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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> <br/> 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> 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"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <span class="p3">(<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>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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="p5">(<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="p4">(<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></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="b8">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Olive" title="b9">m</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> <br/> 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> 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"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <span class="p3">(<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>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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="p5">(<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="p4">(<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></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="b7">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Olive" title="b8">m</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> <br/> 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> 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"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <span class="p3">(<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>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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="p5">(<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="p4">(<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></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="b6">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Olive" title="b7">m</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> <br/> 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> 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"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <span class="p3">(<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>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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="p5">(<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="p4">(<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></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="b5">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Olive" title="b6">m</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> <br/> 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> 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"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <span class="p3">(<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="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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="p5">(<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="p4">(<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></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="b2">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Olive" title="b3">m</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="E2:133"/><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> <br/> 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> 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"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <span class="p3">(<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="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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="p5">(<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="p4">(<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></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="b2">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Olive" title="b3">m</font></span><br/>

<b>let </b><font color="Maroon" title="c12">s</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ; <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="c12">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="c12">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> 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="Maroon" title="c12">s</font>,<span class="p2">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <span class="p3">(<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="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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="p5">(<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="p4">(<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></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">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Olive" title="b2">m</font></span><br/><b>let </b><font color="Maroon" title="c13">n</font>, <font color="Maroon" title="c14">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="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c14">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c14">m</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c12">s</font>,<span class="p2">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <span class="p3">(<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="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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="p5">(<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="p4">(<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></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="c13">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c14">m</font> )</span><br/>



<b>assume </b><a NAME="E3:133"/><i><font color="Green" title="E69">A1</font></i>: 
( <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c14">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c14">m</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c12">s</font>,<span class="p2">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <span class="p3">(<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="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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="p5">(<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="p4">(<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></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="c13">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c14">m</font></span><br/>

<b>set </b><font color="Maroon" title="c15">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="c16">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/>
<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="c12">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="c12">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/>
<a NAME="E6:133"/><i><font color="Green" title="E70">S1</font></i>: 
( <font color="Maroon" title="c17">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="c12">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="c17">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="c12">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="c12">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="c17">s1</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="c12">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="c17">s1</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#E2:133"><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>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="c13">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c14">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="c13">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c14">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/>
<a NAME="E7:133"/><i><font color="Green" title="E71">AB</font></i>: 
( <font color="Maroon" title="c17">s1</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="c14">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E3:133"><i><font color="Green" title="E69">A1</font></i></a>, <a class="txt" href="aofa_i00.html#E6:133"><i><font color="Green" title="E70">S1</font></i></a>, <a class="ref" href="aofa_i00.html#T2" target="_self" title="AOFA_I00:th.2">LemTS</a></i>;<br/>
<a NAME="E8:133"/>
( <font color="Maroon" title="c17">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="c12">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="c17">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="c12">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#E2:133"><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/>
<a NAME="E9:133"/><b>then </b><i><font color="Green" title="E72">II</font></i>: 
<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="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c12">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="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>
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E7:133"><i><font color="Green" title="E71">AB</font></i></a>, <a class="txt" href="aofa_i00.html#E2:133"><i><font color="Green" title="E68">A0</font></i></a>, <a class="txt" href="aofa_i00.html#E3:133"><i><font color="Green" title="E69">A1</font></i></a>, <a class="txt" href="aofa_i00.html#E145"><i><font color="Green" title="E67">LemEAterm</font></i></a></i>;<br/>
<a NAME="E10:133"/><i><font color="Green" title="E73">IA</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">s</font>]
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E3:133"><i><font color="Green" title="E69">A1</font></i></a>, <a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></i>;<br/>
<a NAME="E11:133"/><i><font color="Green" title="E74">ID</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>2</sub></font>[<font color="Olive" title="b1">s</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>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">s</font>] holds <br/><font color="Maroon">S<sub>2</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"><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>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1"><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>2</sub></font>[<font color="Maroon" title="c19">s</font>] &amp; <font color="Maroon" title="c19">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; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c19">s</font>] implies <font color="Maroon">S<sub>2</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"><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>] )</span><br/>

<b>assume </b><a NAME="E1:133_1"/><i><font color="Green" title="E75">00</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c19">s</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c19">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  not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c19">s</font>] or <font color="Maroon">S<sub>2</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"><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>] )</span><br/>

<b>reconsider </b><font color="Maroon" title="c20">n'</font> = <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>, <font color="Maroon" title="c21">m'</font> = <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> as    <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>  <b>by </b><i><a class="txt" href="aofa_i00.html#E1:133_1"><i><font color="Green" title="E75">00</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<b>assume </b><a NAME="E3:133_1"/><i><font color="Green" title="E76">01</font></i>: 
( <font color="Maroon" title="c19">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; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c19">s</font>] )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</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"><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>]</span><br/>

<b>reconsider </b><font color="Maroon" title="c22">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/>
<a NAME="E5:133_1"/><i><font color="Green" title="E77">04</font></i>: 
( <font color="Maroon" title="c22">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="c19">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="c22">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#E2:133"><i><font color="Green" title="E68">A0</font></i></a>, <a class="txt" href="aofa_i00.html#E145"><i><font color="Green" title="E67">LemEAterm</font></i></a></i>;<br/>
<a NAME="E6:133_1"/>
( <font color="Maroon" title="c13">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c14">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c20">n'</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c21">m'</font> &amp; <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">n'</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c21">m'</font> )
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E1:133_1"><i><font color="Green" title="E75">00</font></i></a>, <a class="ref" href="nat_d.html#T11" title="NAT_D:th.11">NAT_D:11</a></i>;<br/>
<b>hence </b><a NAME="E7:133_1"/>
<font color="Maroon">S<sub>2</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"><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>]
 <b>by </b><i><a class="txt" href="aofa_i00.html#E3:133_1"><i><font color="Green" title="E76">01</font></i></a>, <a class="txt" href="aofa_i00.html#E1:133_1"><i><font color="Green" title="E75">00</font></i></a>, <a class="txt" href="aofa_i00.html#E5:133_1"><i><font color="Green" title="E77">04</font></i></a>, <a class="ref" href="newton.html#T79" title="NEWTON:th.79">NEWTON:79</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="E12:133"/><i><font color="Green" title="E75">IE</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>2</sub></font>[<font color="Olive" title="b1">s</font>] holds <br/>( <font color="Maroon">S<sub>2</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"><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>] &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"><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> <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"><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>] ) &amp; ( <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"><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>] 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"><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> <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>  ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_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>2</sub></font>[<font color="Maroon" title="c19">s</font>] implies ( <font color="Maroon">S<sub>2</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"><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>] &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"><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> <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"><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>] ) &amp; ( <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"><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>] 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"><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> <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>  ) ) )</span><br/>

<b>assume </b><a NAME="E1:133_2"/><i><font color="Green" title="E76">01</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c19">s</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>2</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"><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>] &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"><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> <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"><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>] ) &amp; ( <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"><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>] 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"><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> <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>  ) )</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"><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="E3:133_2"/><i><font color="Green" title="E77">02</font></i>: 
( <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> <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="c8">x</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="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> &amp; ( <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>  implies <font color="Maroon" title="c20">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="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#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c20">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#E2:133"><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="E4:133_2"/>
<font color="Maroon">S<sub>2</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"><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>]
 <b>by </b><i><a class="txt" href="aofa_i00.html#E1:133_2"><i><font color="Green" title="E76">01</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="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"><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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c19">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>] )</span><br/>

<b>thus </b><a NAME="E5:133_2"/>
( <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"><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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c19">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>] )
 <b>by </b><i><a class="txt" href="aofa_i00.html#E3:133_2"><i><font color="Green" title="E77">02</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"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E13:133"/><i><font color="Green" title="E76">08</font></i>: 
( <font color="Maroon">S<sub>2</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="c12">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>] &amp;  not <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="c12">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>] )
 
<b>from </b><i><a class="ref" href="aofa_000.html#S5" title="AOFA_000:sch.5">AOFA_000:sch 5</a>(<a class="txt" href="aofa_i00.html#E10:133"><i><font color="Green" title="E73">IA</font></i></a>, <a class="txt" href="aofa_i00.html#E9:133"><i><font color="Green" title="E72">II</font></i></a>, <a class="txt" href="aofa_i00.html#E11:133"><i><font color="Green" title="E74">ID</font></i></a>, <a class="txt" href="aofa_i00.html#E12:133"><i><font color="Green" title="E75">IE</font></i></a>);<br/></i>
<a NAME="E14:133"/><b>then </b>
( <font color="Maroon" title="c13">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c14">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <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> &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="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c18">fin</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="c18">fin</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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
;<br/>
<a NAME="E15:133"/><b>then </b><i><font color="Green" title="E77">cA</font></i>: 
<font color="Maroon">S<sub>3</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="c12">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>]
 
<b>by </b><i><a class="ref" href="int_2.html#T16" title="INT_2:th.16">INT_2:16</a></i>;<br/>
<a NAME="E16:133"/><i><font color="Green" title="E78">cD</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>3</sub></font>[<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b1">s</font>,<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>,<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>] &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"><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> <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">S<sub>3</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"><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>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_3"><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>3</sub></font>[<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c19">s</font>,<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>,<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>] &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"><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> <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>3</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"><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><br/>

<b>assume </b><a NAME="E1:133_3"/><i><font color="Green" title="E79">00</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c19">s</font>,<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>,<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 class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c19">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> <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">S<sub>3</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"><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><br/>

<b>assume </b><a NAME="E2:133_3"/><i><font color="Green" title="E80">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"><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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>3</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"><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><br/>

<b>reconsider </b><font color="Maroon" title="c20">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="c19">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="c21">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="c20">s1</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/>
<a NAME="E5:133_3"/><i><font color="Green" title="E81">02</font></i>: 
( <font color="Maroon" title="c20">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="c19">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="c20">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="c19">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="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>  implies <font color="Maroon" title="c20">s1</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="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#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c20">s1</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#E2:133"><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/>
<a NAME="E6:133_3"/><i><font color="Green" title="E82">03</font></i>: 
( <font color="Maroon" title="c21">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="c20">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="c21">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> <span class="p1">(<span class="default"><font color="Maroon" title="c20">s1</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="c20">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#E2:133"><i><font color="Green" title="E68">A0</font></i></a>, <a class="txt" href="aofa_i00.html#E145"><i><font color="Green" title="E67">LemEAterm</font></i></a></i>;<br/>
<a NAME="E7:133_3"/>
<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K5" title="INT_1:func.5">div</a> <span class="p3">(<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></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">s2</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">s2</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E2:133_3"><i><font color="Green" title="E80">01</font></i></a>, <a class="txt" href="aofa_i00.html#E5:133_3"><i><font color="Green" title="E81">02</font></i></a>, <a class="txt" href="aofa_i00.html#E6:133_3"><i><font color="Green" title="E82">03</font></i></a>, <a class="ref" href="aofa_i00.html#T2" target="_self" title="AOFA_I00:th.2">LemTS</a>, <a class="ref" href="newton.html#T80" title="NEWTON:th.80">NEWTON:80</a></i>;<br/>
<b>hence </b><a NAME="E8:133_3"/>
<font color="Maroon">S<sub>3</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"><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>]
 <b>by </b><i><a class="txt" href="aofa_i00.html#E1:133_3"><i><font color="Green" title="E79">00</font></i></a>, <a class="txt" href="aofa_i00.html#E5:133_3"><i><font color="Green" title="E81">02</font></i></a>, <a class="txt" href="aofa_i00.html#E2:133"><i><font color="Green" title="E68">A0</font></i></a>, <a class="txt" href="aofa_i00.html#E145"><i><font color="Green" title="E67">LemEAterm</font></i></a>, <a class="ref" href="wsierp_1.html#T10" title="WSIERP_1:th.10">WSIERP_1:10</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="E17:133"/><i><font color="Green" title="E79">cE</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>3</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"><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>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">s</font>]
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E2:133"><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/>
<a NAME="E18:133"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c12">s</font>]
 
<b>from </b><i><a class="ref" href="aofa_000.html#S6" title="AOFA_000:sch.6">AOFA_000:sch 6</a>(<a class="txt" href="aofa_i00.html#E15:133"><i><font color="Green" title="E77">cA</font></i></a>, <a class="txt" href="aofa_i00.html#E9:133"><i><font color="Green" title="E72">II</font></i></a>, <a class="txt" href="aofa_i00.html#E16:133"><i><font color="Green" title="E78">cD</font></i></a>, <a class="txt" href="aofa_i00.html#E17:133"><i><font color="Green" title="E79">cE</font></i></a>);<br/></i>
<a NAME="E19:133"/><b>then </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> <font color="Maroon" title="c13">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c14">m</font>
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E3:133"><i><font color="Green" title="E69">A1</font></i></a>, <a class="ref" href="int_2.html#T33" title="INT_2:th.33">INT_2:33</a></i>;<br/>
<a NAME="E20:133"/><b>then </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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c14">m</font> or <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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c14">m</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E13:133"><i><font color="Green" title="E76">08</font></i></a>, <a class="ref" href="int_2.html#T15" title="INT_2:th.15">INT_2:15</a></i>;<br/>
<b>hence </b><a NAME="E21:133"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c12">s</font>,<span class="p2">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <span class="p3">(<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="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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="p5">(<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="p4">(<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></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="c13">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c14">m</font>
 <b>by </b><i><a class="txt" href="aofa_i00.html#E13:133"><i><font color="Green" title="E76">08</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
