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<b>let </b><font color="Maroon" title="c1">A</font> be  <a href="aofa_i00.html#V2" title="AOFA_I00:attr.2">Euclidean</a> <a href="aofa_000.html#NM1" title="AOFA_000:NM.1">preIfWhileAlgebra</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="card_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="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Olive" title="b1">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Olive" title="b1">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Olive" title="b2">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <br/> for <font color="Olive" title="b4">n</font>, <font color="Olive" title="b5">x</font>, <font color="Olive" title="b6">y</font>, <font color="Olive" title="b7">z</font>, <font color="Olive" title="b8">i</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="b9">d</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b9">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="b9">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b9">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Olive" title="b9">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Olive" title="b9">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 &amp; <font color="Olive" title="b9">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 ) holds <br/>for <font color="Olive" title="b9">s</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Olive" title="b1">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> <br/> for <font color="Olive" title="b10">N</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="b10">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b9">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b4">n</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b3">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b9">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b5">x</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Olive" title="b6">y</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</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#K17" title="AOFA_000:func.17">for-do</a> <span class="p4">(<span class="default"><font color="Olive" title="b8">i</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b4">n</font></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b8">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Olive" title="b7">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b5">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p0">(<span class="default"><font color="Olive" title="b5">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b6">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Olive" title="b6">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Olive" title="b7">z</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <font color="Olive" title="b10">N</font></span><br/><b>let </b><font color="Maroon" title="c2">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="card_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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font><br/> for <font color="Olive" title="b2">g</font> being <a href="aofa_i00.html#V1" title="AOFA_I00:attr.1">Euclidean</a>  <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>, <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Olive" title="b1">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <br/> for <font color="Olive" title="b3">n</font>, <font color="Olive" title="b4">x</font>, <font color="Olive" title="b5">y</font>, <font color="Olive" title="b6">z</font>, <font color="Olive" title="b7">i</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="b8">d</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b8">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="b8">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b8">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> 2 &amp; <font color="Olive" title="b8">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> 3 &amp; <font color="Olive" title="b8">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> 4 &amp; <font color="Olive" title="b8">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 ) holds <br/>for <font color="Olive" title="b8">s</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> <br/> for <font color="Olive" title="b9">N</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="b9">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b3">n</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b8">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b4">x</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Olive" title="b5">y</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</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#K17" title="AOFA_000:func.17">for-do</a> <span class="p4">(<span class="default"><font color="Olive" title="b7">i</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b3">n</font></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Olive" title="b6">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b4">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p0">(<span class="default"><font color="Olive" title="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="p5">(<span class="default"><font color="Olive" title="b5">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Olive" title="b6">z</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <font color="Olive" title="b9">N</font></span><br/><b>let </b><font color="Maroon" title="c3">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">g</font> being <a href="aofa_i00.html#V1" title="AOFA_I00:attr.1">Euclidean</a>  <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>, <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <br/> for <font color="Olive" title="b2">n</font>, <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font>, <font color="Olive" title="b5">z</font>, <font color="Olive" title="b6">i</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="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="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="b7">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</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="b3">x</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="b4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &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">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 &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">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 ) holds <br/>for <font color="Olive" title="b7">s</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> <br/> for <font color="Olive" title="b8">N</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="b2">n</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b7">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b3">x</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Olive" title="b4">y</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</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#K17" title="AOFA_000:func.17">for-do</a> <span class="p4">(<span class="default"><font color="Olive" title="b6">i</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b6">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b2">n</font></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b6">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Olive" title="b5">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p0">(<span class="default"><font color="Olive" title="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="p5">(<span class="default"><font color="Olive" title="b4">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Olive" title="b5">z</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <font color="Olive" title="b8">N</font></span><br/><b>let </b><font color="Maroon" title="c4">g</font> be  <a href="aofa_i00.html#V1" title="AOFA_I00:attr.1">Euclidean</a>  <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>, <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font>, <font color="Olive" title="b4">z</font>, <font color="Olive" title="b5">i</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="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="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="b6">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</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="b2">x</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="b3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &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">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 &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">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 ) holds <br/>for <font color="Olive" title="b6">s</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> <br/> for <font color="Olive" title="b7">N</font> 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="b1">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b6">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b2">x</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Olive" title="b3">y</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</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#K17" title="AOFA_000:func.17">for-do</a> <span class="p4">(<span class="default"><font color="Olive" title="b5">i</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b5">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b5">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Olive" title="b4">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p0">(<span class="default"><font color="Olive" title="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="p5">(<span class="default"><font color="Olive" title="b3">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Olive" title="b4">z</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <font color="Olive" title="b7">N</font></span><br/>







<b>set </b><font color="Maroon" title="c5">S</font> =  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ;<br/>
<b>set </b><font color="Maroon" title="c6">T</font> = <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<b>let </b><font color="Maroon" title="c7">n</font>, <font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">y</font>, <font color="Maroon" title="c10">z</font>, <font color="Maroon" title="c11">i</font> be    <a href="aofa_i00.html#M1" title="AOFA_I00:mode.1">Variable</a> of <font color="Maroon" title="c4">g</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">d</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</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">x</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">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Olive" title="b1">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 &amp; <font color="Olive" title="b1">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 ) implies for <font color="Olive" title="b1">s</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> <br/> for <font color="Olive" title="b2">N</font> 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">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b1">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</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#K17" title="AOFA_000:func.17">for-do</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c7">n</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <font color="Olive" title="b2">N</font> )</span><br/>

<b>given </b><font color="Maroon" title="c12">d</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><a NAME="E2:129"/><i><font color="Green" title="E66">A0</font></i>: 
( <font color="Maroon" title="c12">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c12">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c12">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c12">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c12">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 &amp; <font color="Maroon" title="c12">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">s</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> <br/> for <font color="Olive" title="b2">N</font> 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">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b1">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</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#K17" title="AOFA_000:func.17">for-do</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c7">n</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <font color="Olive" title="b2">N</font></span><br/>

<b>let </b><font color="Maroon" title="c13">s</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">N</font> 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="c13">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c13">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</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#K17" title="AOFA_000:func.17">for-do</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c7">n</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <font color="Olive" title="b1">N</font></span><br/><b>let </b><font color="Maroon" title="c14">N</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">n</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c13">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</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#K17" title="AOFA_000:func.17">for-do</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c7">n</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <font color="Maroon" title="c14">N</font> )</span><br/>



<b>assume </b><a NAME="E3:129"/><i><font color="Green" title="E67">A1</font></i>: 
<font color="Maroon" title="c14">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c13">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</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#K17" title="AOFA_000:func.17">for-do</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c7">n</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <font color="Maroon" title="c14">N</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c15">s1</font> = <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c13">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c16">s2</font> = <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c15">s1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c17">s3</font> = <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c16">s2</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K17" title="AOFA_000:func.17">for-do</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c7">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c18">s4</font> = <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c16">s2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  ;<br/>
<b>set </b><font color="Maroon" title="c19">I</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">a</font> =  <a href="aofa_i00.html#K43" title="AOFA_I00:func.43">.</a> 1,<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">g</font> as    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">g</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c21">F</font> =  <a href="aofa_000.html#K17" title="AOFA_000:func.17">for-do</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <font color="Maroon" title="c20">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c7">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/>
<a NAME="E10:129"/>
<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c13">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c16">s2</font>
 
<b>by </b><i><a class="ref" href="aofa_000.html#D29" title="AOFA_000:def.29">AOFA_000:def 29</a></i>;<br/>
<a NAME="E11:129"/><b>then </b><i><font color="Green" title="E68">S0</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c13">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c21">F</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c17">s3</font>
 
<b>by </b><i><a class="ref" href="aofa_000.html#D29" title="AOFA_000:def.29">AOFA_000:def 29</a></i>;<br/>
<a NAME="E12:129"/><i><font color="Green" title="E69">D0</font></i>: 
( <font color="Maroon" title="c7">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c11">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c7">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c11">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c11">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">n</font> &amp; <font color="Maroon" title="c7">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">z</font> &amp; <font color="Maroon" title="c11">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">z</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">z</font> &amp; <font color="Maroon" title="c9">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">z</font> )
 
<b>by </b><i><a class="txt" href="#E2:129"><i><font color="Green" title="E66">A0</font></i></a></i>;<br/>
<a NAME="E13:129"/><b>then </b><i><font color="Green" title="E70">S1</font></i>: 
( <font color="Maroon" title="c15">s1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">n</font> &amp; <font color="Maroon" title="c15">s1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="aofa_i00.html#T25" target="_self" title="AOFA_I00:th.25">Th011</a></i>;<br/>
<a NAME="E14:129"/><i><font color="Green" title="E71">S2</font></i>: 
( <font color="Maroon" title="c16">s2</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">s1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">n</font> &amp; <font color="Maroon" title="c16">s2</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c16">s2</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">s1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> )
 
<b>by </b><i><a class="txt" href="#E12:129"><i><font color="Green" title="E69">D0</font></i></a>, <a class="ref" href="aofa_i00.html#T25" target="_self" title="AOFA_I00:th.25">Th011</a></i>;<br/>
<a NAME="E15:129"/><i><font color="Green" title="E72">S3</font></i>: 
( <font color="Maroon" title="c18">s4</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">s2</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">n</font> &amp; <font color="Maroon" title="c18">s4</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">s2</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c18">s4</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">s2</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c18">s4</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="#E12:129"><i><font color="Green" title="E69">D0</font></i></a>, <a class="ref" href="aofa_i00.html#T25" target="_self" title="AOFA_I00:th.25">Th011</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ] <b>means </b> ex <font color="Olive" title="b1">N</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 <br/>( <font color="Olive" title="b1">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">$1 <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; $1 <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <font color="Olive" title="b1">N</font> &amp; $1 <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">N</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> );<br/>
<a NAME="E16:129"/><i><font color="Green" title="E73">F</font></i>: 
<font color="Maroon" title="c21">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aofa_000.html#K17" title="AOFA_000:func.17">for-do</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <font color="Maroon" title="c20">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c7">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E17:129"/>
( 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
;<br/>
<a NAME="E18:129"/><b>then </b><i><font color="Green" title="E74">A</font></i>: 
<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="c16">s2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <font color="Maroon" title="c20">a</font></span>)</span>]
 
<b>by </b><i><a class="txt" href="#E13:129"><i><font color="Green" title="E70">S1</font></i></a>, <a class="txt" href="#E14:129"><i><font color="Green" title="E71">S2</font></i></a>, <a class="txt" href="#E15:129"><i><font color="Green" title="E72">S3</font></i></a>, <a class="ref" href="pre_ff.html#T1" title="PRE_FF:th.1">PRE_FF:1</a></i>;<br/>
<div><i><font color="Green" title="E75">Si</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="129_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c22">s</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c22">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">i</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="c22">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">n</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="c22">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c22">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font></span>)</span> )</span><br/><b>reconsider </b><font color="Maroon" title="c23">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="c22">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  ;<br/><b>reconsider </b><font color="Maroon" title="c24">s2</font> = <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c23">s1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  ;<br/><b>reconsider </b><font color="Maroon" title="c25">s3</font> = <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c24">s2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  ;<br/><a NAME="E4:129_1"/><i><font color="Green" title="E76">R1</font></i>: 
( <font color="Maroon" title="c23">s1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">n</font> &amp; <font color="Maroon" title="c23">s1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c23">s1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c23">s1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c10">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c23">s1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">i</font> )
 <b>by </b><i><a class="txt" href="#E12:129"><i><font color="Green" title="E69">D0</font></i></a>, <a class="ref" href="aofa_i00.html#T27" target="_self" title="AOFA_I00:th.27">Th211</a></i>;<br/><a NAME="E5:129_1"/><i><font color="Green" title="E77">R2</font></i>: 
( <font color="Maroon" title="c24">s2</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">s1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">n</font> &amp; <font color="Maroon" title="c24">s2</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">s1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c24">s2</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">s1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c24">s2</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c10">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">s1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c10">z</font> &amp; <font color="Maroon" title="c24">s2</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">s1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">i</font> )
 <b>by </b><i><a class="txt" href="#E12:129"><i><font color="Green" title="E69">D0</font></i></a>, <a class="ref" href="aofa_i00.html#T27" target="_self" title="AOFA_I00:th.27">Th211</a></i>;<br/><i><font color="Green" title="E78">R4</font></i>: <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c22">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c22">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span>
<b>by </b><i><a class="ref" href="aofa_000.html#D29" title="AOFA_000:def.29">AOFA_000:def 29</a></i>
<br/>.= 
<font color="Maroon" title="c25">s3</font>
<b>by </b><i><a class="ref" href="aofa_000.html#D29" title="AOFA_000:def.29">AOFA_000:def 29</a></i>
;<br/><b>thus </b><a NAME="E7:129_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="c22">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">i</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="c22">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">n</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="c22">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c22">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E4:129_1"><i><font color="Green" title="E76">R1</font></i></a>, <a class="txt" href="#E5:129_1"><i><font color="Green" title="E77">R2</font></i></a>, <a class="txt" href="#E12:129"><i><font color="Green" title="E69">D0</font></i></a>, <a class="txt" href="#E6:129_1"><i><font color="Green" title="E78">R4</font></i></a>, <a class="ref" href="aofa_i00.html#T30" target="_self" title="AOFA_I00:th.30">Th212</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E20:129"/><i><font color="Green" title="E76">B</font></i>: 
for <font color="Olive" title="b1">s</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">s</font>] holds <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c7">n</font></span>)</span>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="129_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c22">s</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c22">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="c22">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c22">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c7">n</font></span>)</span>] ) )</span><br/>

<b>given </b><font color="Maroon" title="c23">N</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> <b> such that </b><a NAME="E2:129_2"/><i><font color="Green" title="E77">Q0</font></i>: 
( <font color="Maroon" title="c23">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon" title="c22">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <font color="Maroon" title="c23">N</font> &amp; <font color="Maroon" title="c22">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">N</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c22">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c22">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c7">n</font></span>)</span>] )</span><br/>

<b>reconsider </b><font color="Maroon" title="c24">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="c22">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c25">s2</font> = <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c24">s1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  ;<br/>
<a NAME="E5:129_2"/><i><font color="Green" title="E78">Q1</font></i>: 
( <font color="Maroon" title="c24">s1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c24">s1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font></span>)</span> &amp; <font color="Maroon" title="c24">s1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">i</font> )
 
<b>by </b><i><a class="txt" href="#E19:129"><i><font color="Green" title="E75">Si</font></i></a></i>;<br/>
<a NAME="E6:129_2"/><i><font color="Green" title="E79">Q2</font></i>: 
( <font color="Maroon" title="c25">s2</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">s1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c25">s2</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">s1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c25">s2</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">s1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 )
 
<b>by </b><i><a class="txt" href="#E12:129"><i><font color="Green" title="E69">D0</font></i></a>, <a class="ref" href="aofa_i00.html#T28" target="_self" title="AOFA_I00:th.28">Th012</a></i>;<br/>
<b>thus </b><a NAME="E7:129_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="c22">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c22">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c7">n</font></span>)</span>]</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="129_2_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon" title="c26">N1</font> = <font color="Maroon" title="c23">N</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c26">N1</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> <font color="Maroon" title="c22">s</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c22">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</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">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <font color="Maroon" title="c26">N1</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="c22">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</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">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">N1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><br/>

<a NAME="E1:129_2_1"/>
( <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c22">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c25">s2</font> &amp; <font color="Maroon" title="c26">N1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">N</font> )
 
<b>by </b><i><a class="ref" href="aofa_000.html#D29" title="AOFA_000:def.29">AOFA_000:def 29</a></i>;<br/>
<b>hence </b><a NAME="E2:129_2_1"/>
( <font color="Maroon" title="c26">N1</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> <font color="Maroon" title="c22">s</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c22">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</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">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <font color="Maroon" title="c26">N1</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="c22">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</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">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">N1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="#E2:129_2"><i><font color="Green" title="E77">Q0</font></i></a>, <a class="txt" href="#E5:129_2"><i><font color="Green" title="E78">Q1</font></i></a>, <a class="txt" href="#E6:129_2"><i><font color="Green" title="E79">Q2</font></i></a>, <a class="ref" href="pre_ff.html#T1" title="PRE_FF:th.1">PRE_FF:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>take </b>
<font color="Maroon" title="c23">N</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c23">N</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> <font color="Maroon" title="c22">s</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c22">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <font color="Maroon" title="c23">N</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="c22">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">N</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><br/>

<b>thus </b><a NAME="E8:129_2"/>
( <font color="Maroon" title="c23">N</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> <font color="Maroon" title="c22">s</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c22">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <font color="Maroon" title="c23">N</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="c22">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">N</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="#E2:129_2"><i><font color="Green" title="E77">Q0</font></i></a>, <a class="txt" href="#E2:129"><i><font color="Green" title="E66">A0</font></i></a>, <a class="ref" href="aofa_i00.html#T35" target="_self" title="AOFA_I00:th.35">Th114</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E21:129"/><i><font color="Green" title="E77">C</font></i>: 
for <font color="Olive" title="b1">s</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">s</font>] holds <br/>( <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"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">i</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="c11">i</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"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">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">n</font> )
 
<b>by </b><i><a class="txt" href="#E19:129"><i><font color="Green" title="E75">Si</font></i></a></i>;<br/>
<a NAME="E22:129"/><i><font color="Green" title="E78">D</font></i>: 
( <font color="Maroon" title="c7">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c7">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c11">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> )
 
<b>by </b><i><a class="txt" href="#E2:129"><i><font color="Green" title="E66">A0</font></i></a></i>;<br/>
<a NAME="E23:129"/><i><font color="Green" title="E79">E</font></i>: 
( <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="c16">s2</font>,<font color="Maroon" title="c21">F</font>] &amp; ( <font color="Maroon" title="c20">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">s2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">s2</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">n</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c16">s2</font>,<font color="Maroon" title="c21">F</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">s2</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 ) &amp; ( <font color="Maroon" title="c20">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">s2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c16">s2</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">n</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c16">s2</font>,<font color="Maroon" title="c21">F</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">s2</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="c16">s2</font>,<font color="Maroon" title="c21">F</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">s2</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">n</font> )
 
<b>from </b><i><a class="ref" href="aofa_i00.html#S1" target="_self" title="AOFA_I00:sch.1">AOFA_I00:sch 1</a>(<a class="txt" href="#E16:129"><i><font color="Green" title="E73">F</font></i></a>, <a class="txt" href="#E18:129"><i><font color="Green" title="E74">A</font></i></a>, <a class="txt" href="#E20:129"><i><font color="Green" title="E76">B</font></i></a>, <a class="txt" href="#E21:129"><i><font color="Green" title="E77">C</font></i></a>, <a class="txt" href="#E22:129"><i><font color="Green" title="E78">D</font></i></a>);<br/></i>
<a NAME="E24:129"/><i><font color="Green" title="E80">E2</font></i>: 
( (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">N</font> &amp; <font color="Maroon" title="c14">N</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>  ) or <font color="Maroon" title="c14">N</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>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E25:129"/>
<font color="Maroon" title="c20">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<b>hence </b><a NAME="E26:129"/>
<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="c13">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</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#K17" title="AOFA_000:func.17">for-do</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c7">n</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <font color="Maroon" title="c14">N</font>
 <b>by </b><i><a class="txt" href="#E3:129"><i><font color="Green" title="E67">A1</font></i></a>, <a class="txt" href="#E11:129"><i><font color="Green" title="E68">S0</font></i></a>, <a class="txt" href="#E12:129"><i><font color="Green" title="E69">D0</font></i></a>, <a class="ref" href="aofa_i00.html#T25" target="_self" title="AOFA_I00:th.25">Th011</a>, <a class="txt" href="#E14:129"><i><font color="Green" title="E71">S2</font></i></a>, <a class="txt" href="#E23:129"><i><font color="Green" title="E79">E</font></i></a>, <a class="txt" href="#E24:129"><i><font color="Green" title="E80">E2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
