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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">z</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">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) holds <br/>for <font color="Olive" title="b7">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="Olive" title="b1">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  holds <br/> ( <span class="p1">(<span class="default"><font color="Olive" title="b3">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b7">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b6">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b4">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Olive" title="b6">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Olive" title="b5">y</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">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b5">y</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="Olive" title="b5">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b6">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b5">y</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b3">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b7">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b6">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b4">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Olive" title="b6">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Olive" title="b5">y</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">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b5">y</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="Olive" title="b5">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b6">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b4">x</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b5">y</font></span>)</span> &amp; ( for <font color="Olive" title="b8">n</font>, <font color="Olive" title="b9">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b4">x</font> &amp; <font color="Olive" title="b9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b5">y</font> &amp; <font color="Olive" title="b8">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b9">m</font> &amp; ( <font color="Olive" title="b7">s</font> <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="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>  implies <font color="Olive" title="b9">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>  ) &amp; ( <font color="Olive" title="b9">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 <font color="Olive" title="b7">s</font> <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="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>  ) holds <br/><font color="Olive" title="b3">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"><span class="p4">(<span class="default"><font color="Olive" title="b6">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b4">x</font></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">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Olive" title="b5">y</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="Olive" title="b4">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b5">y</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="Olive" title="b5">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b6">z</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="Olive" title="b5">y</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="Olive" title="b7">s</font> ) )</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">z</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">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) holds <br/>for <font color="Olive" title="b6">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>  holds <br/> ( <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b6">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b5">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Olive" title="b5">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Olive" title="b4">y</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">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b4">y</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="Olive" title="b4">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b5">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b4">y</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b6">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b5">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Olive" title="b5">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Olive" title="b4">y</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">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b4">y</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="Olive" title="b4">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b5">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b4">y</font></span>)</span> &amp; ( for <font color="Olive" title="b7">n</font>, <font color="Olive" title="b8">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b3">x</font> &amp; <font color="Olive" title="b8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b4">y</font> &amp; <font color="Olive" title="b7">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b8">m</font> &amp; ( <font color="Olive" title="b6">s</font> <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="Olive" title="b1">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Olive" title="b8">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>  ) &amp; ( <font color="Olive" title="b8">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 <font color="Olive" title="b6">s</font> <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="Olive" title="b1">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) holds <br/><font color="Olive" title="b2">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"><span class="p4">(<span class="default"><font color="Olive" title="b5">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b3">x</font></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">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Olive" title="b4">y</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="Olive" title="b3">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b4">y</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="Olive" title="b4">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b5">z</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="Olive" title="b4">y</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="Olive" title="b6">s</font> ) )</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">z</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">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) holds <br/>for <font color="Olive" title="b5">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>  holds <br/> ( <span class="p1">(<span class="default"><font color="Olive" title="b1">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b5">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b4">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Olive" title="b4">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Olive" title="b3">y</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="b2">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b3">y</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="Olive" title="b3">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b4">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b3">y</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b5">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b4">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Olive" title="b4">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Olive" title="b3">y</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="b2">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b3">y</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="Olive" title="b3">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b4">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b3">y</font></span>)</span> &amp; ( for <font color="Olive" title="b6">n</font>, <font color="Olive" title="b7">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b2">x</font> &amp; <font color="Olive" title="b7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b3">y</font> &amp; <font color="Olive" title="b6">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b7">m</font> &amp; ( <font color="Olive" title="b5">s</font> <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="Olive" title="b7">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>  ) &amp; ( <font color="Olive" title="b7">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 <font color="Olive" title="b5">s</font> <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="Olive" title="b1">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"><span class="p4">(<span class="default"><font color="Olive" title="b4">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b2">x</font></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">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Olive" title="b3">y</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="Olive" title="b2">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b3">y</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="Olive" title="b3">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b4">z</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="Olive" title="b3">y</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="Olive" title="b5">s</font> ) )</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">z</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">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) holds <br/>for <font color="Olive" title="b4">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>  holds <br/> ( <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="b4">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b3">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Olive" title="b3">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Olive" title="b2">y</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="b1">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b2">y</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="Olive" title="b2">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b3">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b2">y</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="Olive" title="b4">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b3">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Olive" title="b3">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Olive" title="b2">y</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="b1">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b2">y</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="Olive" title="b2">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b3">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b2">y</font></span>)</span> &amp; ( for <font color="Olive" title="b5">n</font>, <font color="Olive" title="b6">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b5">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b6">m</font> &amp; ( <font color="Olive" title="b4">s</font> <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="Olive" title="b6">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>  ) &amp; ( <font color="Olive" title="b6">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 <font color="Olive" title="b4">s</font> <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_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"><span class="p4">(<span class="default"><font color="Olive" title="b3">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b1">x</font></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">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Olive" title="b2">y</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="Olive" title="b1">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b2">y</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="Olive" title="b2">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b3">z</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="Olive" title="b2">y</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="Olive" title="b4">s</font> ) )</span><br/>







<b>set </b><font color="Maroon" title="c5">h</font> = <font color="Maroon" title="c4">g</font>;<br/>
<b>set </b><font color="Maroon" title="c6">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="c7">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="c8">x</font>, <font color="Maroon" title="c9">y</font>, <font color="Maroon" title="c10">z</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="c8">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="c9">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="c10">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) implies 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>  holds <br/> ( <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"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">y</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="Olive" title="b1">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font></span>)</span> &amp; ( for <font color="Olive" title="b2">n</font>, <font color="Olive" title="b3">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Olive" title="b3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">y</font> &amp; <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b3">m</font> &amp; ( <font color="Olive" title="b1">s</font> <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="Olive" title="b3">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>  ) &amp; ( <font color="Olive" title="b3">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 <font color="Olive" title="b1">s</font> <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_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"><span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</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">y</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="Olive" title="b1">s</font> ) ) )</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="E2:131"/><i><font color="Green" title="E68">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="c8">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="c9">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="c10">z</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"> 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>  holds <br/> ( <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"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">y</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="Olive" title="b1">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font></span>)</span> &amp; ( for <font color="Olive" title="b2">n</font>, <font color="Olive" title="b3">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Olive" title="b3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">y</font> &amp; <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b3">m</font> &amp; ( <font color="Olive" title="b1">s</font> <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="Olive" title="b3">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>  ) &amp; ( <font color="Olive" title="b3">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 <font color="Olive" title="b1">s</font> <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_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"><span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</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">y</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="Olive" title="b1">s</font> ) )</span><br/>

<a NAME="E3:131"/><i><font color="Green" title="E69">D1</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c9">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">z</font> &amp; <font color="Maroon" title="c10">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">x</font> )
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E2:131"><i><font color="Green" title="E68">A0</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c12">I</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c13">C</font> = <font color="Maroon" title="c9">y</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/>
<div><i><font color="Green" title="E70">IT</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="131_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">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"> ( <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="c14">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</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="c14">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font></span>)</span> )</span><br/><b>reconsider </b><font color="Maroon" title="c15">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="c14">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  ;<br/><b>reconsider </b><font color="Maroon" title="c16">s2</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">s1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</font></span>)</span> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  ;<br/><b>reconsider </b><font color="Maroon" title="c17">s3</font> = <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c16">s2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  ;<br/><b>reconsider </b><font color="Maroon" title="c18">s4</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">s3</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</font></span>)</span> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  ;<br/><i><font color="Green" title="E71">00</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="c14">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"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span> = 
<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="c14">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</font></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/>.= 
<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="c14">s</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</font></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/>.= 
<font color="Maroon" title="c18">s4</font>
<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="E6:131_1"/><i><font color="Green" title="E72">01</font></i>: 
( <font color="Maroon" title="c15">s1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c15">s1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c15">s1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c10">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> )
 <b>by </b><i><a class="txt" href="aofa_i00.html#E3:131"><i><font color="Green" title="E69">D1</font></i></a>, <a class="ref" href="aofa_i00.html#T27" target="_self" title="AOFA_I00:th.27">Th211</a></i>;<br/><a NAME="E7:131_1"/><i><font color="Green" title="E73">02</font></i>: 
( <font color="Maroon" title="c16">s2</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">s1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c16">s2</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">s1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c16">s2</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c10">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">s1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c10">z</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">s1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font></span>)</span> )
 <b>by </b><i><a class="txt" href="aofa_i00.html#E3:131"><i><font color="Green" title="E69">D1</font></i></a>, <a class="ref" href="aofa_i00.html#T44" target="_self" title="AOFA_I00:th.44">Th216</a></i>;<br/><a NAME="E8:131_1"/><i><font color="Green" title="E74">03</font></i>: 
( <font color="Maroon" title="c17">s3</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">s2</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c17">s3</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">s2</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c17">s3</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c10">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">s2</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c10">z</font> )
 <b>by </b><i><a class="txt" href="aofa_i00.html#E3:131"><i><font color="Green" title="E69">D1</font></i></a>, <a class="ref" href="aofa_i00.html#T27" target="_self" title="AOFA_I00:th.27">Th211</a></i>;<br/><b>thus </b><a NAME="E9:131_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="c14">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</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="c14">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font></span>)</span> )
 <b>by </b><i><a class="txt" href="aofa_i00.html#E5:131_1"><i><font color="Green" title="E71">00</font></i></a>, <a class="txt" href="aofa_i00.html#E6:131_1"><i><font color="Green" title="E72">01</font></i></a>, <a class="txt" href="aofa_i00.html#E7:131_1"><i><font color="Green" title="E73">02</font></i></a>, <a class="txt" href="aofa_i00.html#E8:131_1"><i><font color="Green" title="E74">03</font></i></a>, <a class="txt" href="aofa_i00.html#E3:131"><i><font color="Green" title="E69">D1</font></i></a>, <a class="ref" href="aofa_i00.html#T27" target="_self" title="AOFA_I00:th.27">Th211</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>let </b><font color="Maroon" title="c14">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"> ( <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="c14">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</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="c14">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font></span>)</span> &amp; ( for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">m</font> &amp; ( <font color="Maroon" title="c14">s</font> <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="Olive" title="b2">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>  ) &amp; ( <font color="Olive" title="b2">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 <font color="Maroon" title="c14">s</font> <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_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"><span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</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">y</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="c14">s</font> ) )</span><br/>

<b>thus </b><a NAME="E5:131"/>
( <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="c14">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</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="c14">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font></span>)</span> )
 <b>by </b><i><a class="txt" href="aofa_i00.html#E4:131"><i><font color="Green" title="E70">IT</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">m</font> &amp; ( <font color="Maroon" title="c14">s</font> <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="Olive" title="b2">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>  ) &amp; ( <font color="Olive" title="b2">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 <font color="Maroon" title="c14">s</font> <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_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"><span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</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">y</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="c14">s</font></span><br/>

<b>let </b><font color="Maroon" title="c15">n</font>, <font color="Maroon" title="c16">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c16">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c15">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c16">m</font> &amp; ( <font color="Maroon" title="c14">s</font> <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="c16">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>  ) &amp; ( <font color="Maroon" title="c16">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 <font color="Maroon" title="c14">s</font> <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_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"><span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</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">y</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="c14">s</font> )</span><br/>

<b>assume </b><a NAME="E6:131"/><i><font color="Green" title="E71">A1</font></i>: 
( <font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c16">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c15">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c16">m</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c14">s</font> <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; not <font color="Maroon" title="c16">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>  ) or ( <font color="Maroon" title="c16">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>  &amp; not <font color="Maroon" title="c14">s</font> <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_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"><span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</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">y</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="c14">s</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c17">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="c14">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">y</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> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c18">fin</font> = <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<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">y</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"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  ;<br/>
<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">y</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">y</font></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ] <b>means </b>( <font color="Maroon" title="c15">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c16">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> $1 <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c15">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c16">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> $1 <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> &amp; $1 <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> $1 <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> &amp; $1 <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</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/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ] <b>means </b>( <font color="Maroon" title="c18">fin</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> $1 <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c18">fin</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> $1 <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> );<br/>
<b>assume </b><a NAME="E9:131"/>
( <font color="Maroon" title="c14">s</font> <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" title="c16">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" 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"><span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</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">y</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="c14">s</font></span><br/>

<a NAME="E10:131"/><b>then </b><i><font color="Green" title="E72">AB</font></i>: 
( <font color="Maroon" title="c14">s</font> <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="c14">s</font>] )
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E6:131"><i><font color="Green" title="E71">A1</font></i></a></i>;<br/>
<a NAME="E11:131"/><i><font color="Green" title="E73">C</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"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</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">y</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"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</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">y</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"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</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">y</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"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</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">y</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"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</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">y</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="131_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">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="c19">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="c19">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"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</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">y</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="c19">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"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</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">y</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="c19">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"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</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">y</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="c19">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"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</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">y</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="c19">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"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</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">y</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="c19">s</font>) ) )</span><br/>

<b>assume </b><a NAME="E1:131_2"/><i><font color="Green" title="E74">00</font></i>: 
<font color="Maroon" title="c19">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</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="c19">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"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</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">y</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="c19">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"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</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">y</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="c19">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"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</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">y</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="c19">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"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</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">y</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="c19">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"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</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">y</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="c19">s</font>) )</span><br/>

<b>reconsider </b><font color="Maroon" title="c20">s'</font> = <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c19">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"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c21">s''</font> = <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c20">s'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">y</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> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  ;<br/>
<a NAME="E4:131_2"/><i><font color="Green" title="E75">01</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="c19">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"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</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">y</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="c21">s''</font>
 
<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="E5:131_2"/><i><font color="Green" title="E76">02</font></i>: 
( <font color="Maroon" title="c21">s''</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">s'</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c21">s''</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">s'</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> &amp; ( <font color="Maroon" title="c20">s'</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</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 <font color="Maroon" title="c21">s''</font> <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="c20">s'</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</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 <font color="Maroon" title="c21">s''</font> <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#E2:131"><i><font color="Green" title="E68">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="E6:131_2"/>
( <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="c19">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"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</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">y</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="c19">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"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</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">y</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="txt" href="aofa_i00.html#E4:131_2"><i><font color="Green" title="E75">01</font></i></a>, <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="c19">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"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</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">y</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="c19">s</font>)</span><br/>

<a NAME="E7:131_2"/>
<font color="Maroon" title="c20">s'</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E4:131"><i><font color="Green" title="E70">IT</font></i></a></i>;<br/>
<a NAME="E8:131_2"/><b>then </b><i><font color="Green" title="E77">03</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">s'</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c20">s'</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> )
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E1:131_2"><i><font color="Green" title="E74">00</font></i></a>, <a class="ref" href="newton.html#T78" title="NEWTON:th.78">NEWTON:78</a>, <a class="ref" href="newton.html#T79" title="NEWTON:th.79">NEWTON:79</a></i>;<br/>
<a NAME="E9:131_2"/><b>then </b>
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c21">s''</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">s'</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c19">s</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> )
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E5:131_2"><i><font color="Green" title="E76">02</font></i></a>, <a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<b>hence </b><a NAME="E10:131_2"/>
<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="c19">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"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</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">y</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="c19">s</font>)
 <b>by </b><i><a class="txt" href="aofa_i00.html#E8:131_2"><i><font color="Green" title="E77">03</font></i></a>, <a class="ref" href="aofa_000.html#D29" title="AOFA_000:def.29">AOFA_000:def 29</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="E12:131"/>
<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"><span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</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">y</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="c14">s</font>
 
<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#E10:131"><i><font color="Green" title="E72">AB</font></i></a>, <a class="txt" href="aofa_i00.html#E11:131"><i><font color="Green" title="E73">C</font></i></a>);<br/></i>
<b>hence </b><a NAME="E13:131"/>
<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"><span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K83" title="AOFA_I00:func.83">%=</a> <font color="Maroon" title="c9">y</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="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c10">z</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">y</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="c14">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
