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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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_3.html#V4" title="CARD_3:attr.4">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="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Olive" title="b1">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Olive" title="b1">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">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="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Olive" title="b1">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  holds <br/> ( <span class="p1">(<span class="default"><font color="Olive" title="b3">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b7">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b6">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Olive" title="b4">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Olive" title="b5">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Olive" title="b6">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Olive" title="b6">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Olive" title="b4">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b5">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Olive" title="b5">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b6">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b5">y</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b3">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b7">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b6">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Olive" title="b4">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Olive" title="b5">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Olive" title="b6">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Olive" title="b6">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Olive" title="b4">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b5">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Olive" title="b5">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b6">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b7">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Olive" title="b7">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b5">y</font></span>)</span></span>)</span> &amp; ( for <font color="Olive" title="b8">n</font>, <font color="Olive" title="b9">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b4">x</font> &amp; <font color="Olive" title="b9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b5">y</font> &amp; ( <font color="Olive" title="b7">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Olive" title="b1">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Olive" title="b2">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Olive" title="b9">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Olive" title="b9">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Olive" title="b7">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Olive" title="b1">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Olive" title="b2">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) holds <br/><font color="Olive" title="b3">g</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b6">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Olive" title="b4">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Olive" title="b5">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Olive" title="b6">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Olive" title="b6">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Olive" title="b4">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b5">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Olive" title="b5">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b6">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">y</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Olive" title="b7">s</font> ) )</span><br/><span class="kw">let </span><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_3.html#V4" title="CARD_3:attr.4">countable</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">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="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  holds <br/> ( <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b6">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b5">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Olive" title="b4">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Olive" title="b5">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Olive" title="b5">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Olive" title="b3">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b4">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Olive" title="b4">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b5">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b4">y</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b6">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b5">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Olive" title="b4">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Olive" title="b5">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Olive" title="b5">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Olive" title="b3">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b4">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Olive" title="b4">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b5">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b6">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Olive" title="b6">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b4">y</font></span>)</span></span>)</span> &amp; ( for <font color="Olive" title="b7">n</font>, <font color="Olive" title="b8">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b3">x</font> &amp; <font color="Olive" title="b8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b4">y</font> &amp; ( <font color="Olive" title="b6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Olive" title="b1">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Olive" title="b8">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Olive" title="b8">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Olive" title="b6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Olive" title="b1">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) holds <br/><font color="Olive" title="b2">g</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b5">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Olive" title="b4">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Olive" title="b5">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Olive" title="b5">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b4">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b5">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">y</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Olive" title="b6">s</font> ) )</span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">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="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  holds <br/> ( <span class="p1">(<span class="default"><font color="Olive" title="b1">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b5">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b4">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Olive" title="b3">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Olive" title="b4">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Olive" title="b4">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Olive" title="b2">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b3">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b4">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b3">y</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b5">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b4">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Olive" title="b3">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Olive" title="b4">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Olive" title="b4">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Olive" title="b2">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b3">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b4">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Olive" title="b5">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b3">y</font></span>)</span></span>)</span> &amp; ( for <font color="Olive" title="b6">n</font>, <font color="Olive" title="b7">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b2">x</font> &amp; <font color="Olive" title="b7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b3">y</font> &amp; ( <font color="Olive" title="b5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Olive" title="b7">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Olive" title="b7">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Olive" title="b5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) holds <br/><font color="Olive" title="b1">g</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b4">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Olive" title="b3">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Olive" title="b4">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Olive" title="b4">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b3">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b4">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">y</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Olive" title="b5">s</font> ) )</span><br/><span class="kw">let </span><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="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  holds <br/> ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b4">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b3">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Olive" title="b2">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Olive" title="b3">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Olive" title="b3">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b2">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b3">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b2">y</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b4">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b3">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Olive" title="b2">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Olive" title="b3">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Olive" title="b3">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b2">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b3">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b4">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b2">y</font></span>)</span></span>)</span> &amp; ( for <font color="Olive" title="b5">n</font>, <font color="Olive" title="b6">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b2">y</font> &amp; ( <font color="Olive" title="b4">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Olive" title="b6">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Olive" title="b6">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Olive" title="b4">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) holds <br/><font color="Maroon" title="c4">g</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b3">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Olive" title="b2">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Olive" title="b3">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Olive" title="b3">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b2">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b3">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Olive" title="b4">s</font> ) )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">S</font> =  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">T</font> = <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">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/>
<span class="kw">let </span><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font>, <font color="Maroon" title="c9">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">d</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b1">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Olive" title="b1">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">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="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  holds <br/> ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b1">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b1">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span> &amp; ( for <font color="Olive" title="b2">n</font>, <font color="Olive" title="b3">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">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="c8">y</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="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) holds <br/><font color="Maroon" title="c4">g</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">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="c8">y</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Olive" title="b1">s</font> ) ) )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c10">d</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><a NAME="E2:133"/><span class="lab"><font color="Green" title="E67">A1</font></span>: 
<font color="Maroon" title="c10">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> 
 <span class="kw">and </span><br/><a NAME="E3:133"/><span class="lab"><font color="Green" title="E68">A2</font></span>: 
<font color="Maroon" title="c10">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">and </span><br/><a NAME="E4:133"/><span class="lab"><font color="Green" title="E69">A3</font></span>: 
<font color="Maroon" title="c10">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">and </span><br/><a NAME="E5:133"/><span class="lab"><font color="Green" title="E70">A4</font></span>: 
<font color="Maroon" title="c10">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">s</font> being   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  holds <br/> ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b1">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b1">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span> &amp; ( for <font color="Olive" title="b2">n</font>, <font color="Olive" title="b3">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">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="c8">y</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="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) holds <br/><font color="Maroon" title="c4">g</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">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="c8">y</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Olive" title="b1">s</font> ) )</span><br/>

<a NAME="E6:133"/><span class="lab"><font color="Green" title="E71">A5</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:133"><span class="lab"><font color="Green" title="E69">A3</font></span></a>, <a class="txt" href="#E5:133"><span class="lab"><font color="Green" title="E70">A4</font></span></a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c11">s</font> be    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c11">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c11">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span> &amp; ( for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font> &amp; ( <font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) holds <br/><font color="Maroon" title="c4">g</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">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="c8">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="c11">s</font> ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c12">J</font> =  <a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span>;<br/>
<a NAME="E7:133"/><span class="lab"><font color="Green" title="E72">A6</font></span>: 
<font color="Maroon" title="c4">g</font> <a href="aofa_000.html#R2" title="AOFA_000:pred.2">complies_with_if_wrt</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">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 class="kw">by</span> <span class="lab"><a class="ref" href="aofa_000.html#D32" title="AOFA_000:def.32">AOFA_000:def 32</a></span>;<br/>
<a NAME="E8:133"/><span class="lab"><font color="Green" title="E73">A7</font></span>: 
<font color="Maroon" title="c9">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:133"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="txt" href="#E5:133"><span class="lab"><font color="Green" title="E70">A4</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">I</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p5">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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="c8">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span>;<br/>
<a NAME="E9:133"/><span class="lab"><font color="Green" title="E74">A8</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:133"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="txt" href="#E4:133"><span class="lab"><font color="Green" title="E69">A3</font></span></a></span>;<br/>
<div><span class="lab"><font color="Green" title="E75">A9</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">s</font> be    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span> )</span><br/><span class="kw">set </span><font color="Maroon" title="c15">s1</font> = <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p3">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c16">s2</font> = <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p4">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c17">q</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="c14">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p4">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c18">qz</font> = <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p5">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span>;<br/><a NAME="E1:133_1"/><span class="lab"><font color="Green" title="E76">A10</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">z</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">z</font></span>)</span>
 ;<br/><span class="kw">set </span><font color="Maroon" title="c19">s3</font> = <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p5">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">y</font></span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c20">s4</font> = <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span>;<br/><span class="lab"><font color="Green" title="E77">A11</font></span>: <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_000.html#D29" title="AOFA_000:def.29">AOFA_000:def 29</a></span>
<br/>.= 
<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_000.html#D29" title="AOFA_000:def.29">AOFA_000:def 29</a></span>
<br/>.= 
<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_000.html#D29" title="AOFA_000:def.29">AOFA_000:def 29</a></span>
;<br/><a NAME="E3:133_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p5">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p4">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</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="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/><a NAME="E4:133_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A12</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p5">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p4">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p5">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:133"><span class="lab"><font color="Green" title="E72">A6</font></span></a>, <a class="ref" href="aofa_000.html#D30" title="AOFA_000:def.30">AOFA_000:def 30</a></span>;<br/><a NAME="E5:133_1"/><span class="lab"><font color="Green" title="E79">A13</font></span>: 
<span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_i00.html#T22" target="_self" title="AOFA_I00:th.22">Th22</a></span>;<br/><a NAME="E6:133_1"/><span class="lab"><font color="Green" title="E80">A14</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p5">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:133"><span class="lab"><font color="Green" title="E71">A5</font></span></a>, <a class="ref" href="aofa_i00.html#T31" target="_self" title="AOFA_I00:th.31">Th31</a></span>;<br/><a NAME="E7:133_1"/><span class="lab"><font color="Green" title="E81">A15</font></span>: 
<span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_i00.html#T22" target="_self" title="AOFA_I00:th.22">Th22</a></span>;<br/><a NAME="E8:133_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p2">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">s</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_i00.html#D11" target="_self" title="AOFA_I00:def.11">Def11</a></span>;<br/><a NAME="E9:133_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A16</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p4">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</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="c9">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:133_1"><span class="lab"><font color="Green" title="E79">A13</font></span></a>, <a class="txt" href="#E7:133_1"><span class="lab"><font color="Green" title="E81">A15</font></span></a>, <a class="ref" href="aofa_i00.html#T26" target="_self" title="AOFA_I00:th.26">Th26</a></span>;<br/><a NAME="E10:133_1"/><span class="lab"><font color="Green" title="E83">A17</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p4">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</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="c9">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p5">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_i00.html#T38" target="_self" title="AOFA_I00:th.38">Th38</a></span>;<br/><a NAME="E11:133_1"/><span class="lab"><font color="Green" title="E84">A18</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p5">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p4">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</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="c9">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:133"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="txt" href="#E5:133"><span class="lab"><font color="Green" title="E70">A4</font></span></a>, <a class="ref" href="aofa_i00.html#T38" target="_self" title="AOFA_I00:th.38">Th38</a></span>;<br/><a NAME="E12:133_1"/><span class="lab"><font color="Green" title="E85">A19</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">z</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_i00.html#T31" target="_self" title="AOFA_I00:th.31">Th31</a></span>;<br/><a NAME="E13:133_1"/><span class="lab"><font color="Green" title="E86">A20</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p5">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:133"><span class="lab"><font color="Green" title="E73">A7</font></span></a>, <a class="ref" href="aofa_i00.html#T27" target="_self" title="AOFA_I00:th.27">Th27</a></span>;<br/><a NAME="E14:133_1"/><span class="lab"><font color="Green" title="E87">A21</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_i00.html#T27" target="_self" title="AOFA_I00:th.27">Th27</a></span>;<br/><a NAME="E15:133_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p5">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p4">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">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 class="kw">by</span> <span class="lab"><a class="ref" href="aofa_i00.html#T2" target="_self" title="AOFA_I00:th.2">Th2</a></span>;<br/><a NAME="E16:133_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A22</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p5">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p4">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p4">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:133"><span class="lab"><font color="Green" title="E72">A6</font></span></a>, <a class="ref" href="aofa_000.html#T80" title="AOFA_000:th.80">AOFA_000:80</a></span>;<br/><a NAME="E17:133_1"/><span class="lab"><font color="Green" title="E89">A23</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p4">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</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="c9">z</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>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p5">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_i00.html#T38" target="_self" title="AOFA_I00:th.38">Th38</a></span>;<br/><a NAME="E18:133_1"/><span class="lab"><font color="Green" title="E90">A24</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p4">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</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">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:133"><span class="lab"><font color="Green" title="E71">A5</font></span></a>, <a class="ref" href="aofa_i00.html#T26" target="_self" title="AOFA_I00:th.26">Th26</a></span>;<br/><a NAME="E19:133_1"/><span class="lab"><font color="Green" title="E91">A25</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p5">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</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">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_i00.html#T27" target="_self" title="AOFA_I00:th.27">Th27</a></span>;<br/><a NAME="E20:133_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p5">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p4">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</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">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:133"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="txt" href="#E4:133"><span class="lab"><font color="Green" title="E69">A3</font></span></a>, <a class="ref" href="aofa_i00.html#T38" target="_self" title="AOFA_I00:th.38">Th38</a></span>;<br/><span class="kw">hence </span><a NAME="E21:133_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:133"><span class="lab"><font color="Green" title="E74">A8</font></span></a>, <a class="txt" href="#E16:133_1"><span class="lab"><font color="Green" title="E88">A22</font></span></a>, <a class="txt" href="#E4:133_1"><span class="lab"><font color="Green" title="E78">A12</font></span></a>, <a class="txt" href="#E11:133_1"><span class="lab"><font color="Green" title="E84">A18</font></span></a>, <a class="txt" href="#E10:133_1"><span class="lab"><font color="Green" title="E83">A17</font></span></a>, <a class="txt" href="#E17:133_1"><span class="lab"><font color="Green" title="E89">A23</font></span></a>, <a class="txt" href="#E6:133_1"><span class="lab"><font color="Green" title="E80">A14</font></span></a>, <a class="txt" href="#E12:133_1"><span class="lab"><font color="Green" title="E85">A19</font></span></a>, <a class="txt" href="#E1:133_1"><span class="lab"><font color="Green" title="E76">A10</font></span></a>, <a class="txt" href="#E2:133_1"><span class="lab"><font color="Green" title="E77">A11</font></span></a>, <a class="txt" href="#E18:133_1"><span class="lab"><font color="Green" title="E90">A24</font></span></a>, <a class="txt" href="#E9:133_1"><span class="lab"><font color="Green" title="E82">A16</font></span></a>, <a class="txt" href="#E19:133_1"><span class="lab"><font color="Green" title="E91">A25</font></span></a>, <a class="txt" href="#E13:133_1"><span class="lab"><font color="Green" title="E86">A20</font></span></a>, <a class="txt" href="#E14:133_1"><span class="lab"><font color="Green" title="E87">A21</font></span></a>, <a class="ref" href="aofa_i00.html#T27" target="_self" title="AOFA_I00:th.27">Th27</a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E11:133"/>
( <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="c11">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c11">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c11">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font> &amp; ( <font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) holds <br/><font color="Maroon" title="c4">g</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">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="c8">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="c11">s</font></span><br/>

<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ) <span class="kw">-&gt; </span>   <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="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p1">(<span class="default">$1 <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default"><a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p2">(<span class="default">$1 <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">x</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2,<span class="p2">(<span class="default"><a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p3">(<span class="default">$1 <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">x</font></span>)</span>,<span class="p3">(<span class="default">$1 <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font></span>)</span>,1,<span class="p3">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <span class="p4">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p0">(<span class="default">$1 <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">$1 <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span></span>)</span></span>)</span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ] <span class="kw">means </span>( $1 <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">x</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>  &amp; $1 <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">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/>
<span class="kw">set </span><font color="Maroon" title="c14">C</font> = <font color="Maroon" title="c8">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/>
<a NAME="E12:133"/><span class="lab"><font color="Green" title="E76">A26</font></span>: 
for <font color="Olive" title="b1">s</font> being   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">s</font>] holds <br/>( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>] implies <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>] ) &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">s</font>) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">s</font> be    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">s</font>] implies ( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>] implies <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">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="c15">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>] ) &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">s</font>) ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:133_2"/><span class="lab"><font color="Green" title="E77">A27</font></span>: 
<font color="Maroon" title="c15">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E2:133_2"/><span class="lab"><font color="Green" title="E78">A28</font></span>: 
<font color="Maroon" title="c15">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>] implies <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">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="c15">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>] ) &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">s</font>) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c16">s9</font> = <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">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="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">s99</font> = <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c16">s9</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">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="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  ;<br/>
<a NAME="E5:133_2"/><span class="lab"><font color="Green" title="E79">A29</font></span>: 
<font color="Maroon" title="c16">s9</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:133"><span class="lab"><font color="Green" title="E75">A9</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">nx</font> = <font color="Maroon" title="c15">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c19">ny</font> = <font color="Maroon" title="c15">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font>, <font color="Maroon" title="c20">nn</font> = <font color="Maroon" title="c17">s99</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:133"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="txt" href="#E4:133"><span class="lab"><font color="Green" title="E69">A3</font></span></a>, <a class="txt" href="#E1:133_2"><span class="lab"><font color="Green" title="E77">A27</font></span></a>, <a class="txt" href="#E2:133_2"><span class="lab"><font color="Green" title="E78">A28</font></span></a>, <a class="ref" href="aofa_i00.html#T38" target="_self" title="AOFA_I00:th.38">Th38</a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></span>;<br/>
<a NAME="E7:133_2"/><span class="lab"><font color="Green" title="E80">A30</font></span>: 
<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c17">s99</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_000.html#D29" title="AOFA_000:def.29">AOFA_000:def 29</a></span>;<br/>
<a NAME="E8:133_2"/><span class="lab"><font color="Green" title="E81">A31</font></span>: 
<font color="Maroon" title="c17">s99</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">s9</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:133"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="txt" href="#E3:133"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="ref" href="aofa_i00.html#T38" target="_self" title="AOFA_I00:th.38">Th38</a></span>;<br/>
<a NAME="E9:133_2"/><span class="lab"><font color="Green" title="E82">A32</font></span>: 
( <font color="Maroon" title="c16">s9</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">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">s99</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>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_i00.html#T38" target="_self" title="AOFA_I00:th.38">Th38</a></span>;<br/>
<a NAME="E10:133_2"/><span class="lab"><font color="Green" title="E83">A33</font></span>: 
( <font color="Maroon" title="c16">s9</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">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">s99</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 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_i00.html#T38" target="_self" title="AOFA_I00:th.38">Th38</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:133_2"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>] iff <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E10:133"><span class="lab"><font color="Green" title="E75">A9</font></span></a>, <a class="txt" href="#E2:133_2"><span class="lab"><font color="Green" title="E78">A28</font></span></a>, <a class="txt" href="#E7:133_2"><span class="lab"><font color="Green" title="E80">A30</font></span></a>, <a class="txt" href="#E8:133_2"><span class="lab"><font color="Green" title="E81">A31</font></span></a>, <a class="txt" href="#E9:133_2"><span class="lab"><font color="Green" title="E82">A32</font></span></a>, <a class="ref" href="aofa_i00.html#T2" target="_self" title="AOFA_I00:th.2">Th2</a>, <a class="ref" href="aofa_i00.html#T38" target="_self" title="AOFA_I00:th.38">Th38</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">s</font>)</span><br/>

<a NAME="E12:133_2"/><span class="lab"><font color="Green" title="E84">A34</font></span>: 
<font color="Maroon" title="c16">s9</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:133"><span class="lab"><font color="Green" title="E75">A9</font></span></a></span>;<br/>
<a NAME="E13:133_2"/>
(  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">ny</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">nn</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">ny</font> or  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">ny</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">nn</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">nn</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T16" title="XXREAL_0:th.16">XXREAL_0:16</a></span>;<br/>
<a NAME="E14:133_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A35</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">s99</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c20">nn</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default"><a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c19">ny</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2,<span class="p2">(<span class="default"><a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c19">ny</font>,<font color="Maroon" title="c20">nn</font>,1,<span class="p3">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">ny</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">nn</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:133_2"><span class="lab"><font color="Green" title="E81">A31</font></span></a>, <a class="txt" href="#E12:133_2"><span class="lab"><font color="Green" title="E84">A34</font></span></a>, <a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></span>;<br/>
<a NAME="E15:133_2"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">ny</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">ny</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E16:133_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A36</font></span>: 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c18">nx</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">ny</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">ny</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T30" title="XXREAL_0:th.30">XXREAL_0:30</a></span>;<br/>
<a NAME="E17:133_2"/><span class="lab"><font color="Green" title="E87">A37</font></span>: 
<font color="Maroon" title="c17">s99</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">s9</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:133"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="txt" href="#E4:133"><span class="lab"><font color="Green" title="E69">A3</font></span></a>, <a class="ref" href="aofa_i00.html#T38" target="_self" title="AOFA_I00:th.38">Th38</a></span>;<br/>
<a NAME="E18:133_2"/>
(  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c18">nx</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">ny</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c18">nx</font> or  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c18">nx</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">ny</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">ny</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T16" title="XXREAL_0:th.16">XXREAL_0:16</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E88">A38</font></span>: <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">s</font>) = 
 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c19">ny</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default"><a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c18">nx</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2,<span class="p2">(<span class="default"><a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c18">nx</font>,<font color="Maroon" title="c19">ny</font>,1,<span class="p3">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c18">nx</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">ny</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></span>
<br/>.= 
 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c18">nx</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2,<span class="p1">(<span class="default"><a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c18">nx</font>,<font color="Maroon" title="c19">ny</font>,1,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c18">nx</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">ny</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:133_2"><span class="lab"><font color="Green" title="E78">A28</font></span></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="133_2_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:133_2_2"/>
( <font color="Maroon" title="c18">nx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">ny</font> or <font color="Maroon" title="c18">nx</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c19">ny</font> or ( <font color="Maroon" title="c18">nx</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">ny</font> &amp; <font color="Maroon" title="c18">nx</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or <font color="Maroon" title="c18">nx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="133_2_2_1"><span class="kw">suppose </span></a><a NAME="E1:133_2_2_1"/><span class="lab"><font color="Green" title="E89">A39</font></span>: 
<font color="Maroon" title="c18">nx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">ny</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">s</font>)</span><br/><div class="add"><a NAME="E2:133_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A40</font></span>: 
 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c18">nx</font>,<font color="Maroon" title="c19">ny</font>,1,<span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c18">nx</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">ny</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>;<br/><a NAME="E3:133_2_2_1"/><span class="lab"><font color="Green" title="E91">A41</font></span>: 
<font color="Maroon" title="c20">nn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:133_2"><span class="lab"><font color="Green" title="E87">A37</font></span></a>, <a class="txt" href="#E5:133_2"><span class="lab"><font color="Green" title="E79">A29</font></span></a>, <a class="txt" href="#E1:133_2_2_1"><span class="lab"><font color="Green" title="E89">A39</font></span></a>, <a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></span>;<br/><a NAME="E4:133_2_2_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">s</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c18">nx</font>,<font color="Maroon" title="c19">ny</font>,1,<span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c18">nx</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">ny</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:133_2"><span class="lab"><font color="Green" title="E78">A28</font></span></a>, <a class="txt" href="#E19:133_2"><span class="lab"><font color="Green" title="E88">A38</font></span></a>, <a class="txt" href="#E1:133_2_2_1"><span class="lab"><font color="Green" title="E89">A39</font></span></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>;<br/><span class="kw">hence </span><a NAME="E5:133_2_2_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">s</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:133_2"><span class="lab"><font color="Green" title="E80">A30</font></span></a>, <a class="txt" href="#E2:133_2_2_1"><span class="lab"><font color="Green" title="E90">A40</font></span></a>, <a class="txt" href="#E3:133_2_2_1"><span class="lab"><font color="Green" title="E91">A41</font></span></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="133_2_2_2"><span class="kw">suppose </span></a><a NAME="E1:133_2_2_2"/><span class="lab"><font color="Green" title="E89">A42</font></span>: 
<font color="Maroon" title="c18">nx</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c19">ny</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">s</font>)</span><br/><div class="add"><a NAME="E2:133_2_2_2"/><span class="kw">then </span>
 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c18">nx</font>,<font color="Maroon" title="c19">ny</font>,1,<span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c18">nx</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">ny</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c18">nx</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">ny</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>;<br/><a NAME="E3:133_2_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A43</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">s</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c18">nx</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">ny</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:133_2"><span class="lab"><font color="Green" title="E88">A38</font></span></a>, <a class="txt" href="#E1:133_2_2_2"><span class="lab"><font color="Green" title="E89">A42</font></span></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>;<br/><a NAME="E4:133_2_2_2"/><span class="lab"><font color="Green" title="E91">A44</font></span>: 
<font color="Maroon" title="c18">nx</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">ny</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:133_2_2_2"><span class="lab"><font color="Green" title="E89">A42</font></span></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A45</font></span>: <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">s99</font>) = 
 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c19">ny</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2,<span class="p1">(<span class="default"><a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c19">ny</font>,<font color="Maroon" title="c20">nn</font>,1,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">ny</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">nn</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:133_2"><span class="lab"><font color="Green" title="E87">A37</font></span></a>, <a class="txt" href="#E10:133_2"><span class="lab"><font color="Green" title="E83">A33</font></span></a>, <a class="txt" href="#E9:133_2"><span class="lab"><font color="Green" title="E82">A32</font></span></a>, <a class="txt" href="#E5:133_2"><span class="lab"><font color="Green" title="E79">A29</font></span></a>, <a class="txt" href="#E14:133_2"><span class="lab"><font color="Green" title="E85">A35</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>
<br/>.= 
 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c19">ny</font>,<font color="Maroon" title="c20">nn</font>,1,<span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">ny</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">nn</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:133_2"><span class="lab"><font color="Green" title="E78">A28</font></span></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>
;<br/><a NAME="E6:133_2_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A46</font></span>: 
( <font color="Maroon" title="c19">ny</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">nn</font> implies <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">s99</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>;<br/><a NAME="E7:133_2_2_2"/>
<font color="Maroon" title="c20">nn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">nx</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">ny</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:133_2"><span class="lab"><font color="Green" title="E87">A37</font></span></a>, <a class="txt" href="#E5:133_2"><span class="lab"><font color="Green" title="E79">A29</font></span></a>, <a class="txt" href="#E4:133_2_2_2"><span class="lab"><font color="Green" title="E91">A44</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E8:133_2_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">nn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">nx</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:133_2"><span class="lab"><font color="Green" title="E78">A28</font></span></a>, <a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">XREAL_1:46</a></span>;<br/><a NAME="E9:133_2_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">nn</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">nx</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E10:133_2_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A47</font></span>: 
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">nn</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c18">nx</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></span>;<br/><a NAME="E11:133_2_2_2"/><span class="lab"><font color="Green" title="E95">A48</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">nn</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E12:133_2_2_2"/><span class="lab"><font color="Green" title="E96">A49</font></span>: 
( <font color="Maroon" title="c19">ny</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c20">nn</font> implies <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">s99</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">ny</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">nn</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:133_2_2_2"><span class="lab"><font color="Green" title="E92">A45</font></span></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>;<br/><a NAME="E13:133_2_2_2"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">nn</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">nn</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><a NAME="E14:133_2_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">nn</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c18">nx</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:133_2_2_2"><span class="lab"><font color="Green" title="E94">A47</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E15:133_2_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">nn</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">s</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:133_2_2_2"><span class="lab"><font color="Green" title="E90">A43</font></span></a>, <a class="ref" href="xxreal_0.html#T30" title="XXREAL_0:th.30">XXREAL_0:30</a></span>;<br/><span class="kw">hence </span><a NAME="E16:133_2_2_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">s</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:133_2"><span class="lab"><font color="Green" title="E80">A30</font></span></a>, <a class="txt" href="#E16:133_2"><span class="lab"><font color="Green" title="E86">A36</font></span></a>, <a class="txt" href="#E3:133_2_2_2"><span class="lab"><font color="Green" title="E90">A43</font></span></a>, <a class="txt" href="#E6:133_2_2_2"><span class="lab"><font color="Green" title="E93">A46</font></span></a>, <a class="txt" href="#E12:133_2_2_2"><span class="lab"><font color="Green" title="E96">A49</font></span></a>, <a class="txt" href="#E11:133_2_2_2"><span class="lab"><font color="Green" title="E95">A48</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="ref" href="xxreal_0.html#T29" title="XXREAL_0:th.29">XXREAL_0:29</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="133_2_2_3"><span class="kw">suppose </span></a><a NAME="E1:133_2_2_3"/><span class="lab"><font color="Green" title="E89">A50</font></span>: 
( <font color="Maroon" title="c18">nx</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">ny</font> &amp; <font color="Maroon" title="c18">nx</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">s</font>)</span><br/><div class="add"><a NAME="E2:133_2_2_3"/><span class="lab"><font color="Green" title="E90">A51</font></span>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">nx</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">ny</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">ny</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">nx</font>
 ;<br/><a NAME="E3:133_2_2_3"/><span class="lab"><font color="Green" title="E91">A52</font></span>: 
<font color="Maroon" title="c18">nx</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">ny</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:133_2_2_3"><span class="lab"><font color="Green" title="E89">A50</font></span></a>, <a class="ref" href="xreal_1.html#T51" title="XREAL_1:th.51">XREAL_1:51</a></span>;<br/><a NAME="E4:133_2_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A53</font></span>: 
<font color="Maroon" title="c20">nn</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="c18">nx</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">ny</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:133_2"><span class="lab"><font color="Green" title="E87">A37</font></span></a>, <a class="txt" href="#E5:133_2"><span class="lab"><font color="Green" title="E79">A29</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E5:133_2_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A54</font></span>: 
<font color="Maroon" title="c20">nn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">ny</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:133_2_2_3"><span class="lab"><font color="Green" title="E89">A50</font></span></a>, <a class="txt" href="#E2:133_2_2_3"><span class="lab"><font color="Green" title="E90">A51</font></span></a>, <a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">XREAL_1:46</a></span>;<br/><a NAME="E6:133_2_2_3"/>
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">nn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">ny</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:133_2_2_3"><span class="lab"><font color="Green" title="E89">A50</font></span></a>, <a class="txt" href="#E4:133_2_2_3"><span class="lab"><font color="Green" title="E92">A53</font></span></a>, <a class="txt" href="#E2:133_2_2_3"><span class="lab"><font color="Green" title="E90">A51</font></span></a>, <a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">XREAL_1:46</a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></span>;<br/><a NAME="E7:133_2_2_3"/><span class="kw">then </span>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">nn</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">ny</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E8:133_2_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A55</font></span>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">nn</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c18">nx</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">ny</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T30" title="XXREAL_0:th.30">XXREAL_0:30</a></span>;<br/><a NAME="E9:133_2_2_3"/>
<font color="Maroon" title="c20">nn</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:133_2"><span class="lab"><font color="Green" title="E87">A37</font></span></a>, <a class="txt" href="#E5:133_2"><span class="lab"><font color="Green" title="E79">A29</font></span></a>, <a class="txt" href="#E3:133_2_2_3"><span class="lab"><font color="Green" title="E91">A52</font></span></a>, <a class="txt" href="#E2:133_2_2_3"><span class="lab"><font color="Green" title="E90">A51</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A56</font></span>: <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">s99</font>) = 
 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c19">ny</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2,<span class="p1">(<span class="default"><a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c19">ny</font>,<font color="Maroon" title="c20">nn</font>,1,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">ny</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">nn</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:133_2"><span class="lab"><font color="Green" title="E85">A35</font></span></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>
<br/>.= 
 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c19">ny</font>,<font color="Maroon" title="c20">nn</font>,1,<span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">ny</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">nn</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:133_2"><span class="lab"><font color="Green" title="E78">A28</font></span></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>
<br/>.= 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">ny</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">nn</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:133_2_2_3"><span class="lab"><font color="Green" title="E93">A54</font></span></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>
;<br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">s</font>) = 
 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c18">nx</font>,<font color="Maroon" title="c19">ny</font>,1,<span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c18">nx</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">ny</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:133_2"><span class="lab"><font color="Green" title="E88">A38</font></span></a>, <a class="txt" href="#E1:133_2_2_3"><span class="lab"><font color="Green" title="E89">A50</font></span></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>
<br/>.= 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c18">nx</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">ny</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:133_2_2_3"><span class="lab"><font color="Green" title="E89">A50</font></span></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>
;<br/><span class="kw">hence </span><a NAME="E12:133_2_2_3"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">s</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:133_2"><span class="lab"><font color="Green" title="E80">A30</font></span></a>, <a class="txt" href="#E16:133_2"><span class="lab"><font color="Green" title="E86">A36</font></span></a>, <a class="txt" href="#E8:133_2_2_3"><span class="lab"><font color="Green" title="E94">A55</font></span></a>, <a class="txt" href="#E10:133_2_2_3"><span class="lab"><font color="Green" title="E95">A56</font></span></a>, <a class="ref" href="xxreal_0.html#T29" title="XXREAL_0:th.29">XXREAL_0:29</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="133_2_2_4"><span class="kw">suppose </span></a><a NAME="E1:133_2_2_4"/><span class="lab"><font color="Green" title="E89">A57</font></span>: 
<font color="Maroon" title="c18">nx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">s</font>)</span><br/><div class="add"><a NAME="E2:133_2_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A58</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">s</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:133_2"><span class="lab"><font color="Green" title="E88">A38</font></span></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>;<br/><span class="lab"><font color="Green" title="E91">A59</font></span>: <font color="Maroon" title="c20">nn</font> = 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">nx</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">ny</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:133_2"><span class="lab"><font color="Green" title="E78">A28</font></span></a>, <a class="txt" href="#E17:133_2"><span class="lab"><font color="Green" title="E87">A37</font></span></a>, <a class="txt" href="#E5:133_2"><span class="lab"><font color="Green" title="E79">A29</font></span></a>, <a class="txt" href="#E1:133_2_2_4"><span class="lab"><font color="Green" title="E89">A57</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c19">ny</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:133_2_2_4"><span class="lab"><font color="Green" title="E89">A57</font></span></a></span>
;<br/><span class="kw">then </span><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">s99</font>) = 
 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c19">ny</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2,<span class="p1">(<span class="default"><a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c19">ny</font>,<font color="Maroon" title="c20">nn</font>,1,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">ny</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">nn</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:133_2"><span class="lab"><font color="Green" title="E78">A28</font></span></a>, <a class="txt" href="#E14:133_2"><span class="lab"><font color="Green" title="E85">A35</font></span></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>
<br/>.= 
 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c19">ny</font>,<font color="Maroon" title="c20">nn</font>,1,<span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">ny</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">nn</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:133_2"><span class="lab"><font color="Green" title="E78">A28</font></span></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:133_2_2_4"><span class="lab"><font color="Green" title="E91">A59</font></span></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:133_2_2_4"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="aofa_i00.html#K78" title="AOFA_I00:func.78">gt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">s</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:133_2"><span class="lab"><font color="Green" title="E80">A30</font></span></a>, <a class="txt" href="#E2:133_2_2_4"><span class="lab"><font color="Green" title="E90">A58</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c15">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="c11">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="c8">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="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span></span>)</span> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">s1</font> = <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c11">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">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="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  ;<br/>
<span class="kw">let </span><font color="Maroon" title="c17">n</font>, <font color="Maroon" title="c18">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c18">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font> &amp; ( <font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c18">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Maroon" title="c18">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) implies <font color="Maroon" title="c4">g</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">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="c8">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="c11">s</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E15:133"/><span class="lab"><font color="Green" title="E77">A60</font></span>: 
<font color="Maroon" title="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">and </span><br/><a NAME="E16:133"/><span class="lab"><font color="Green" title="E78">A61</font></span>: 
<font color="Maroon" title="c18">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; not <font color="Maroon" title="c18">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <font color="Maroon" title="c18">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; not <font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or <font color="Maroon" title="c4">g</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">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="c8">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="c11">s</font> )</span><br/>

<span class="kw">assume </span><a NAME="E17:133"/>
( <font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">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="c18">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">g</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">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="c8">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="c11">s</font></span><br/>

<a NAME="E18:133"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A62</font></span>: 
( <font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">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="c11">s</font>] )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:133"><span class="lab"><font color="Green" title="E77">A60</font></span></a>, <a class="txt" href="#E16:133"><span class="lab"><font color="Green" title="E78">A61</font></span></a></span>;<br/>
<a NAME="E19:133"/>
<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="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">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="c8">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="c11">s</font>
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="aofa_000.html#S3" title="AOFA_000:sch.3">AOFA_000:sch 3</a>(<a class="txt" href="#E18:133"><span class="lab"><font color="Green" title="E79">A62</font></span></a>, <a class="txt" href="#E12:133"><span class="lab"><font color="Green" title="E76">A26</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E20:133"/>
<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="c9">z</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="aofa_i00.html#K32" title="AOFA_I00:func.32">-</a> <span class="p0">(<span class="default"><a href="aofa_i00.html#K42" title="AOFA_I00:func.42">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><a href="aofa_000.html#K15" title="AOFA_000:func.15">if-then</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K79" title="AOFA_I00:func.79">lt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="aofa_i00.html#K58" title="AOFA_I00:func.58">*=</a> <span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">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">y</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">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="c8">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="c11">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ] <span class="kw">means </span>( <font color="Maroon" title="c17">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c18">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="c7">x</font> &amp; <font color="Maroon" title="c17">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c18">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">y</font> &amp; $1 <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; $1 <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">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>  &amp; ( for <font color="Olive" title="b1">c</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">c</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="c7">x</font> &amp; <font color="Olive" title="b1">c</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">y</font> holds <br/><font color="Olive" title="b1">c</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c17">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c18">m</font> ) );<br/>

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