</font></i></a>, <a class="ref" href="aofa_000.html#D30" title="AOFA_000:def.30">AOFA_000:def 30</a></i>;<br/>
<a NAME="E6:137_4_1"/>
<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p2">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="aofa_000.html#D29" title="AOFA_000:def.29">AOFA_000:def 29</a></i>;<br/>
<a NAME="E7:137_4_1"/><b>then </b>
<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p3">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="aofa_000.html#D29" title="AOFA_000:def.29">AOFA_000:def 29</a></i>;<br/>
<b>then </b><i><font color="Green" title="E83">02</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p3">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font>
<b>by </b><i><a class="txt" href="aofa_i00.html#E3:137"><i><font color="Green" title="E70">D0</font></i></a>, <a class="ref" href="aofa_i00.html#T33" target="_self" title="AOFA_I00:th.33">Th213</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p3">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2
<b>by </b><i><a class="ref" href="aofa_i00.html#T45" target="_self" title="AOFA_I00:th.45">Th017</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2
<b>by </b><i><a class="txt" href="aofa_i00.html#E3:137"><i><font color="Green" title="E70">D0</font></i></a>, <a class="txt" href="aofa_i00.html#E5:137_4_1"><i><font color="Green" title="E82">03</font></i></a>, <a class="ref" href="aofa_i00.html#T33" target="_self" title="AOFA_I00:th.33">Th213</a>, <a class="ref" href="aofa_000.html#D28" title="AOFA_000:def.28">AOFA_000:def 28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2
<b>by </b><i><a class="txt" href="aofa_i00.html#E2:137"><i><font color="Green" title="E69">A0</font></i></a>, <a class="ref" href="aofa_i00.html#T49" target="_self" title="AOFA_I00:th.49">Th218</a></i>
;<br/>
<a NAME="E9:137_4_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c17">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E1:137_4_1"><i><font color="Green" title="E80">00</font></i></a>, <a class="txt" href="aofa_i00.html#E2:137_4_1"><i><font color="Green" title="E81">01</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a>, <a class="ref" href="int_1.html#T88" title="INT_1:th.88">INT_1:88</a></i>;<br/>
<a NAME="E10:137_4_1"/><b>then </b>
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">s</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> )
 
<b>by </b><i><a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></i>;<br/>
<b>hence </b><a NAME="E11:137_4_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">s</font>)
 <b>by </b><i><a class="txt" href="aofa_i00.html#E1:137_4_1"><i><font color="Green" title="E80">00</font></i></a>, <a class="txt" href="aofa_i00.html#E2:137_4_1"><i><font color="Green" title="E81">01</font></i></a>, <a class="txt" href="aofa_i00.html#E8:137_4_1"><i><font color="Green" title="E83">02</font></i></a>, <a class="ref" href="int_1.html#T83" title="INT_1:th.83">INT_1:83</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>thus </b><a NAME="E5:137_4"/>
<font color="Maroon" title="c4">g</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s0</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <b>from </b><i><a class="ref" href="aofa_000.html#S3" title="AOFA_000:sch.3">AOFA_000:sch 3</a>(<a class="txt" href="aofa_i00.html#E3:137_4"><i><font color="Green" title="E78">B0</font></i></a>, <a class="txt" href="aofa_i00.html#E4:137_4"><i><font color="Green" title="E79">B1</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>
<a NAME="E11:137"/><b>then </b>
 <a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#R6" title="AOFA_000:pred.6">is_terminating_wrt</a> <font color="Maroon" title="c4">g</font>,<span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  : <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> 
 
<b>by </b><i><a class="ref" href="aofa_000.html#T104" title="AOFA_000:th.104">AOFA_000:104</a>, <a class="txt" href="aofa_i00.html#E6:137"><i><font color="Green" title="E73">A4</font></i></a>, <a class="txt" href="aofa_i00.html#E7:137"><i><font color="Green" title="E74">A5</font></i></a>, <a class="txt" href="aofa_i00.html#E8:137"><i><font color="Green" title="E75">A6</font></i></a>, <a class="ref" href="aofa_000.html#T118" title="AOFA_000:th.118">AOFA_000:118</a></i>;<br/>
<b>hence </b><a NAME="E12:137"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K72" title="AOFA_I00:func.72">is_odd</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="aofa_i00.html#K60" title="AOFA_I00:func.60">/=</a> 2</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#R6" title="AOFA_000:pred.6">is_terminating_wrt</a> <font color="Maroon" title="c4">g</font>,<span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</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="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  : <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> 
 <b>by </b><i><a class="txt" href="aofa_i00.html#E4:137"><i><font color="Green" title="E71">A1</font></i></a>, <a class="txt" href="aofa_i00.html#E5:137"><i><font color="Green" title="E72">A2</font></i></a>, <a class="ref" href="aofa_000.html#T111" title="AOFA_000:th.111">AOFA_000:111</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>
