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<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   <a href="aofa_i00.html#V2" title="AOFA_I00:attr.2">Euclidean</a>  <a href="aofa_000.html#NM1" title="AOFA_000:NM.1">preIfWhileAlgebra</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">X</font><br/> for <font color="Olive" title="b3">g</font> being  <a href="aofa_i00.html#V1" title="AOFA_I00:attr.1">Euclidean</a>   <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>, <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Olive" title="b1">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Olive" title="b1">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Olive" title="b2">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <br/> for <font color="Olive" title="b4">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="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Olive" title="b1">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <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#K11" title="NEWTON:func.11">!</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">X</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font><br/> for <font color="Olive" title="b2">g</font> being  <a href="aofa_i00.html#V1" title="AOFA_I00:attr.1">Euclidean</a>   <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>, <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Olive" title="b1">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <br/> for <font color="Olive" title="b3">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="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <br/> 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#K11" title="NEWTON:func.11">!</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">g</font> being  <a href="aofa_i00.html#V1" title="AOFA_I00:attr.1">Euclidean</a>   <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>, <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <br/> for <font color="Olive" title="b2">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="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <br/> 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#K11" title="NEWTON:func.11">!</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">g</font> be   <a href="aofa_i00.html#V1" title="AOFA_I00:attr.1">Euclidean</a>   <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>, <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">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="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <br/> 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#K11" title="NEWTON:func.11">!</a> </span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c4">g</font>;<br/>
<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">d</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="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="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <br/> 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#K11" title="NEWTON:func.11">!</a>  )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c9">d</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><a NAME="E2:125"/><span class="lab"><font color="Green" title="E65">A1</font></span>: 
<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
 <span class="kw">and </span><br/><a NAME="E3:125"/><span class="lab"><font color="Green" title="E66">A2</font></span>: 
<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
 <span class="kw">and </span><br/><a NAME="E4:125"/><span class="lab"><font color="Green" title="E67">A3</font></span>: 
<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
 <span class="kw">and </span><br/><a NAME="E5:125"/><span class="lab"><font color="Green" title="E68">A4</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">q</font> being   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <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#K11" title="NEWTON:func.11">!</a> </span><br/>

<a NAME="E6:125"/><span class="lab"><font color="Green" title="E69">A5</font></span>: 
( <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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:125"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="txt" href="#E4:125"><span class="lab"><font color="Green" title="E67">A3</font></span></a>, <a class="txt" href="#E5:125"><span class="lab"><font color="Green" title="E68">A4</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">S</font> =  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c11">q</font> be    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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="c11">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="c11">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#K11" title="NEWTON:func.11">!</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c12">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="c11">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="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  ;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ] <span class="kw">means </span> 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#K11" title="NEWTON:func.11">!</a>  );<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">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/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">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="c12">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="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  ;<br/>
<a NAME="E10:125"/><span class="lab"><font color="Green" title="E70">A6</font></span>: 
<font color="Maroon" title="c14">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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_i00.html#T25" target="_self" title="AOFA_I00:th.25">Th25</a></span>;<br/>
<a NAME="E11:125"/><span class="lab"><font color="Green" title="E71">A7</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:125"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="txt" href="#E4:125"><span class="lab"><font color="Green" title="E67">A3</font></span></a></span>;<br/>
<a NAME="E12:125"/><span class="kw">then </span>
<font color="Maroon" title="c14">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="c12">q1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_i00.html#T25" target="_self" title="AOFA_I00:th.25">Th25</a></span>;<br/>
<a NAME="E13:125"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A8</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c12">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="c13">a</font></span>)</span>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:125"><span class="lab"><font color="Green" title="E70">A6</font></span></a>, <a class="ref" href="aofa_i00.html#T25" target="_self" title="AOFA_I00:th.25">Th25</a>, <a class="ref" href="newton.html#T19" title="NEWTON:th.19">NEWTON:19</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">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/>
<a NAME="E14:125"/><span class="lab"><font color="Green" title="E73">A9</font></span>: 
<font color="Maroon" title="c13">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<a NAME="E15:125"/><span class="lab"><font color="Green" title="E74">A10</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:125"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="txt" href="#E5:125"><span class="lab"><font color="Green" title="E68">A4</font></span></a></span>;<br/>
<a NAME="E16:125"/><span class="lab"><font color="Green" title="E75">A11</font></span>: 
for <font color="Olive" title="b1">q</font> being   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">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="125_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">q</font> be    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c16">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="c16">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="c16">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/>

<span class="kw">given </span><font color="Maroon" title="c17">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> <span class="kw"> such that </span><a NAME="E2:125_1"/><span class="lab"><font color="Green" title="E76">A12</font></span>: 
<font color="Maroon" title="c17">Ki</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">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 class="kw">and </span><br/><a NAME="E3:125_1"/><span class="lab"><font color="Green" title="E77">A13</font></span>: 
<font color="Maroon" title="c16">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="c17">Ki</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c16">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="c16">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/>

<span class="kw">reconsider </span><font color="Maroon" title="c18">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="c16">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="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">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="c18">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="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  ;<br/>
<a NAME="E6:125_1"/><span class="lab"><font color="Green" title="E78">A14</font></span>: 
<font color="Maroon" title="c18">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="c16">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="c16">q</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_i00.html#T33" target="_self" title="AOFA_I00:th.33">Th33</a></span>;<br/>
<a NAME="E7:125_1"/>
<font color="Maroon" title="c19">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="c18">q3</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:125"><span class="lab"><font color="Green" title="E71">A7</font></span></a>, <a class="ref" href="aofa_i00.html#T28" target="_self" title="AOFA_I00:th.28">Th28</a></span>;<br/>
<a NAME="E8:125_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A15</font></span>: 
<font color="Maroon" title="c19">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="c17">Ki</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K11" title="NEWTON:func.11">!</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:125_1"><span class="lab"><font color="Green" title="E76">A12</font></span></a>, <a class="txt" href="#E3:125_1"><span class="lab"><font color="Green" title="E77">A13</font></span></a>, <a class="txt" href="#E6:125_1"><span class="lab"><font color="Green" title="E78">A14</font></span></a>, <a class="ref" href="newton.html#T21" title="NEWTON:th.21">NEWTON:21</a></span>;<br/>
<a NAME="E9:125_1"/><span class="lab"><font color="Green" title="E80">A16</font></span>: 
<font color="Maroon" title="c19">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="c16">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_000.html#D29" title="AOFA_000:def.29">AOFA_000:def 29</a></span>;<br/>
<a NAME="E10:125_1"/>
<font color="Maroon" title="c19">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="c18">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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_i00.html#T28" target="_self" title="AOFA_I00:th.28">Th28</a></span>;<br/>
<a NAME="E11:125_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">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="c19">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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:125"><span class="lab"><font color="Green" title="E71">A7</font></span></a>, <a class="txt" href="#E2:125_1"><span class="lab"><font color="Green" title="E76">A12</font></span></a>, <a class="ref" href="aofa_i00.html#T33" target="_self" title="AOFA_I00:th.33">Th33</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:125_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="c16">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>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:125_1"><span class="lab"><font color="Green" title="E80">A16</font></span></a>, <a class="txt" href="#E8:125_1"><span class="lab"><font color="Green" title="E79">A15</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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">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/>

<span class="kw">reconsider </span><font color="Maroon" title="c20">q9</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">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="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  ;<br/>
<a NAME="E14:125_1"/><span class="lab"><font color="Green" title="E81">A17</font></span>: 
<font color="Maroon" title="c20">q9</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="c16">q</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c8">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:125"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="ref" href="aofa_i00.html#T35" target="_self" title="AOFA_I00:th.35">Th35</a></span>;<br/>
<a NAME="E15:125_1"/>
<font color="Maroon" title="c20">q9</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="c16">q</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:125"><span class="lab"><font color="Green" title="E74">A10</font></span></a>, <a class="ref" href="aofa_i00.html#T35" target="_self" title="AOFA_I00:th.35">Th35</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:125_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="c16">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:125_1"><span class="lab"><font color="Green" title="E76">A12</font></span></a>, <a class="txt" href="#E3:125_1"><span class="lab"><font color="Green" title="E77">A13</font></span></a>, <a class="txt" href="#E14:125_1"><span class="lab"><font color="Green" title="E81">A17</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c16">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="c13">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/>
<span class="kw">set </span><font color="Maroon" title="c17">T</font> = <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c18">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">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="c11">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="c18">N</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E18:125"/><span class="lab"><font color="Green" title="E76">A18</font></span>: 
<font color="Maroon" title="c18">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c11">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="c18">N</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a> </span><br/>

<a NAME="E19:125"/><span class="lab"><font color="Green" title="E77">A19</font></span>: 
<font color="Maroon" title="c16">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="c13">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="E20:125"/><span class="lab"><font color="Green" title="E78">A20</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:125"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="txt" href="#E3:125"><span class="lab"><font color="Green" title="E66">A2</font></span></a></span>;<br/>
<a NAME="E21:125"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A21</font></span>: 
<font color="Maroon" title="c12">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="c11">q</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c6">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_i00.html#T25" target="_self" title="AOFA_I00:th.25">Th25</a></span>;<br/>
<a NAME="E22:125"/><span class="lab"><font color="Green" title="E80">A22</font></span>: 
for <font color="Olive" title="b1">q</font> being   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:125"><span class="lab"><font color="Green" title="E78">A20</font></span></a>, <a class="txt" href="#E11:125"><span class="lab"><font color="Green" title="E71">A7</font></span></a>, <a class="ref" href="aofa_i00.html#T32" target="_self" title="AOFA_I00:th.32">Th32</a></span>;<br/>
<a NAME="E23:125"/><span class="lab"><font color="Green" title="E81">A23</font></span>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c12">q1</font>,<font color="Maroon" title="c16">F</font>] &amp; ( <font color="Maroon" title="c13">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">q1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">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="c12">q1</font>,<font color="Maroon" title="c16">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="c12">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="c13">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">q1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c12">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="c12">q1</font>,<font color="Maroon" title="c16">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="c13">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">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="c12">q1</font>,<font color="Maroon" title="c16">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="c12">q1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c6">n</font> )
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="aofa_i00.html#S1" target="_self" title="AOFA_I00:sch.1">AOFA_I00:sch 1</a>(<a class="txt" href="#E19:125"><span class="lab"><font color="Green" title="E77">A19</font></span></a>, <a class="txt" href="#E13:125"><span class="lab"><font color="Green" title="E72">A8</font></span></a>, <a class="txt" href="#E16:125"><span class="lab"><font color="Green" title="E75">A11</font></span></a>, <a class="txt" href="#E22:125"><span class="lab"><font color="Green" title="E80">A22</font></span></a>, <a class="txt" href="#E6:125"><span class="lab"><font color="Green" title="E69">A5</font></span></a>);<br/></span>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c11">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="c12">q1</font>,<font color="Maroon" title="c16">F</font></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">s</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_000.html#D29" title="AOFA_000:def.29">AOFA_000:def 29</a></span>
<br/>.= 
<font color="Maroon" title="c18">N</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:125"><span class="lab"><font color="Green" title="E76">A18</font></span></a>, <a class="txt" href="#E21:125"><span class="lab"><font color="Green" title="E79">A21</font></span></a>, <a class="txt" href="#E23:125"><span class="lab"><font color="Green" title="E81">A23</font></span></a>, <a class="txt" href="#E14:125"><span class="lab"><font color="Green" title="E73">A9</font></span></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></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
