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<b>let </b><font color="Maroon" title="c1">A</font> be  <a href="aofa_i00.html#V2" title="AOFA_I00:attr.2">Euclidean</a> <a href="aofa_000.html#NM1" title="AOFA_000:NM.1">preIfWhileAlgebra</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">X</font><br/> for <font color="Olive" title="b3">g</font> being <a href="aofa_i00.html#V1" title="AOFA_I00:attr.1">Euclidean</a>  <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>, <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Olive" title="b1">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Olive" title="b1">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Olive" title="b2">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <br/> for <font color="Olive" title="b4">n</font>, <font color="Olive" title="b5">s</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="b3">g</font>  st  ex <font color="Olive" title="b7">d</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b7">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">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="b5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Olive" title="b7">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">i</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="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 ) holds <br/>for <font color="Olive" title="b7">q</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Olive" title="b1">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> <br/> for <font color="Olive" title="b8">N</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b8">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">q</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="b7">q</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b5">s</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="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> 2</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="b4">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"><font color="Olive" title="b5">s</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Olive" title="b6">i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">N</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span><br/><b>let </b><font color="Maroon" title="c2">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font><br/> for <font color="Olive" title="b2">g</font> being <a href="aofa_i00.html#V1" title="AOFA_I00:attr.1">Euclidean</a>  <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>, <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Olive" title="b1">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <br/> for <font color="Olive" title="b3">n</font>, <font color="Olive" title="b4">s</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="Olive" title="b2">g</font>  st  ex <font color="Olive" title="b6">d</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b6">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">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="b4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Olive" title="b6">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</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="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 ) holds <br/>for <font color="Olive" title="b6">q</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="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b7">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">q</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="b6">q</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b4">s</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="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> 2</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="b3">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"><font color="Olive" title="b4">s</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Olive" title="b5">i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">N</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </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">s</font>, <font color="Olive" title="b4">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="b5">d</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b5">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b5">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Olive" title="b5">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Olive" title="b5">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 ) holds <br/>for <font color="Olive" title="b5">q</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> <br/> for <font color="Olive" title="b6">N</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b6">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">q</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="b5">q</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b3">s</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="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="b4">i</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 2</span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b4">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="b4">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b3">s</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Olive" title="b4">i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">N</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </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">s</font>, <font color="Olive" title="b3">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="b4">d</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b4">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b4">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Olive" title="b4">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Olive" title="b4">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 ) holds <br/>for <font color="Olive" title="b4">q</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> <br/> for <font color="Olive" title="b5">N</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b5">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">q</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="b4">q</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b2">s</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="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="b3">i</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 2</span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b3">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="b3">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b2">s</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Olive" title="b3">i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">N</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span><br/>







<b>set </b><font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c4">g</font>;<br/>
<b>let </b><font color="Maroon" title="c6">n</font>, <font color="Maroon" title="c7">s</font>, <font color="Maroon" title="c8">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="c6">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="c7">s</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="c8">i</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="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 ) implies for <font color="Olive" title="b1">q</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="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b2">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c6">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">q</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="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="c8">i</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 2</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c6">n</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c8">i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">N</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a>  )</span><br/>

<b>given </b><font color="Maroon" title="c9">d</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><a NAME="E2:126"/><i><font color="Green" title="E67">A0</font></i>: 
( <font color="Maroon" title="c9">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c9">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c9">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c9">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> 4 )
 ; <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">q</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="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b2">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c6">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">q</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="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="c8">i</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 2</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c6">n</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c8">i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">N</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span><br/>

<b>let </b><font color="Maroon" title="c10">q</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="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c6">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="c10">q</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="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="c8">i</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 2</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c6">n</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c8">i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">N</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span><br/><b>let </b><font color="Maroon" title="c11">N</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c6">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="c10">q</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="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="c8">i</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 2</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c6">n</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c8">i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">N</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a>  )</span><br/>



<b>assume </b><a NAME="E3:126"/><i><font color="Green" title="E68">A1</font></i>: 
<font color="Maroon" title="c11">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c6">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="c10">q</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="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="c8">i</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 2</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c6">n</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c8">i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">N</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span><br/>

<b>set </b><font color="Maroon" title="c12">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="c13">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>reconsider </b><font color="Maroon" title="c14">q1</font> = <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c10">q</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">s</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="c15">q2</font> = <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">q1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 2</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="c16">I</font> = <font color="Maroon" title="c7">s</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c8">i</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">a</font> =  <a href="aofa_i00.html#K43" title="AOFA_I00:func.43">.</a> 2,<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="c18">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="c8">i</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <font color="Maroon" title="c17">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c6">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c8">i</font></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/>
<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">K</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">K</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="c8">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="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">K</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a>  );<br/>
<a NAME="E8:126"/><i><font color="Green" title="E69">F</font></i>: 
<font color="Maroon" title="c18">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="c8">i</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <font color="Maroon" title="c17">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c6">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c8">i</font></span>)</span>
 
;<br/>
<a NAME="E9:126"/><i><font color="Green" title="E70">A2</font></i>: 
( <font color="Maroon" title="c7">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">n</font> &amp; <font color="Maroon" title="c7">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c7">s</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="aofa_i00.html#E2:126"><i><font color="Green" title="E67">A0</font></i></a></i>;<br/>
<a NAME="E10:126"/><i><font color="Green" title="E71">D</font></i>: 
( <font color="Maroon" title="c6">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c6">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="c8">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="aofa_i00.html#E2:126"><i><font color="Green" title="E67">A0</font></i></a></i>;<br/>
<a NAME="E11:126"/><i><font color="Green" title="E72">Q1</font></i>: 
( <font color="Maroon" title="c14">q1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c6">n</font> &amp; <font color="Maroon" title="c14">q1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">q</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <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="c10">q</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c6">n</font> )
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E9:126"><i><font color="Green" title="E70">A2</font></i></a>, <a class="ref" href="aofa_i00.html#T25" target="_self" title="AOFA_I00:th.25">Th011</a></i>;<br/>
<a NAME="E12:126"/><i><font color="Green" title="E73">Q2</font></i>: 
( <font color="Maroon" title="c15">q2</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">q1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c6">n</font> &amp; <font color="Maroon" title="c15">q2</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">q1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">s</font> &amp; <font color="Maroon" title="c15">q2</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; 2 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E9:126"><i><font color="Green" title="E70">A2</font></i></a>, <a class="txt" href="aofa_i00.html#E10:126"><i><font color="Green" title="E71">D</font></i></a>, <a class="ref" href="aofa_i00.html#T25" target="_self" title="AOFA_I00:th.25">Th011</a></i>;<br/>
<a NAME="E13:126"/><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="c14">q1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <font color="Maroon" title="c17">a</font></span>)</span>]
 
<b>by </b><i><a class="ref" href="aofa_i00.html#T25" target="_self" title="AOFA_I00:th.25">Th011</a>, <a class="txt" href="aofa_i00.html#E12:126"><i><font color="Green" title="E73">Q2</font></i></a>, <a class="ref" href="newton.html#T19" title="NEWTON:th.19">NEWTON:19</a></i>;<br/>
<a NAME="E14:126"/><i><font color="Green" title="E75">C</font></i>: 
for <font color="Olive" title="b1">q</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">q</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">q</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c8">i</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">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">q</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c8">i</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c6">n</font> )
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E9:126"><i><font color="Green" title="E70">A2</font></i></a>, <a class="ref" href="aofa_i00.html#T32" target="_self" title="AOFA_I00:th.32">Th113</a></i>;<br/>
<a NAME="E15:126"/><i><font color="Green" title="E76">B</font></i>: 
for <font color="Olive" title="b1">q</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">q</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">q</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c8">i</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">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">q</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c6">n</font></span>)</span>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="126_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">q</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c19">q</font>] implies ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c19">q</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c8">i</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">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="c19">q</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c6">n</font></span>)</span>] ) )</span><br/>

<b>given </b><font color="Maroon" title="c20">Ki</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><a NAME="E2:126_1"/><i><font color="Green" title="E77">A3</font></i>: 
( <font color="Maroon" title="c20">Ki</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">q</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon" title="c19">q</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">Ki</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c19">q</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c8">i</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">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="c19">q</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c6">n</font></span>)</span>] )</span><br/>

<b>reconsider </b><font color="Maroon" title="c21">q3</font> = <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c19">q</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c8">i</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="c22">q4</font> = <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c21">q3</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">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/>
<b>reconsider </b><font color="Maroon" title="c23">q'</font> = <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c19">q</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c6">n</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="E6:126_1"/><i><font color="Green" title="E78">A4</font></i>: 
<font color="Maroon" title="c22">q4</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c19">q</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c8">i</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">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="aofa_000.html#D29" title="AOFA_000:def.29">AOFA_000:def 29</a></i>;<br/>
<a NAME="E7:126_1"/>
( <font color="Maroon" title="c21">q3</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">q</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c21">q3</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">q</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">q</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">i</font></span>)</span> &amp; <font color="Maroon" title="c22">q4</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">q3</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">s</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">q</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <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="c19">q</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">i</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">q</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">q</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c22">q4</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">q3</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">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="aofa_i00.html#E9:126"><i><font color="Green" title="E70">A2</font></i></a>, <a class="ref" href="aofa_i00.html#T28" target="_self" title="AOFA_I00:th.28">Th012</a>, <a class="ref" href="aofa_i00.html#T33" target="_self" title="AOFA_I00:th.33">Th213</a></i>;<br/>
<a NAME="E8:126_1"/><b>then </b>
( <font color="Maroon" title="c20">Ki</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">q4</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon" title="c22">q4</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">Ki</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K12" title="NEWTON:func.12">!</a>  )
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E2:126_1"><i><font color="Green" title="E77">A3</font></i></a>, <a class="ref" href="newton.html#T21" title="NEWTON:th.21">NEWTON:21</a></i>;<br/>
<b>hence </b><a NAME="E9:126_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c19">q</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c8">i</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">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span>]
 <b>by </b><i><a class="txt" href="aofa_i00.html#E6:126_1"><i><font color="Green" title="E78">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c19">q</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c6">n</font></span>)</span>]</span><br/>

<a NAME="E10:126_1"/>
( <font color="Maroon" title="c23">q'</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">q</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">s</font> &amp; <font color="Maroon" title="c23">q'</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">q</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">i</font> )
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E9:126"><i><font color="Green" title="E70">A2</font></i></a>, <a class="txt" href="aofa_i00.html#E10:126"><i><font color="Green" title="E71">D</font></i></a>, <a class="ref" href="aofa_i00.html#T35" target="_self" title="AOFA_I00:th.35">Th114</a></i>;<br/>
<b>hence </b><a NAME="E11:126_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c19">q</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c6">n</font></span>)</span>]
 <b>by </b><i><a class="txt" href="aofa_i00.html#E2:126_1"><i><font color="Green" title="E77">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E16:126"/><i><font color="Green" title="E77">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="c14">q1</font>,<font color="Maroon" title="c18">F</font>] &amp; ( <font color="Maroon" title="c17">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">q1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">q1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c6">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="c14">q1</font>,<font color="Maroon" title="c18">F</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">q1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 ) &amp; ( <font color="Maroon" title="c17">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">q1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c14">q1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c6">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="c14">q1</font>,<font color="Maroon" title="c18">F</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">q1</font> ) &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">q1</font>,<font color="Maroon" title="c18">F</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">q1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c6">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="aofa_i00.html#E8:126"><i><font color="Green" title="E69">F</font></i></a>, <a class="txt" href="aofa_i00.html#E13:126"><i><font color="Green" title="E74">A</font></i></a>, <a class="txt" href="aofa_i00.html#E15:126"><i><font color="Green" title="E76">B</font></i></a>, <a class="txt" href="aofa_i00.html#E14:126"><i><font color="Green" title="E75">C</font></i></a>, <a class="txt" href="aofa_i00.html#E10:126"><i><font color="Green" title="E71">D</font></i></a>);<br/></i>
<a NAME="E17:126"/><i><font color="Green" title="E78">M</font></i>: 
<font color="Maroon" title="c17">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c10">q</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="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="c8">i</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 2</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c6">n</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="aofa_i00.html#K82" title="AOFA_I00:func.82">*=</a> <font color="Maroon" title="c8">i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">s</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c14">q1</font>,<font color="Maroon" title="c18">F</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">s</font>
<b>by </b><i><a class="ref" href="aofa_000.html#D29" title="AOFA_000:def.29">AOFA_000:def 29</a></i>
<br/>.= 
<font color="Maroon" title="c11">N</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> 
<b>by </b><i><a class="txt" href="aofa_i00.html#E17:126"><i><font color="Green" title="E78">M</font></i></a>, <a class="txt" href="aofa_i00.html#E3:126"><i><font color="Green" title="E68">A1</font></i></a>, <a class="txt" href="aofa_i00.html#E11:126"><i><font color="Green" title="E72">Q1</font></i></a>, <a class="txt" href="aofa_i00.html#E16:126"><i><font color="Green" title="E77">E</font></i></a>, <a class="ref" href="nat_1.html#T27" title="NAT_1:th.27">NAT_1:27</a>, <a class="ref" href="newton.html#T18" title="NEWTON:th.18">NEWTON:18</a>, <a class="ref" href="newton.html#T19" title="NEWTON:th.19">NEWTON:19</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>
