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<span class="kw">reconsider </span><font color="Maroon" title="c3">b1</font> = <font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="nat_1.html#T20" title="NAT_1:th.20">NAT_1:20</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">d</font>, <font color="Maroon" title="c5">e</font> be   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="numeral1.html#K1" title="NUMERAL1:func.1">value</a> <font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">d</font> holds <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> ) ) &amp;  <a href="numeral1.html#K1" title="NUMERAL1:func.1">value</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">e</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c5">e</font> holds <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> ) ) implies <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> ) &amp; ( not <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> &amp; <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> implies <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E2:8_1_2"/>
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="numeral1.html#K1" title="NUMERAL1:func.1">value</a> <font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">d</font> holds <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> ) ) &amp;  <a href="numeral1.html#K1" title="NUMERAL1:func.1">value</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">e</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c5">e</font> holds <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> ) ) implies <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> &amp; <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> implies <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:8_1_2_1"/>
 <a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="power.html#K6" title="POWER:func.6">log</a> 2,1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="power.html#T65" title="POWER:th.65">POWER:65</a></span>;<br/>
<a NAME="E2:8_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E6">A33</font></span>: 
 <a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T59" title="POWER:th.59">POWER:59</a></span>;<br/>
<a NAME="E3:8_1_2_1"/>
 <a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="power.html#K6" title="POWER:func.6">log</a> 2,1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="power.html#T65" title="POWER:th.65">POWER:65</a></span>;<br/>
<a NAME="E4:8_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A34</font></span>: 
 <a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T59" title="POWER:th.59">POWER:59</a></span>;<br/>
<a NAME="E5:8_1_2_1"/><span class="lab"><font color="Green" title="E8">A35</font></span>: 
1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a></span>;<br/>
<a NAME="E6:8_1_2_1"/><span class="lab"><font color="Green" title="E9">A36</font></span>: 
1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">g</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font></span>)</span> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="numeral1.html#T3" target="_self" title="NUMERAL1:th.3">Th3</a></span>;<br/>
<span class="kw">assume </span><a NAME="E8:8_1_2_1"/><span class="lab"><font color="Green" title="E10">A37</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not  <a href="numeral1.html#K1" title="NUMERAL1:func.1">value</a> <font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> or  not <font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  ex <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">d</font> &amp; not <!--HUMANRECFAILED-->(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> ) ) or  not  <a href="numeral1.html#K1" title="NUMERAL1:func.1">value</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> or  not <font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">e</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  ex <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c5">e</font> &amp; not <!--HUMANRECFAILED-->(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> ) ) or <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E9:8_1_2_1"/><span class="lab"><font color="Green" title="E11">A38</font></span>: 
 <a href="numeral1.html#K1" title="NUMERAL1:func.1">value</a> <font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E10:8_1_2_1"/><span class="lab"><font color="Green" title="E12">A39</font></span>: 
<font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E11:8_1_2_1"/><span class="lab"><font color="Green" title="E13">A40</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">d</font> holds <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not  <a href="numeral1.html#K1" title="NUMERAL1:func.1">value</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> or  not <font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">e</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  ex <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c5">e</font> &amp; not <!--HUMANRECFAILED-->(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> ) ) or <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">D</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E13:8_1_2_1"/><span class="lab"><font color="Green" title="E14">A41</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">d</font>
 <span class="kw">and </span><br/><a NAME="E14:8_1_2_1"/><span class="lab"><font color="Green" title="E15">A42</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">D</font> holds <br/><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b1">i</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E15:8_1_2_1"/><span class="lab"><font color="Green" title="E16">A43</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c7">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:8_1_2_1"><span class="lab"><font color="Green" title="E11">A38</font></span></a>, <a class="ref" href="numeral1.html#D1" target="_self" title="NUMERAL1:def.1">Def1</a></span>;<br/>
<a NAME="E16:8_1_2_1"/><span class="lab"><font color="Green" title="E17">A44</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c7">D</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:8_1_2_1"><span class="lab"><font color="Green" title="E12">A39</font></span></a>, <a class="txt" href="#E13:8_1_2_1"><span class="lab"><font color="Green" title="E14">A41</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>;<br/>
<a NAME="E17:8_1_2_1"/><span class="lab"><font color="Green" title="E18">A45</font></span>: 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:8_1_2_1"><span class="lab"><font color="Green" title="E12">A39</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">k</font> = <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E19:8_1_2_1"/><span class="lab"><font color="Green" title="E19">A46</font></span>: 
1 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c8">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:8_1_2_1"><span class="lab"><font color="Green" title="E12">A39</font></span></a>, <a class="ref" href="nat_1.html#T4" title="NAT_1:th.4">NAT_1:4</a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/>
<a NAME="E20:8_1_2_1"/><span class="lab"><font color="Green" title="E20">A47</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="prepower.html#T13" title="PREPOWER:th.13">PREPOWER:13</a></span>;<br/>
<a NAME="E21:8_1_2_1"/>
<font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c8">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:8_1_2_1"><span class="lab"><font color="Green" title="E14">A41</font></span></a>, <a class="txt" href="#E14:8_1_2_1"><span class="lab"><font color="Green" title="E15">A42</font></span></a>, <a class="txt" href="#E17:8_1_2_1"><span class="lab"><font color="Green" title="E18">A45</font></span></a></span>;<br/>
<a NAME="E22:8_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:8_1_2_1"><span class="lab"><font color="Green" title="E14">A41</font></span></a>, <a class="txt" href="#E15:8_1_2_1"><span class="lab"><font color="Green" title="E16">A43</font></span></a>, <a class="txt" href="#E17:8_1_2_1"><span class="lab"><font color="Green" title="E18">A45</font></span></a>, <a class="txt" href="#E16:8_1_2_1"><span class="lab"><font color="Green" title="E17">A44</font></span></a>, <a class="ref" href="afinsq_2.html#T73" title="AFINSQ_2:th.73">AFINSQ_2:73</a></span>;<br/>
<a NAME="E23:8_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:8_1_2_1"><span class="lab"><font color="Green" title="E19">A46</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E24:8_1_2_1"/><span class="kw">then </span>
 <a href="power.html#K5" title="POWER:func.5">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:8_1_2_1"><span class="lab"><font color="Green" title="E20">A47</font></span></a>, <a class="ref" href="pre_ff.html#T12" title="PRE_FF:th.12">PRE_FF:12</a></span>;<br/>
<a NAME="E25:8_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">k</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="power.html#T63" title="POWER:th.63">POWER:63</a></span>;<br/>
<a NAME="E26:8_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1_2_1"><span class="lab"><font color="Green" title="E7">A34</font></span></a>, <a class="ref" href="xreal_1.html#T74" title="XREAL_1:th.74">XREAL_1:74</a></span>;<br/>
<a NAME="E27:8_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A48</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1_2_1"><span class="lab"><font color="Green" title="E7">A34</font></span></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></span>;<br/>
<a NAME="E28:8_1_2_1"/>
<font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E22">A49</font></span>:  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c6">g</font> = 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T68" title="AFINSQ_2:th.68">AFINSQ_2:68</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#T12" title="SERIES_1:th.12">SERIES_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="series_1.html#T26" title="SERIES_1:th.26">SERIES_1:26</a></span>
<br/>.= 
 <a href="binop_2.html#K13" title="BINOP_2:func.13">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="binop_2.html#K17" title="BINOP_2:func.17">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
 <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:8_1_2_1"><span class="lab"><font color="Green" title="E9">A36</font></span></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1

;<br/>
<a NAME="E30:8_1_2_1"/><span class="lab"><font color="Green" title="E23">A50</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E31:8_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A51</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c7">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c6">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:8_1_2_1"><span class="lab"><font color="Green" title="E14">A41</font></span></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>;<br/>
<a NAME="E32:8_1_2_1"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:8_1_2_1"><span class="lab"><font color="Green" title="E23">A50</font></span></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/>
<a NAME="E33:8_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A52</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>;<br/>
<a NAME="E34:8_1_2_1"/><span class="lab"><font color="Green" title="E26">A53</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">D</font> holds <br/><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">D</font> implies <font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">i</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c10">I</font> = <font color="Maroon" title="c9">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:8_1_2_1_2"/><span class="lab"><font color="Green" title="E27">A54</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span><br/>

<a NAME="E3:8_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A55</font></span>: 
<font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c9">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:8_1_2_1"><span class="lab"><font color="Green" title="E15">A42</font></span></a></span>;<br/>
<a NAME="E4:8_1_2_1_2"/>
<font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">b1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:8_1_2_1"><span class="lab"><font color="Green" title="E13">A40</font></span></a>, <a class="txt" href="#E13:8_1_2_1"><span class="lab"><font color="Green" title="E14">A41</font></span></a>, <a class="txt" href="#E2:8_1_2_1_2"><span class="lab"><font color="Green" title="E27">A54</font></span></a></span>;<br/>
<a NAME="E5:8_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A56</font></span>: 
<font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<font color="Maroon" title="c6">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">I</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">I</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:8_1_2_1"><span class="lab"><font color="Green" title="E14">A41</font></span></a>, <a class="txt" href="#E33:8_1_2_1"><span class="lab"><font color="Green" title="E25">A52</font></span></a>, <a class="txt" href="#E2:8_1_2_1_2"><span class="lab"><font color="Green" title="E27">A54</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">I</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c3">b1</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c10">I</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#D1" title="PREPOWER:def.1">PREPOWER:def 1</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E7:8_1_2_1_2"/>
<font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_2_1_2"><span class="lab"><font color="Green" title="E29">A56</font></span></a>, <a class="txt" href="#E3:8_1_2_1_2"><span class="lab"><font color="Green" title="E28">A55</font></span></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</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">assume </span><span class="kw">that </span><br/><a NAME="E35:8_1_2_1"/><span class="lab"><font color="Green" title="E27">A57</font></span>: 
 <a href="numeral1.html#K1" title="NUMERAL1:func.1">value</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E36:8_1_2_1"/><span class="lab"><font color="Green" title="E28">A58</font></span>: 
<font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">e</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E37:8_1_2_1"/><span class="lab"><font color="Green" title="E29">A59</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c5">e</font> holds <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c9">E</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E39:8_1_2_1"/><span class="lab"><font color="Green" title="E30">A60</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c9">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c5">e</font>
 <span class="kw">and </span><br/><a NAME="E40:8_1_2_1"/><span class="lab"><font color="Green" title="E31">A61</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c9">E</font> holds <br/><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b1">i</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E41:8_1_2_1"/><span class="lab"><font color="Green" title="E32">A62</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c9">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:8_1_2_1"><span class="lab"><font color="Green" title="E27">A57</font></span></a>, <a class="ref" href="numeral1.html#D1" target="_self" title="NUMERAL1:def.1">Def1</a></span>;<br/>
<a NAME="E42:8_1_2_1"/><span class="lab"><font color="Green" title="E33">A63</font></span>: 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">e</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c5">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:8_1_2_1"><span class="lab"><font color="Green" title="E28">A58</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">l</font> = <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">e</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E44:8_1_2_1"/><span class="lab"><font color="Green" title="E34">A64</font></span>: 
1 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c10">l</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">l</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c10">l</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:8_1_2_1"><span class="lab"><font color="Green" title="E28">A58</font></span></a>, <a class="ref" href="nat_1.html#T4" title="NAT_1:th.4">NAT_1:4</a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">g</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">e</font></span>)</span> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="numeral1.html#T3" target="_self" title="NUMERAL1:th.3">Th3</a></span>;<br/>
<a NAME="E46:8_1_2_1"/>
<font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E35">A65</font></span>:  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c11">g</font> = 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">l</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T68" title="AFINSQ_2:th.68">AFINSQ_2:68</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">l</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#T12" title="SERIES_1:th.12">SERIES_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">l</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="series_1.html#T26" title="SERIES_1:th.26">SERIES_1:26</a></span>
<br/>.= 
 <a href="binop_2.html#K13" title="BINOP_2:func.13">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="binop_2.html#K17" title="BINOP_2:func.17">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
 <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_2_1"><span class="lab"><font color="Green" title="E8">A35</font></span></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1

;<br/>
<a NAME="E48:8_1_2_1"/><span class="lab"><font color="Green" title="E36">A66</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E49:8_1_2_1"/><span class="kw">then </span>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/>
<a NAME="E50:8_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A67</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>;<br/>
<a NAME="E51:8_1_2_1"/><span class="lab"><font color="Green" title="E38">A68</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c9">E</font> holds <br/><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c9">E</font> implies <font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">i</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c13">I</font> = <font color="Maroon" title="c12">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:8_1_2_1_4"/><span class="lab"><font color="Green" title="E39">A69</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c9">E</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span><br/>

<a NAME="E3:8_1_2_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A70</font></span>: 
<font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c12">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:8_1_2_1"><span class="lab"><font color="Green" title="E31">A61</font></span></a></span>;<br/>
<a NAME="E4:8_1_2_1_4"/>
<font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">b1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:8_1_2_1"><span class="lab"><font color="Green" title="E29">A59</font></span></a>, <a class="txt" href="#E39:8_1_2_1"><span class="lab"><font color="Green" title="E30">A60</font></span></a>, <a class="txt" href="#E2:8_1_2_1_4"><span class="lab"><font color="Green" title="E39">A69</font></span></a></span>;<br/>
<a NAME="E5:8_1_2_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A71</font></span>: 
<font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<font color="Maroon" title="c11">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">I</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">I</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:8_1_2_1"><span class="lab"><font color="Green" title="E30">A60</font></span></a>, <a class="txt" href="#E50:8_1_2_1"><span class="lab"><font color="Green" title="E37">A67</font></span></a>, <a class="txt" href="#E2:8_1_2_1_4"><span class="lab"><font color="Green" title="E39">A69</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">I</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c3">b1</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">I</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#D1" title="PREPOWER:def.1">PREPOWER:def 1</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E7:8_1_2_1_4"/>
<font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_2_1_4"><span class="lab"><font color="Green" title="E41">A71</font></span></a>, <a class="txt" href="#E3:8_1_2_1_4"><span class="lab"><font color="Green" title="E40">A70</font></span></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</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>
<a NAME="E52:8_1_2_1"/><span class="lab"><font color="Green" title="E39">A72</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c9">E</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:8_1_2_1"><span class="lab"><font color="Green" title="E28">A58</font></span></a>, <a class="txt" href="#E39:8_1_2_1"><span class="lab"><font color="Green" title="E30">A60</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>;<br/>
<a NAME="E53:8_1_2_1"/><span class="lab"><font color="Green" title="E40">A74</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c10">l</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="prepower.html#T13" title="PREPOWER:th.13">PREPOWER:13</a></span>;<br/>
<a NAME="E54:8_1_2_1"/>
<font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">l</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c10">l</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:8_1_2_1"><span class="lab"><font color="Green" title="E30">A60</font></span></a>, <a class="txt" href="#E40:8_1_2_1"><span class="lab"><font color="Green" title="E31">A61</font></span></a>, <a class="txt" href="#E42:8_1_2_1"><span class="lab"><font color="Green" title="E33">A63</font></span></a></span>;<br/>
<a NAME="E55:8_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">l</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c10">l</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:8_1_2_1"><span class="lab"><font color="Green" title="E30">A60</font></span></a>, <a class="txt" href="#E41:8_1_2_1"><span class="lab"><font color="Green" title="E32">A62</font></span></a>, <a class="txt" href="#E42:8_1_2_1"><span class="lab"><font color="Green" title="E33">A63</font></span></a>, <a class="txt" href="#E52:8_1_2_1"><span class="lab"><font color="Green" title="E39">A72</font></span></a>, <a class="ref" href="afinsq_2.html#T73" title="AFINSQ_2:th.73">AFINSQ_2:73</a></span>;<br/>
<a NAME="E56:8_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c10">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:8_1_2_1"><span class="lab"><font color="Green" title="E34">A64</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E57:8_1_2_1"/><span class="kw">then </span>
 <a href="power.html#K5" title="POWER:func.5">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c10">l</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:8_1_2_1"><span class="lab"><font color="Green" title="E40">A74</font></span></a>, <a class="ref" href="pre_ff.html#T12" title="PRE_FF:th.12">PRE_FF:12</a></span>;<br/>
<a NAME="E58:8_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">l</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="power.html#T63" title="POWER:th.63">POWER:63</a></span>;<br/>
<a NAME="E59:8_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">l</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1_2_1"><span class="lab"><font color="Green" title="E6">A33</font></span></a>, <a class="ref" href="xreal_1.html#T74" title="XREAL_1:th.74">XREAL_1:74</a></span>;<br/>
<a NAME="E60:8_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A75</font></span>: 
<font color="Maroon" title="c10">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1_2_1"><span class="lab"><font color="Green" title="E6">A33</font></span></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></span>;<br/>
<a NAME="E61:8_1_2_1"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c9">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:8_1_2_1"><span class="lab"><font color="Green" title="E30">A60</font></span></a>, <a class="txt" href="#E48:8_1_2_1"><span class="lab"><font color="Green" title="E36">A66</font></span></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>;<br/>
<a NAME="E62:8_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:8_1_2_1"><span class="lab"><font color="Green" title="E32">A62</font></span></a>, <a class="txt" href="#E47:8_1_2_1"><span class="lab"><font color="Green" title="E35">A65</font></span></a>, <a class="txt" href="#E51:8_1_2_1"><span class="lab"><font color="Green" title="E38">A68</font></span></a>, <a class="ref" href="afinsq_2.html#T69" title="AFINSQ_2:th.69">AFINSQ_2:69</a>, <a class="ref" href="xreal_1.html#T41" title="XREAL_1:th.41">XREAL_1:41</a></span>;<br/>
<a NAME="E63:8_1_2_1"/><span class="kw">then </span>
 <a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="power.html#K5" title="POWER:func.5">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:8_1_2_1"><span class="lab"><font color="Green" title="E10">A37</font></span></a>, <a class="ref" href="power.html#T65" title="POWER:th.65">POWER:65</a></span>;<br/>
<a NAME="E64:8_1_2_1"/><span class="kw">then </span>
 <a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="power.html#T63" title="POWER:th.63">POWER:63</a></span>;<br/>
<a NAME="E65:8_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1_2_1"><span class="lab"><font color="Green" title="E6">A33</font></span></a>, <a class="ref" href="xreal_1.html#T76" title="XREAL_1:th.76">XREAL_1:76</a></span>;<br/>
<a NAME="E66:8_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1_2_1"><span class="lab"><font color="Green" title="E6">A33</font></span></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></span>;<br/>
<a NAME="E67:8_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:8_1_2_1"><span class="lab"><font color="Green" title="E21">A48</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E68:8_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A76</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E69:8_1_2_1"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:8_1_2_1"><span class="lab"><font color="Green" title="E16">A43</font></span></a>, <a class="txt" href="#E29:8_1_2_1"><span class="lab"><font color="Green" title="E22">A49</font></span></a>, <a class="txt" href="#E31:8_1_2_1"><span class="lab"><font color="Green" title="E24">A51</font></span></a>, <a class="txt" href="#E34:8_1_2_1"><span class="lab"><font color="Green" title="E26">A53</font></span></a>, <a class="ref" href="afinsq_2.html#T69" title="AFINSQ_2:th.69">AFINSQ_2:69</a>, <a class="ref" href="xreal_1.html#T41" title="XREAL_1:th.41">XREAL_1:41</a></span>;<br/>
<a NAME="E70:8_1_2_1"/><span class="kw">then </span>
 <a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="power.html#K5" title="POWER:func.5">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:8_1_2_1"><span class="lab"><font color="Green" title="E10">A37</font></span></a>, <a class="ref" href="power.html#T65" title="POWER:th.65">POWER:65</a></span>;<br/>
<a NAME="E71:8_1_2_1"/><span class="kw">then </span>
 <a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="power.html#T63" title="POWER:th.63">POWER:63</a></span>;<br/>
<a NAME="E72:8_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1_2_1"><span class="lab"><font color="Green" title="E7">A34</font></span></a>, <a class="ref" href="xreal_1.html#T76" title="XREAL_1:th.76">XREAL_1:76</a></span>;<br/>
<a NAME="E73:8_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1_2_1"><span class="lab"><font color="Green" title="E7">A34</font></span></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></span>;<br/>
<a NAME="E74:8_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:8_1_2_1"><span class="lab"><font color="Green" title="E41">A75</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E75:8_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E76:8_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A77</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E68:8_1_2_1"><span class="lab"><font color="Green" title="E42">A76</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">a</font> be    <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"> ( <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">d</font> implies <font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:8_1_2_1_5"/><span class="lab"><font color="Green" title="E44">A78</font></span>: 
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">o</font> = <font color="Maroon" title="c12">a</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E3:8_1_2_1_5"/>
<font color="Maroon" title="c13">o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_2_1_5"><span class="lab"><font color="Green" title="E44">A78</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E4:8_1_2_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A79</font></span>: 
<font color="Maroon" title="c13">o</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:8_1_2_1_5"/><span class="lab"><font color="Green" title="E46">A80</font></span>: 
<font color="Maroon" title="c13">o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_2_1_5"><span class="lab"><font color="Green" title="E44">A78</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">oo</font> = <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c13">o</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_2_1_5_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:8_1_2_1_5_1"/>
( ( <font color="Maroon" title="c13">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c13">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> ) or ( <font color="Maroon" title="c13">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c13">o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k</font> ) or ( <font color="Maroon" title="c13">o</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c13">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> ) or ( <font color="Maroon" title="c13">o</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c13">o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1_2_1_5"><span class="lab"><font color="Green" title="E45">A79</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_2_1_5_1_1"><span class="kw">suppose </span></a><a NAME="E1:8_1_2_1_5_1_1"/><span class="lab"><font color="Green" title="E47">A81</font></span>: 
( <font color="Maroon" title="c13">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c13">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:8_1_2_1_5_1_1"/><span class="kw">then </span>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c9">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:8_1_2_1"><span class="lab"><font color="Green" title="E30">A60</font></span></a>, <a class="txt" href="#E76:8_1_2_1"><span class="lab"><font color="Green" title="E43">A77</font></span></a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A82</font></span>: <font color="Maroon" title="c9">E</font> = 
<span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a></span>
<br/>.= 
<span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:8_1_2_1"><span class="lab"><font color="Green" title="E30">A60</font></span></a>, <a class="txt" href="#E40:8_1_2_1"><span class="lab"><font color="Green" title="E31">A61</font></span></a>, <a class="txt" href="#E76:8_1_2_1"><span class="lab"><font color="Green" title="E43">A77</font></span></a>, <a class="txt" href="#E1:8_1_2_1_5"><span class="lab"><font color="Green" title="E44">A78</font></span></a>, <a class="txt" href="#E1:8_1_2_1_5_1_1"><span class="lab"><font color="Green" title="E47">A81</font></span></a></span>
<br/>.= 
<span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 1</span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></span>
;<br/><a NAME="E4:8_1_2_1_5_1_1"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c7">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:8_1_2_1"><span class="lab"><font color="Green" title="E14">A41</font></span></a>, <a class="txt" href="#E1:8_1_2_1_5_1_1"><span class="lab"><font color="Green" title="E47">A81</font></span></a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c7">D</font> = 
<span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a></span>
<br/>.= 
<span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:8_1_2_1"><span class="lab"><font color="Green" title="E14">A41</font></span></a>, <a class="txt" href="#E14:8_1_2_1"><span class="lab"><font color="Green" title="E15">A42</font></span></a>, <a class="txt" href="#E1:8_1_2_1_5"><span class="lab"><font color="Green" title="E44">A78</font></span></a>, <a class="txt" href="#E1:8_1_2_1_5_1_1"><span class="lab"><font color="Green" title="E47">A81</font></span></a></span>
<br/>.= 
<span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 1</span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></span>
;<br/><span class="kw">hence </span><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">a</font> = 
<font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:8_1_2_1"><span class="lab"><font color="Green" title="E16">A43</font></span></a>, <a class="txt" href="#E1:8_1_2_1_5_1_1"><span class="lab"><font color="Green" title="E47">A81</font></span></a>, <a class="ref" href="afinsq_2.html#T65" title="AFINSQ_2:th.65">AFINSQ_2:65</a></span>
<br/>.= 
<font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:8_1_2_1"><span class="lab"><font color="Green" title="E32">A62</font></span></a>, <a class="txt" href="#E1:8_1_2_1_5_1_1"><span class="lab"><font color="Green" title="E47">A81</font></span></a>, <a class="txt" href="#E3:8_1_2_1_5_1_1"><span class="lab"><font color="Green" title="E48">A82</font></span></a>, <a class="ref" href="afinsq_2.html#T65" title="AFINSQ_2:th.65">AFINSQ_2:65</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_2_1_5_1_2"><span class="kw">suppose </span></a><a NAME="E1:8_1_2_1_5_1_2"/><span class="lab"><font color="Green" title="E47">A83</font></span>: 
( <font color="Maroon" title="c13">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c13">o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c15">do</font> =  <a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E3:8_1_2_1_5_1_2"/><span class="lab"><font color="Green" title="E48">A84</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c15">do</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E47">A83</font></span></a></span>;<br/><a NAME="E4:8_1_2_1_5_1_2"/>
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c15">do</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E5:8_1_2_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A85</font></span>: 
<span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c15">do</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_2.html#T4" title="NAT_2:th.4">NAT_2:4</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c16">d9</font> = <font color="Maroon" title="c7">D</font>;<br/><a NAME="E6:8_1_2_1_5_1_2"/><span class="lab"><font color="Green" title="E50">A86</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c15">do</font> holds <br/><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 ;<br/><a NAME="E7:8_1_2_1_5_1_2"/>
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c15">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c7">D</font></span>)</span> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">D</font> holds <br/><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c15">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E47">A83</font></span></a>, <a class="txt" href="#E3:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E48">A84</font></span></a></span>;<br/><a NAME="E8:8_1_2_1_5_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">do</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c7">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E50">A86</font></span></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></span>;<br/><a NAME="E9:8_1_2_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A87</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c15">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c7">D</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:8_1_2_1"><span class="lab"><font color="Green" title="E16">A43</font></span></a>, <a class="ref" href="afinsq_2.html#T67" title="AFINSQ_2:th.67">AFINSQ_2:67</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">bo</font> = <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><a NAME="E11:8_1_2_1_5_1_2"/><span class="lab"><font color="Green" title="E52">A88</font></span>: 
<font color="Maroon" title="c17">bo</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="prepower.html#T12" title="PREPOWER:th.12">PREPOWER:12</a></span>;<br/><div><span class="lab"><font color="Green" title="E53">A89</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1_5_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c18">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> implies <font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c18">k</font> )</span><br/><span class="kw">assume </span><a NAME="E1:8_1_2_1_5_1_2_1"/>
<font color="Maroon" title="c18">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</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" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c18">k</font></span><br/><a NAME="E2:8_1_2_1_5_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T36" title="AFINSQ_1:th.36">AFINSQ_1:36</a></span>;<br/><a NAME="E3:8_1_2_1_5_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E4:8_1_2_1_5_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/><span class="kw">hence </span><a NAME="E5:8_1_2_1_5_1_2_1"/>
<font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c18">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E47">A83</font></span></a>, <a class="ref" href="afinsq_1.html#D5" title="AFINSQ_1:def.5">AFINSQ_1:def 5</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="c18">o1</font> = <font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E14:8_1_2_1_5_1_2"/>
<font color="Maroon" title="c18">o1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E47">A83</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E15:8_1_2_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A90</font></span>: 
<font color="Maroon" title="c18">o1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T149" title="XREAL_1:th.149">XREAL_1:149</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">oo1</font> = <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c18">o1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E47">A83</font></span></a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c18">o1</font></span>)</span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">do1</font> = <font color="Maroon" title="c7">D</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c18">o1</font> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E18:8_1_2_1_5_1_2"/><span class="lab"><font color="Green" title="E55">A91</font></span>: 
for <font color="Olive" title="b1">u</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">oo1</font> holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">x</font>]
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c21">d91</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E20:8_1_2_1_5_1_2"/><span class="lab"><font color="Green" title="E56">A92</font></span>: 
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c21">d91</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">oo1</font> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">oo1</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c21">d91</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="stirl2_1.html#S5" title="STIRL2_1:sch.5">STIRL2_1:sch 5</a>(<a class="txt" href="#E18:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E55">A91</font></span></a>);<br/></span><a NAME="E21:8_1_2_1_5_1_2"/><span class="lab"><font color="Green" title="E57">A93</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c7">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:8_1_2_1"><span class="lab"><font color="Green" title="E14">A41</font></span></a></span>;<br/><a NAME="E22:8_1_2_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A94</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c20">do1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">o1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E54">A90</font></span></a>, <a class="ref" href="afinsq_1.html#T14" title="AFINSQ_1:th.14">AFINSQ_1:14</a></span>;<br/><a NAME="E23:8_1_2_1_5_1_2"/><span class="lab"><font color="Green" title="E59">A96</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c20">do1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c21">d91</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:8_1_2_1"><span class="lab"><font color="Green" title="E14">A41</font></span></a>, <a class="txt" href="#E20:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E56">A92</font></span></a>, <a class="txt" href="#E22:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E58">A94</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1_5_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c22">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c21">d91</font> implies <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c21">d91</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">k</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c23">K</font> = <font color="Maroon" title="c22">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">assume </span><a NAME="E2:8_1_2_1_5_1_2_2"/><span class="lab"><font color="Green" title="E60">A97</font></span>: 
<font color="Maroon" title="c22">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c21">d91</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c21">d91</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">k</font></span><br/><a NAME="E3:8_1_2_1_5_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c22">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c21">d91</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E4:8_1_2_1_5_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">o1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c22">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E58">A94</font></span></a>, <a class="txt" href="#E23:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E59">A96</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E5:8_1_2_1_5_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">o1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c22">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">D</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E6:8_1_2_1_5_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">o1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c22">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">o1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">o1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:8_1_2_1"><span class="lab"><font color="Green" title="E15">A42</font></span></a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c21">d91</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">K</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">o1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">o1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E56">A92</font></span></a>, <a class="txt" href="#E2:8_1_2_1_5_1_2_2"><span class="lab"><font color="Green" title="E60">A97</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">o1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">o1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span>

;<br/><span class="kw">hence </span><a NAME="E8:8_1_2_1_5_1_2_2"/>
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c21">d91</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</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><a NAME="E25:8_1_2_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A98</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c21">d91</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="numeral1.html#T5" target="_self" title="NUMERAL1:th.5">Th5</a></span>;<br/><div><span class="lab"><font color="Green" title="E61">A99</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1_5_1_2_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c22">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c21">d91</font> implies <font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c22">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">d91</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">k</font> )</span><br/><span class="kw">assume </span><a NAME="E1:8_1_2_1_5_1_2_3"/><span class="lab"><font color="Green" title="E62">A100</font></span>: 
<font color="Maroon" title="c22">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c21">d91</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c22">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">d91</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">k</font></span><br/><span class="kw">thus </span><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c22">k</font></span>)</span> = 
<font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c20">do1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c22">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E47">A83</font></span></a>, <a class="txt" href="#E22:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E58">A94</font></span></a>, <a class="ref" href="afinsq_1.html#T36" title="AFINSQ_1:th.36">AFINSQ_1:36</a></span>
<br/>.= 
<font color="Maroon" title="c21">d91</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E56">A92</font></span></a>, <a class="txt" href="#E22:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E58">A94</font></span></a>, <a class="txt" href="#E1:8_1_2_1_5_1_2_3"><span class="lab"><font color="Green" title="E62">A100</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> <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">D</font> = 
1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c21">d91</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:8_1_2_1"><span class="lab"><font color="Green" title="E14">A41</font></span></a>, <a class="txt" href="#E1:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E47">A83</font></span></a>, <a class="txt" href="#E20:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E56">A92</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c21">d91</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a></span>
;<br/><a NAME="E28:8_1_2_1_5_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c21">d91</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E53">A89</font></span></a>, <a class="txt" href="#E26:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E61">A99</font></span></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></span>;<br/><a NAME="E29:8_1_2_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A101</font></span>: 
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c7">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c21">d91</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T67" title="AFINSQ_2:th.67">AFINSQ_2:67</a></span>;<br/><a NAME="E30:8_1_2_1_5_1_2"/>
( ( for <font color="Olive" title="b1">x</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c20">do1</font> holds <br/><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c21">d91</font> holds <br/><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c20">do1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">d91</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E56">A92</font></span></a>, <a class="txt" href="#E22:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E58">A94</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/><a NAME="E31:8_1_2_1_5_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">do1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c21">d91</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E59">A96</font></span></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></span>;<br/><a NAME="E32:8_1_2_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A102</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c20">do1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c21">d91</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:8_1_2_1"><span class="lab"><font color="Green" title="E16">A43</font></span></a>, <a class="ref" href="afinsq_2.html#T67" title="AFINSQ_2:th.67">AFINSQ_2:67</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c22">bo1</font> = <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c23">ok1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E35:8_1_2_1_5_1_2"/><span class="lab"><font color="Green" title="E64">A103</font></span>: 
<font color="Maroon" title="c23">ok1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">o1</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1_5_1_2_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c24">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">D</font> implies <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">k</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c25">K</font> = <font color="Maroon" title="c24">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">assume </span><a NAME="E2:8_1_2_1_5_1_2_5"/>
<font color="Maroon" title="c24">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span><br/><span class="kw">then </span><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c25">K</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c24">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c24">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:8_1_2_1"><span class="lab"><font color="Green" title="E15">A42</font></span></a>, <a class="txt" href="#E1:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E47">A83</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c24">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c24">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c24">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c24">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span>

;<br/><span class="kw">hence </span><a NAME="E4:8_1_2_1_5_1_2_5"/>
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</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><a NAME="E37:8_1_2_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A104</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c7">D</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="numeral1.html#T5" target="_self" title="NUMERAL1:th.5">Th5</a></span>;<br/><a NAME="E38:8_1_2_1_5_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c7">D</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E52">A88</font></span></a>, <a class="ref" href="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></span>;<br/><a NAME="E39:8_1_2_1_5_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c7">D</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c15">do</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E51">A87</font></span></a>, <a class="ref" href="nat_d.html#T19" title="NAT_D:th.19">NAT_D:19</a></span>;<br/><a NAME="E40:8_1_2_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A105</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c7">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E49">A85</font></span></a>, <a class="txt" href="#E37:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E65">A104</font></span></a>, <a class="ref" href="nat_d.html#T3" title="NAT_D:th.3">NAT_D:3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c24">do</font> =  <a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E42:8_1_2_1_5_1_2"/><span class="lab"><font color="Green" title="E67">A106</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c24">do</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E47">A83</font></span></a></span>;<br/><a NAME="E43:8_1_2_1_5_1_2"/>
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c24">do</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E44:8_1_2_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A107</font></span>: 
<span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c24">do</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_2.html#T4" title="NAT_2:th.4">NAT_2:4</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c25">d9</font> = <font color="Maroon" title="c9">E</font>;<br/><a NAME="E45:8_1_2_1_5_1_2"/><span class="lab"><font color="Green" title="E69">A108</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c24">do</font> holds <br/><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 ;<br/><a NAME="E46:8_1_2_1_5_1_2"/>
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c9">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c24">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c9">E</font></span>)</span> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c9">E</font> holds <br/><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c24">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E47">A83</font></span></a>, <a class="txt" href="#E42:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E67">A106</font></span></a></span>;<br/><a NAME="E47:8_1_2_1_5_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">do</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c9">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E69">A108</font></span></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></span>;<br/><a NAME="E48:8_1_2_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A109</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c24">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c9">E</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:8_1_2_1"><span class="lab"><font color="Green" title="E32">A62</font></span></a>, <a class="ref" href="afinsq_2.html#T67" title="AFINSQ_2:th.67">AFINSQ_2:67</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c26">bo</font> = <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><a NAME="E50:8_1_2_1_5_1_2"/><span class="lab"><font color="Green" title="E71">A110</font></span>: 
<font color="Maroon" title="c26">bo</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="prepower.html#T12" title="PREPOWER:th.12">PREPOWER:12</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c27">ok1</font> = <font color="Maroon" title="c23">ok1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c28">g1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">ok1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="numeral1.html#T3" target="_self" title="NUMERAL1:th.3">Th3</a></span>;<br/><a NAME="E53:8_1_2_1_5_1_2"/><span class="lab"><font color="Green" title="E72">A111</font></span>: 
1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a></span>;<br/><a NAME="E54:8_1_2_1_5_1_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E55:8_1_2_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A112</font></span>: 
<font color="Maroon" title="c18">o1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/><a NAME="E56:8_1_2_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A113</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c20">do1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c28">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E58">A94</font></span></a>, <a class="txt" href="#E35:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E64">A103</font></span></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>;<br/><a NAME="E57:8_1_2_1_5_1_2"/><span class="lab"><font color="Green" title="E75">A114</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c28">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">o1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E64">A103</font></span></a>, <a class="txt" href="#E55:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E73">A112</font></span></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>;<br/><a NAME="E58:8_1_2_1_5_1_2"/><span class="lab"><font color="Green" title="E76">A115</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c20">do1</font> holds <br/><font color="Maroon" title="c20">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1_5_1_2_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c29">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c29">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c20">do1</font> implies <font color="Maroon" title="c20">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">i</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c30">I</font> = <font color="Maroon" title="c29">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:8_1_2_1_5_1_2_6"/><span class="lab"><font color="Green" title="E77">A116</font></span>: 
<font color="Maroon" title="c29">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c20">do1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">i</font></span><br/>

<a NAME="E3:8_1_2_1_5_1_2_6"/><span class="kw">then </span>
<font color="Maroon" title="c29">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">o1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E54">A90</font></span></a>, <a class="txt" href="#E21:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E57">A93</font></span></a>, <a class="ref" href="afinsq_1.html#T14" title="AFINSQ_1:th.14">AFINSQ_1:14</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E78">A117</font></span>: <font color="Maroon" title="c28">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c30">I</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">I</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E75">A114</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">I</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c3">b1</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c30">I</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#D1" title="PREPOWER:def.1">PREPOWER:def 1</a></span>
;<br/>
<a NAME="E5:8_1_2_1_5_1_2_6"/><span class="lab"><font color="Green" title="E79">A118</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c20">do1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></span>;<br/>
<a NAME="E6:8_1_2_1_5_1_2_6"/><span class="kw">then </span>
<font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">b1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:8_1_2_1"><span class="lab"><font color="Green" title="E13">A40</font></span></a>, <a class="txt" href="#E13:8_1_2_1"><span class="lab"><font color="Green" title="E14">A41</font></span></a>, <a class="txt" href="#E2:8_1_2_1_5_1_2_6"><span class="lab"><font color="Green" title="E77">A116</font></span></a></span>;<br/>
<a NAME="E7:8_1_2_1_5_1_2_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A119</font></span>: 
<font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<font color="Maroon" title="c20">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">i</font> = 
<font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1_2_1_5_1_2_6"><span class="lab"><font color="Green" title="E77">A116</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">i</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:8_1_2_1"><span class="lab"><font color="Green" title="E15">A42</font></span></a>, <a class="txt" href="#E2:8_1_2_1_5_1_2_6"><span class="lab"><font color="Green" title="E77">A116</font></span></a>, <a class="txt" href="#E5:8_1_2_1_5_1_2_6"><span class="lab"><font color="Green" title="E79">A118</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E9:8_1_2_1_5_1_2_6"/>
<font color="Maroon" title="c20">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1_2_1_5_1_2_6"><span class="lab"><font color="Green" title="E78">A117</font></span></a>, <a class="txt" href="#E7:8_1_2_1_5_1_2_6"><span class="lab"><font color="Green" title="E80">A119</font></span></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</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> <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c28">g1</font> = 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">ok1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T68" title="AFINSQ_2:th.68">AFINSQ_2:68</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">ok1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#T12" title="SERIES_1:th.12">SERIES_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">ok1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c27">ok1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="series_1.html#T26" title="SERIES_1:th.26">SERIES_1:26</a></span>
<br/>.= 
 <a href="binop_2.html#K13" title="BINOP_2:func.13">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="binop_2.html#K17" title="BINOP_2:func.17">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E64">A103</font></span></a></span>
<br/>.= 
 <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E72">A111</font></span></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1

;<br/><a NAME="E60:8_1_2_1_5_1_2"/><span class="kw">then </span>
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c20">do1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E74">A113</font></span></a>, <a class="txt" href="#E58:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E76">A115</font></span></a>, <a class="ref" href="afinsq_2.html#T69" title="AFINSQ_2:th.69">AFINSQ_2:69</a>, <a class="ref" href="xreal_1.html#T147" title="XREAL_1:th.147">XREAL_1:147</a></span>;<br/><a NAME="E61:8_1_2_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A120</font></span>: 
<span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c20">do1</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T27" title="NAT_D:th.27">NAT_D:27</a></span>;<br/><a NAME="E62:8_1_2_1_5_1_2"/>
<font color="Maroon" title="c22">bo1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="prepower.html#T12" title="PREPOWER:th.12">PREPOWER:12</a></span>;<br/><a NAME="E63:8_1_2_1_5_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c21">d91</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E60">A98</font></span></a>, <a class="ref" href="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></span>;<br/><a NAME="E64:8_1_2_1_5_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c21">d91</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c20">do1</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E63">A102</font></span></a>, <a class="ref" href="nat_d.html#T19" title="NAT_D:th.19">NAT_D:19</a></span>;<br/><a NAME="E65:8_1_2_1_5_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c21">d91</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E61:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E77">A120</font></span></a>, <a class="txt" href="#E25:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E60">A98</font></span></a>, <a class="ref" href="nat_d.html#T3" title="NAT_D:th.3">NAT_D:3</a></span>;<br/><a NAME="E66:8_1_2_1_5_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E66">A105</font></span></a>, <a class="txt" href="#E29:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E62">A101</font></span></a>, <a class="ref" href="afinsq_2.html#T65" title="AFINSQ_2:th.65">AFINSQ_2:65</a></span>;<br/><a NAME="E67:8_1_2_1_5_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:8_1_2_1"><span class="lab"><font color="Green" title="E14">A41</font></span></a>, <a class="txt" href="#E14:8_1_2_1"><span class="lab"><font color="Green" title="E15">A42</font></span></a>, <a class="txt" href="#E1:8_1_2_1_5"><span class="lab"><font color="Green" title="E44">A78</font></span></a></span>;<br/><a NAME="E68:8_1_2_1_5_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 1</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></span>;<br/><a NAME="E69:8_1_2_1_5_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 1</span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E70:8_1_2_1_5_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></span>;<br/><a NAME="E71:8_1_2_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A121</font></span>: 
<font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E52">A88</font></span></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c29">o1</font> = <font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c30">do1</font> = <font color="Maroon" title="c9">E</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c29">o1</font> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E74:8_1_2_1_5_1_2"/><span class="lab"><font color="Green" title="E79">A122</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c9">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:8_1_2_1"><span class="lab"><font color="Green" title="E30">A60</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1_5_1_2_8"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c31">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c31">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c9">E</font> implies <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c31">k</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c32">K</font> = <font color="Maroon" title="c31">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">assume </span><a NAME="E2:8_1_2_1_5_1_2_8"/>
<font color="Maroon" title="c31">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c9">E</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c31">k</font></span><br/><span class="kw">then </span><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c32">K</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c31">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c31">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:8_1_2_1"><span class="lab"><font color="Green" title="E31">A61</font></span></a>, <a class="txt" href="#E1:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E47">A83</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c31">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c31">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c31">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c31">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span>

;<br/><span class="kw">hence </span><a NAME="E4:8_1_2_1_5_1_2_8"/>
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c31">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</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><a NAME="E76:8_1_2_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A123</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c9">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="numeral1.html#T5" target="_self" title="NUMERAL1:th.5">Th5</a></span>;<br/><a NAME="E77:8_1_2_1_5_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c9">E</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E71">A110</font></span></a>, <a class="ref" href="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></span>;<br/><a NAME="E78:8_1_2_1_5_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c9">E</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c24">do</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E70">A109</font></span></a>, <a class="ref" href="nat_d.html#T19" title="NAT_D:th.19">NAT_D:19</a></span>;<br/><a NAME="E79:8_1_2_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A124</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c9">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E68">A107</font></span></a>, <a class="txt" href="#E76:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E80">A123</font></span></a>, <a class="ref" href="nat_d.html#T3" title="NAT_D:th.3">NAT_D:3</a></span>;<br/><div><span class="lab"><font color="Green" title="E82">A125</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1_5_1_2_9"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c31">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c31">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> implies <font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c31">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c31">k</font> )</span><br/><span class="kw">assume </span><a NAME="E1:8_1_2_1_5_1_2_9"/>
<font color="Maroon" title="c31">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</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" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c31">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c31">k</font></span><br/><a NAME="E2:8_1_2_1_5_1_2_9"/><span class="kw">then </span>
<font color="Maroon" title="c31">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T36" title="AFINSQ_1:th.36">AFINSQ_1:36</a></span>;<br/><a NAME="E3:8_1_2_1_5_1_2_9"/><span class="kw">then </span>
<font color="Maroon" title="c31">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E4:8_1_2_1_5_1_2_9"/><span class="kw">then </span>
<font color="Maroon" title="c31">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/><span class="kw">hence </span><a NAME="E5:8_1_2_1_5_1_2_9"/>
<font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c31">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c31">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E47">A83</font></span></a>, <a class="ref" href="afinsq_1.html#D5" title="AFINSQ_1:def.5">AFINSQ_1:def 5</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="c31">oo1</font> = <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c29">o1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E47">A83</font></span></a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c29">o1</font></span>)</span>;<br/><a NAME="E82:8_1_2_1_5_1_2"/><span class="lab"><font color="Green" title="E83">A126</font></span>: 
for <font color="Olive" title="b1">u</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">oo1</font> holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">x</font>]
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c32">d91</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E84:8_1_2_1_5_1_2"/><span class="lab"><font color="Green" title="E84">A127</font></span>: 
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c32">d91</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">oo1</font> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">oo1</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c32">d91</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="stirl2_1.html#S5" title="STIRL2_1:sch.5">STIRL2_1:sch 5</a>(<a class="txt" href="#E82:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E83">A126</font></span></a>);<br/></span><a NAME="E85:8_1_2_1_5_1_2"/>
<font color="Maroon" title="c29">o1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E47">A83</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E86:8_1_2_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A128</font></span>: 
<font color="Maroon" title="c29">o1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T149" title="XREAL_1:th.149">XREAL_1:149</a></span>;<br/><a NAME="E87:8_1_2_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A129</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c30">do1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">o1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E76:8_1_2_1"><span class="lab"><font color="Green" title="E43">A77</font></span></a>, <a class="txt" href="#E74:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E79">A122</font></span></a>, <a class="ref" href="afinsq_1.html#T14" title="AFINSQ_1:th.14">AFINSQ_1:14</a></span>;<br/><div><span class="lab"><font color="Green" title="E87">A131</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1_5_1_2_10"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c33">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c33">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c32">d91</font> implies <font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c33">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">d91</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c33">k</font> )</span><br/><span class="kw">assume </span><a NAME="E1:8_1_2_1_5_1_2_10"/><span class="lab"><font color="Green" title="E88">A132</font></span>: 
<font color="Maroon" title="c33">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c32">d91</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c33">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">d91</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c33">k</font></span><br/><span class="kw">thus </span><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c33">k</font></span>)</span> = 
<font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c30">do1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c33">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E47">A83</font></span></a>, <a class="txt" href="#E87:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E86">A129</font></span></a>, <a class="ref" href="afinsq_1.html#T36" title="AFINSQ_1:th.36">AFINSQ_1:36</a></span>
<br/>.= 
<font color="Maroon" title="c32">d91</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c33">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E84:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E84">A127</font></span></a>, <a class="txt" href="#E87:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E86">A129</font></span></a>, <a class="txt" href="#E1:8_1_2_1_5_1_2_10"><span class="lab"><font color="Green" title="E88">A132</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="c33">bo1</font> = <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c34">ok1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E91:8_1_2_1_5_1_2"/><span class="lab"><font color="Green" title="E88">A133</font></span>: 
<font color="Maroon" title="c34">ok1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">o1</font>
 ;<br/><a NAME="E92:8_1_2_1_5_1_2"/><span class="lab"><font color="Green" title="E89">A134</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c9">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c30">do1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c32">d91</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:8_1_2_1"><span class="lab"><font color="Green" title="E30">A60</font></span></a>, <a class="txt" href="#E76:8_1_2_1"><span class="lab"><font color="Green" title="E43">A77</font></span></a>, <a class="txt" href="#E84:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E84">A127</font></span></a>, <a class="txt" href="#E87:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E86">A129</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1_5_1_2_11"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c35">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c35">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c32">d91</font> implies <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c32">d91</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">k</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c36">K</font> = <font color="Maroon" title="c35">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">assume </span><a NAME="E2:8_1_2_1_5_1_2_11"/><span class="lab"><font color="Green" title="E90">A135</font></span>: 
<font color="Maroon" title="c35">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c32">d91</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c32">d91</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">k</font></span><br/><a NAME="E3:8_1_2_1_5_1_2_11"/><span class="kw">then </span>
<font color="Maroon" title="c35">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c32">d91</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E4:8_1_2_1_5_1_2_11"/><span class="kw">then </span>
<font color="Maroon" title="c29">o1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c35">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c9">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E87:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E86">A129</font></span></a>, <a class="txt" href="#E92:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E89">A134</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E5:8_1_2_1_5_1_2_11"/><span class="kw">then </span>
<font color="Maroon" title="c29">o1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c35">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c9">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E6:8_1_2_1_5_1_2_11"/><span class="kw">then </span>
<font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">o1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c35">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">o1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c35">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">o1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c35">k</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:8_1_2_1"><span class="lab"><font color="Green" title="E31">A61</font></span></a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c32">d91</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c36">K</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">o1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c35">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">o1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c35">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E84:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E84">A127</font></span></a>, <a class="txt" href="#E2:8_1_2_1_5_1_2_11"><span class="lab"><font color="Green" title="E90">A135</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">o1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c35">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c35">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">o1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c35">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c35">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span>

;<br/><span class="kw">hence </span><a NAME="E8:8_1_2_1_5_1_2_11"/>
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c32">d91</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</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><a NAME="E94:8_1_2_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A136</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c32">d91</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="numeral1.html#T5" target="_self" title="NUMERAL1:th.5">Th5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c35">ok1</font> = <font color="Maroon" title="c34">ok1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c36">g1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c35">ok1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="numeral1.html#T3" target="_self" title="NUMERAL1:th.3">Th3</a></span>;<br/><a NAME="E97:8_1_2_1_5_1_2"/><span class="lab"><font color="Green" title="E91">A137</font></span>: 
1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a></span>;<br/><a NAME="E98:8_1_2_1_5_1_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E99:8_1_2_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A138</font></span>: 
<font color="Maroon" title="c29">o1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/><a NAME="E100:8_1_2_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A139</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c30">do1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c36">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E87:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E86">A129</font></span></a>, <a class="txt" href="#E91:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E88">A133</font></span></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>;<br/><a NAME="E101:8_1_2_1_5_1_2"/><span class="lab"><font color="Green" title="E94">A140</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c36">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">o1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E91:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E88">A133</font></span></a>, <a class="txt" href="#E99:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E92">A138</font></span></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>;<br/><a NAME="E102:8_1_2_1_5_1_2"/><span class="lab"><font color="Green" title="E95">A141</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c30">do1</font> holds <br/><font color="Maroon" title="c30">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1_5_1_2_12"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c37">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c37">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c30">do1</font> implies <font color="Maroon" title="c30">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c37">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c37">i</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c38">I</font> = <font color="Maroon" title="c37">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:8_1_2_1_5_1_2_12"/><span class="lab"><font color="Green" title="E96">A142</font></span>: 
<font color="Maroon" title="c37">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c30">do1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c30">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c37">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c37">i</font></span><br/>

<a NAME="E3:8_1_2_1_5_1_2_12"/><span class="kw">then </span>
<font color="Maroon" title="c37">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">o1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E76:8_1_2_1"><span class="lab"><font color="Green" title="E43">A77</font></span></a>, <a class="txt" href="#E86:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E85">A128</font></span></a>, <a class="txt" href="#E74:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E79">A122</font></span></a>, <a class="ref" href="afinsq_1.html#T14" title="AFINSQ_1:th.14">AFINSQ_1:14</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E97">A143</font></span>: <font color="Maroon" title="c36">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c38">I</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c38">I</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E101:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E94">A140</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c38">I</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c3">b1</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c38">I</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#D1" title="PREPOWER:def.1">PREPOWER:def 1</a></span>
;<br/>
<a NAME="E5:8_1_2_1_5_1_2_12"/><span class="lab"><font color="Green" title="E98">A144</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c30">do1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c9">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></span>;<br/>
<a NAME="E6:8_1_2_1_5_1_2_12"/><span class="kw">then </span>
<font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c37">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">b1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:8_1_2_1"><span class="lab"><font color="Green" title="E29">A59</font></span></a>, <a class="txt" href="#E39:8_1_2_1"><span class="lab"><font color="Green" title="E30">A60</font></span></a>, <a class="txt" href="#E2:8_1_2_1_5_1_2_12"><span class="lab"><font color="Green" title="E96">A142</font></span></a></span>;<br/>
<a NAME="E7:8_1_2_1_5_1_2_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A145</font></span>: 
<font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c37">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<font color="Maroon" title="c30">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c37">i</font> = 
<font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c37">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1_2_1_5_1_2_12"><span class="lab"><font color="Green" title="E96">A142</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c37">i</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c37">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:8_1_2_1"><span class="lab"><font color="Green" title="E31">A61</font></span></a>, <a class="txt" href="#E2:8_1_2_1_5_1_2_12"><span class="lab"><font color="Green" title="E96">A142</font></span></a>, <a class="txt" href="#E5:8_1_2_1_5_1_2_12"><span class="lab"><font color="Green" title="E98">A144</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E9:8_1_2_1_5_1_2_12"/>
<font color="Maroon" title="c30">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c37">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c37">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1_2_1_5_1_2_12"><span class="lab"><font color="Green" title="E97">A143</font></span></a>, <a class="txt" href="#E7:8_1_2_1_5_1_2_12"><span class="lab"><font color="Green" title="E99">A145</font></span></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</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> <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c36">g1</font> = 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c35">ok1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T68" title="AFINSQ_2:th.68">AFINSQ_2:68</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c35">ok1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#T12" title="SERIES_1:th.12">SERIES_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c35">ok1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c35">ok1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="series_1.html#T26" title="SERIES_1:th.26">SERIES_1:26</a></span>
<br/>.= 
 <a href="binop_2.html#K13" title="BINOP_2:func.13">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="binop_2.html#K17" title="BINOP_2:func.17">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E91:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E88">A133</font></span></a></span>
<br/>.= 
 <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E97:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E91">A137</font></span></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1

;<br/><a NAME="E104:8_1_2_1_5_1_2"/><span class="kw">then </span>
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c30">do1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E100:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E93">A139</font></span></a>, <a class="txt" href="#E102:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E95">A141</font></span></a>, <a class="ref" href="afinsq_2.html#T69" title="AFINSQ_2:th.69">AFINSQ_2:69</a>, <a class="ref" href="xreal_1.html#T147" title="XREAL_1:th.147">XREAL_1:147</a></span>;<br/><a NAME="E105:8_1_2_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A146</font></span>: 
<span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c30">do1</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T27" title="NAT_D:th.27">NAT_D:27</a></span>;<br/><a NAME="E106:8_1_2_1_5_1_2"/>
( ( for <font color="Olive" title="b1">x</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c30">do1</font> holds <br/><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c32">d91</font> holds <br/><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c30">do1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">d91</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E84:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E84">A127</font></span></a>, <a class="txt" href="#E87:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E86">A129</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/><a NAME="E107:8_1_2_1_5_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">do1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c32">d91</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E92:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E89">A134</font></span></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></span>;<br/><a NAME="E108:8_1_2_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A147</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c30">do1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c32">d91</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:8_1_2_1"><span class="lab"><font color="Green" title="E32">A62</font></span></a>, <a class="ref" href="afinsq_2.html#T67" title="AFINSQ_2:th.67">AFINSQ_2:67</a></span>;<br/><a NAME="E109:8_1_2_1_5_1_2"/>
<font color="Maroon" title="c33">bo1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="prepower.html#T12" title="PREPOWER:th.12">PREPOWER:12</a></span>;<br/><a NAME="E110:8_1_2_1_5_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c32">d91</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E94:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E90">A136</font></span></a>, <a class="ref" href="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></span>;<br/><a NAME="E111:8_1_2_1_5_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c32">d91</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c30">do1</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E108:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E97">A147</font></span></a>, <a class="ref" href="nat_d.html#T19" title="NAT_D:th.19">NAT_D:19</a></span>;<br/><a NAME="E112:8_1_2_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A148</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c32">d91</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E105:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E96">A146</font></span></a>, <a class="txt" href="#E94:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E90">A136</font></span></a>, <a class="ref" href="nat_d.html#T3" title="NAT_D:th.3">NAT_D:3</a></span>;<br/> <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c9">E</font> = 
1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c32">d91</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:8_1_2_1"><span class="lab"><font color="Green" title="E30">A60</font></span></a>, <a class="txt" href="#E76:8_1_2_1"><span class="lab"><font color="Green" title="E43">A77</font></span></a>, <a class="txt" href="#E1:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E47">A83</font></span></a>, <a class="txt" href="#E84:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E84">A127</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c32">d91</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a></span>
;<br/><a NAME="E114:8_1_2_1_5_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c32">d91</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E80:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E82">A125</font></span></a>, <a class="txt" href="#E88:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E87">A131</font></span></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></span>;<br/><a NAME="E115:8_1_2_1_5_1_2"/><span class="kw">then </span>
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c9">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c32">d91</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T67" title="AFINSQ_2:th.67">AFINSQ_2:67</a></span>;<br/><a NAME="E116:8_1_2_1_5_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E79:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E81">A124</font></span></a>, <a class="txt" href="#E112:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E98">A148</font></span></a>, <a class="ref" href="afinsq_2.html#T65" title="AFINSQ_2:th.65">AFINSQ_2:65</a></span>;<br/><a NAME="E117:8_1_2_1_5_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:8_1_2_1"><span class="lab"><font color="Green" title="E30">A60</font></span></a>, <a class="txt" href="#E40:8_1_2_1"><span class="lab"><font color="Green" title="E31">A61</font></span></a>, <a class="txt" href="#E76:8_1_2_1"><span class="lab"><font color="Green" title="E43">A77</font></span></a>, <a class="txt" href="#E1:8_1_2_1_5"><span class="lab"><font color="Green" title="E44">A78</font></span></a></span>;<br/><a NAME="E118:8_1_2_1_5_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 1</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></span>;<br/><a NAME="E119:8_1_2_1_5_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 1</span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E120:8_1_2_1_5_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></span>;<br/><span class="kw">hence </span><a NAME="E121:8_1_2_1_5_1_2"/>
<font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E71:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E78">A121</font></span></a>, <a class="txt" href="#E50:8_1_2_1_5_1_2"><span class="lab"><font color="Green" title="E71">A110</font></span></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_2_1_5_1_3"><span class="kw">suppose </span></a><a NAME="E1:8_1_2_1_5_1_3"/><span class="lab"><font color="Green" title="E47">A149</font></span>: 
( <font color="Maroon" title="c13">o</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c13">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c15">d9</font> = <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">do</font> = <font color="Maroon" title="c7">D</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c13">o</font> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E3:8_1_2_1_5_1_3"/><span class="lab"><font color="Green" title="E48">A150</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c7">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:8_1_2_1"><span class="lab"><font color="Green" title="E14">A41</font></span></a></span>;<br/><a NAME="E4:8_1_2_1_5_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A151</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c16">do</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_2_1_5"><span class="lab"><font color="Green" title="E46">A80</font></span></a>, <a class="ref" href="afinsq_1.html#T14" title="AFINSQ_1:th.14">AFINSQ_1:14</a></span>;<br/><a NAME="E5:8_1_2_1_5_1_3"/><span class="lab"><font color="Green" title="E50">A153</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">oo</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E47">A149</font></span></a>, <a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a></span>;<br/><a NAME="E6:8_1_2_1_5_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A154</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c16">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:8_1_2_1"><span class="lab"><font color="Green" title="E14">A41</font></span></a>, <a class="txt" href="#E4:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E49">A151</font></span></a></span>;<br/><a NAME="E7:8_1_2_1_5_1_3"/><span class="lab"><font color="Green" title="E52">A155</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> holds <br/><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c16">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1_5_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">x</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> implies <font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c16">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:8_1_2_1_5_1_3_1"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</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" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c16">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">x</font></span><br/>

<a NAME="E2:8_1_2_1_5_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E47">A149</font></span></a>, <a class="txt" href="#E5:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E50">A153</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/>
<a NAME="E3:8_1_2_1_5_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:8_1_2_1_5_1_3_1"/>
<font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c16">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E49">A151</font></span></a>, <a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1_5_1_3_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c17">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> implies <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c18">K</font> = <font color="Maroon" title="c17">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">assume </span><a NAME="E2:8_1_2_1_5_1_3_2"/><span class="lab"><font color="Green" title="E53">A156</font></span>: 
<font color="Maroon" title="c17">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</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" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font></span><br/><a NAME="E3:8_1_2_1_5_1_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E4:8_1_2_1_5_1_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E49">A151</font></span></a>, <a class="txt" href="#E6:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E51">A154</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E5:8_1_2_1_5_1_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c17">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">D</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E6:8_1_2_1_5_1_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c17">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c17">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c17">k</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:8_1_2_1"><span class="lab"><font color="Green" title="E15">A42</font></span></a></span>;<br/><span class="kw">then </span><span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c18">K</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c17">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c17">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E49">A151</font></span></a>, <a class="txt" href="#E7:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E52">A155</font></span></a>, <a class="txt" href="#E2:8_1_2_1_5_1_3_2"><span class="lab"><font color="Green" title="E53">A156</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c17">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c17">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c17">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c17">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span>

;<br/><span class="kw">hence </span><a NAME="E8:8_1_2_1_5_1_3_2"/>
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</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><a NAME="E9:8_1_2_1_5_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A157</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="numeral1.html#T5" target="_self" title="NUMERAL1:th.5">Th5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">bo</font> = <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><a NAME="E11:8_1_2_1_5_1_3"/><span class="lab"><font color="Green" title="E54">A158</font></span>: 
1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c18">ok</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E13:8_1_2_1_5_1_3"/><span class="lab"><font color="Green" title="E55">A159</font></span>: 
<font color="Maroon" title="c18">ok</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E47">A149</font></span></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">ok</font> = <font color="Maroon" title="c18">ok</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">g</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">ok</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="numeral1.html#T3" target="_self" title="NUMERAL1:th.3">Th3</a></span>;<br/><span class="lab"><font color="Green" title="E56">A160</font></span>:  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c20">g</font> = 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">ok</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T68" title="AFINSQ_2:th.68">AFINSQ_2:68</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">ok</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#T12" title="SERIES_1:th.12">SERIES_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">ok</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">ok</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="series_1.html#T26" title="SERIES_1:th.26">SERIES_1:26</a></span>
<br/>.= 
 <a href="binop_2.html#K13" title="BINOP_2:func.13">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="binop_2.html#K17" title="BINOP_2:func.17">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E55">A159</font></span></a></span>
<br/>.= 
 <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E54">A158</font></span></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1

;<br/><span class="kw">consider </span><font color="Maroon" title="c21">ok</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E18:8_1_2_1_5_1_3"/><span class="lab"><font color="Green" title="E57">A161</font></span>: 
<font color="Maroon" title="c21">ok</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E47">A149</font></span></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></span>;<br/><a NAME="E19:8_1_2_1_5_1_3"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E20:8_1_2_1_5_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A162</font></span>: 
<font color="Maroon" title="c13">o</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/><a NAME="E21:8_1_2_1_5_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A163</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c20">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E55">A159</font></span></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>;<br/><a NAME="E22:8_1_2_1_5_1_3"/><span class="lab"><font color="Green" title="E60">A164</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c16">do</font> holds <br/><font color="Maroon" title="c16">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1_5_1_3_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c22">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c16">do</font> implies <font color="Maroon" title="c16">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c23">I</font> = <font color="Maroon" title="c22">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:8_1_2_1_5_1_3_4"/><span class="lab"><font color="Green" title="E61">A165</font></span>: 
<font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c16">do</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span><br/>

<a NAME="E3:8_1_2_1_5_1_3_4"/><span class="kw">then </span>
<font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_2_1_5"><span class="lab"><font color="Green" title="E46">A80</font></span></a>, <a class="txt" href="#E3:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E48">A150</font></span></a>, <a class="ref" href="afinsq_1.html#T14" title="AFINSQ_1:th.14">AFINSQ_1:14</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E62">A166</font></span>: <font color="Maroon" title="c20">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">I</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">I</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E59">A163</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">I</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c3">b1</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c23">I</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#D1" title="PREPOWER:def.1">PREPOWER:def 1</a></span>
;<br/>
<a NAME="E5:8_1_2_1_5_1_3_4"/><span class="lab"><font color="Green" title="E63">A167</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c16">do</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></span>;<br/>
<a NAME="E6:8_1_2_1_5_1_3_4"/><span class="kw">then </span>
<font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">b1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:8_1_2_1"><span class="lab"><font color="Green" title="E13">A40</font></span></a>, <a class="txt" href="#E13:8_1_2_1"><span class="lab"><font color="Green" title="E14">A41</font></span></a>, <a class="txt" href="#E2:8_1_2_1_5_1_3_4"><span class="lab"><font color="Green" title="E61">A165</font></span></a></span>;<br/>
<a NAME="E7:8_1_2_1_5_1_3_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A168</font></span>: 
<font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<font color="Maroon" title="c16">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font> = 
<font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1_2_1_5_1_3_4"><span class="lab"><font color="Green" title="E61">A165</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c22">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:8_1_2_1"><span class="lab"><font color="Green" title="E15">A42</font></span></a>, <a class="txt" href="#E2:8_1_2_1_5_1_3_4"><span class="lab"><font color="Green" title="E61">A165</font></span></a>, <a class="txt" href="#E5:8_1_2_1_5_1_3_4"><span class="lab"><font color="Green" title="E63">A167</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E9:8_1_2_1_5_1_3_4"/>
<font color="Maroon" title="c16">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1_2_1_5_1_3_4"><span class="lab"><font color="Green" title="E62">A166</font></span></a>, <a class="txt" href="#E7:8_1_2_1_5_1_3_4"><span class="lab"><font color="Green" title="E64">A168</font></span></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</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><a NAME="E23:8_1_2_1_5_1_3"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c16">do</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c20">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E49">A151</font></span></a>, <a class="txt" href="#E13:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E55">A159</font></span></a>, <a class="txt" href="#E20:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E58">A162</font></span></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>;<br/><a NAME="E24:8_1_2_1_5_1_3"/><span class="kw">then </span>
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c16">do</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E56">A160</font></span></a>, <a class="txt" href="#E22:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E60">A164</font></span></a>, <a class="ref" href="afinsq_2.html#T69" title="AFINSQ_2:th.69">AFINSQ_2:69</a>, <a class="ref" href="xreal_1.html#T147" title="XREAL_1:th.147">XREAL_1:147</a></span>;<br/><a NAME="E25:8_1_2_1_5_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A169</font></span>: 
<span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c16">do</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T27" title="NAT_D:th.27">NAT_D:27</a></span>;<br/><a NAME="E26:8_1_2_1_5_1_3"/>
for <font color="Olive" title="b1">x</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c16">do</font> holds <br/><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/><a NAME="E27:8_1_2_1_5_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">do</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E51">A154</font></span></a>, <a class="txt" href="#E7:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E52">A155</font></span></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></span>;<br/><a NAME="E28:8_1_2_1_5_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A170</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c16">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:8_1_2_1"><span class="lab"><font color="Green" title="E16">A43</font></span></a>, <a class="ref" href="afinsq_2.html#T67" title="AFINSQ_2:th.67">AFINSQ_2:67</a></span>;<br/><a NAME="E29:8_1_2_1_5_1_3"/><span class="lab"><font color="Green" title="E63">A171</font></span>: 
<font color="Maroon" title="c17">bo</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="prepower.html#T12" title="PREPOWER:th.12">PREPOWER:12</a></span>;<br/><a NAME="E30:8_1_2_1_5_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E53">A157</font></span></a>, <a class="ref" href="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></span>;<br/><a NAME="E31:8_1_2_1_5_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p3"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c16">do</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E62">A170</font></span></a>, <a class="ref" href="nat_d.html#T19" title="NAT_D:th.19">NAT_D:19</a></span>;<br/><a NAME="E32:8_1_2_1_5_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E61">A169</font></span></a>, <a class="txt" href="#E9:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E53">A157</font></span></a>, <a class="ref" href="nat_d.html#T3" title="NAT_D:th.3">NAT_D:3</a></span>;<br/><a NAME="E33:8_1_2_1_5_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T65" title="AFINSQ_2:th.65">AFINSQ_2:65</a></span>;<br/><a NAME="E34:8_1_2_1_5_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:8_1_2_1"><span class="lab"><font color="Green" title="E14">A41</font></span></a>, <a class="txt" href="#E14:8_1_2_1"><span class="lab"><font color="Green" title="E15">A42</font></span></a>, <a class="txt" href="#E1:8_1_2_1_5"><span class="lab"><font color="Green" title="E44">A78</font></span></a></span>;<br/><a NAME="E35:8_1_2_1_5_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A172</font></span>: 
<font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E63">A171</font></span></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c22">d9</font> = <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c23">do</font> = <font color="Maroon" title="c9">E</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c13">o</font> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E37:8_1_2_1_5_1_3"/><span class="lab"><font color="Green" title="E65">A173</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c9">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:8_1_2_1"><span class="lab"><font color="Green" title="E30">A60</font></span></a></span>;<br/><a NAME="E38:8_1_2_1_5_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A174</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c23">do</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E76:8_1_2_1"><span class="lab"><font color="Green" title="E43">A77</font></span></a>, <a class="txt" href="#E5:8_1_2_1_5"><span class="lab"><font color="Green" title="E46">A80</font></span></a>, <a class="ref" href="afinsq_1.html#T14" title="AFINSQ_1:th.14">AFINSQ_1:14</a></span>;<br/><a NAME="E39:8_1_2_1_5_1_3"/><span class="lab"><font color="Green" title="E67">A176</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">oo</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E47">A149</font></span></a>, <a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a></span>;<br/><a NAME="E40:8_1_2_1_5_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A177</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c9">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c23">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:8_1_2_1"><span class="lab"><font color="Green" title="E30">A60</font></span></a>, <a class="txt" href="#E76:8_1_2_1"><span class="lab"><font color="Green" title="E43">A77</font></span></a>, <a class="txt" href="#E38:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E66">A174</font></span></a></span>;<br/><a NAME="E41:8_1_2_1_5_1_3"/><span class="lab"><font color="Green" title="E69">A178</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> holds <br/><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c23">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1_5_1_3_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">x</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> implies <font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c23">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c24">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:8_1_2_1_5_1_3_5"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</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" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c23">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c24">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">x</font></span><br/>

<a NAME="E2:8_1_2_1_5_1_3_5"/><span class="kw">then </span>
<font color="Maroon" title="c24">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E47">A149</font></span></a>, <a class="txt" href="#E39:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E67">A176</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/>
<a NAME="E3:8_1_2_1_5_1_3_5"/><span class="kw">then </span>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:8_1_2_1_5_1_3_5"/>
<font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c23">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c24">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E66">A174</font></span></a>, <a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1_5_1_3_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c24">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> implies <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">k</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c25">K</font> = <font color="Maroon" title="c24">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">assume </span><a NAME="E2:8_1_2_1_5_1_3_6"/><span class="lab"><font color="Green" title="E70">A179</font></span>: 
<font color="Maroon" title="c24">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</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" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span><br/><a NAME="E3:8_1_2_1_5_1_3_6"/><span class="kw">then </span>
<font color="Maroon" title="c24">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E4:8_1_2_1_5_1_3_6"/><span class="kw">then </span>
<font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c24">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c9">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E66">A174</font></span></a>, <a class="txt" href="#E40:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E68">A177</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E5:8_1_2_1_5_1_3_6"/><span class="kw">then </span>
<font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c24">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c9">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E6:8_1_2_1_5_1_3_6"/><span class="kw">then </span>
<font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c24">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c24">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c24">k</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:8_1_2_1"><span class="lab"><font color="Green" title="E31">A61</font></span></a></span>;<br/><span class="kw">then </span><span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c25">K</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c24">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c24">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E66">A174</font></span></a>, <a class="txt" href="#E41:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E69">A178</font></span></a>, <a class="txt" href="#E2:8_1_2_1_5_1_3_6"><span class="lab"><font color="Green" title="E70">A179</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c24">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c24">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c24">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c24">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span>

;<br/><span class="kw">hence </span><a NAME="E8:8_1_2_1_5_1_3_6"/>
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</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><a NAME="E43:8_1_2_1_5_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A180</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="numeral1.html#T5" target="_self" title="NUMERAL1:th.5">Th5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c24">ok</font> = <font color="Maroon" title="c21">ok</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c25">g</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">ok</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="numeral1.html#T3" target="_self" title="NUMERAL1:th.3">Th3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c26">bo</font> = <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><a NAME="E47:8_1_2_1_5_1_3"/><span class="lab"><font color="Green" title="E71">A181</font></span>: 
1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a></span>;<br/><a NAME="E48:8_1_2_1_5_1_3"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E49:8_1_2_1_5_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A182</font></span>: 
<font color="Maroon" title="c13">o</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/><a NAME="E50:8_1_2_1_5_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A183</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c23">do</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c25">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E66">A174</font></span></a>, <a class="txt" href="#E18:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E57">A161</font></span></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>;<br/><a NAME="E51:8_1_2_1_5_1_3"/><span class="lab"><font color="Green" title="E74">A184</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c25">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E57">A161</font></span></a>, <a class="txt" href="#E49:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E72">A182</font></span></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>;<br/><a NAME="E52:8_1_2_1_5_1_3"/><span class="lab"><font color="Green" title="E75">A185</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c23">do</font> holds <br/><font color="Maroon" title="c23">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1_5_1_3_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c27">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c27">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c23">do</font> implies <font color="Maroon" title="c23">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">i</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c28">I</font> = <font color="Maroon" title="c27">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:8_1_2_1_5_1_3_7"/><span class="lab"><font color="Green" title="E76">A186</font></span>: 
<font color="Maroon" title="c27">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c23">do</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">i</font></span><br/>

<a NAME="E3:8_1_2_1_5_1_3_7"/><span class="kw">then </span>
<font color="Maroon" title="c27">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E76:8_1_2_1"><span class="lab"><font color="Green" title="E43">A77</font></span></a>, <a class="txt" href="#E5:8_1_2_1_5"><span class="lab"><font color="Green" title="E46">A80</font></span></a>, <a class="txt" href="#E37:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E65">A173</font></span></a>, <a class="ref" href="afinsq_1.html#T14" title="AFINSQ_1:th.14">AFINSQ_1:14</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E77">A187</font></span>: <font color="Maroon" title="c25">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c28">I</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">I</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E74">A184</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">I</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c3">b1</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c28">I</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#D1" title="PREPOWER:def.1">PREPOWER:def 1</a></span>
;<br/>
<a NAME="E5:8_1_2_1_5_1_3_7"/><span class="lab"><font color="Green" title="E78">A188</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c23">do</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c9">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></span>;<br/>
<a NAME="E6:8_1_2_1_5_1_3_7"/><span class="kw">then </span>
<font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">b1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:8_1_2_1"><span class="lab"><font color="Green" title="E29">A59</font></span></a>, <a class="txt" href="#E39:8_1_2_1"><span class="lab"><font color="Green" title="E30">A60</font></span></a>, <a class="txt" href="#E2:8_1_2_1_5_1_3_7"><span class="lab"><font color="Green" title="E76">A186</font></span></a></span>;<br/>
<a NAME="E7:8_1_2_1_5_1_3_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A189</font></span>: 
<font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<font color="Maroon" title="c23">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">i</font> = 
<font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1_2_1_5_1_3_7"><span class="lab"><font color="Green" title="E76">A186</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">i</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c27">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:8_1_2_1"><span class="lab"><font color="Green" title="E31">A61</font></span></a>, <a class="txt" href="#E2:8_1_2_1_5_1_3_7"><span class="lab"><font color="Green" title="E76">A186</font></span></a>, <a class="txt" href="#E5:8_1_2_1_5_1_3_7"><span class="lab"><font color="Green" title="E78">A188</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E9:8_1_2_1_5_1_3_7"/>
<font color="Maroon" title="c23">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1_2_1_5_1_3_7"><span class="lab"><font color="Green" title="E77">A187</font></span></a>, <a class="txt" href="#E7:8_1_2_1_5_1_3_7"><span class="lab"><font color="Green" title="E79">A189</font></span></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</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> <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c25">g</font> = 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">ok</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T68" title="AFINSQ_2:th.68">AFINSQ_2:68</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">ok</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#T12" title="SERIES_1:th.12">SERIES_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">ok</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c24">ok</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="series_1.html#T26" title="SERIES_1:th.26">SERIES_1:26</a></span>
<br/>.= 
 <a href="binop_2.html#K13" title="BINOP_2:func.13">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="binop_2.html#K17" title="BINOP_2:func.17">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E57">A161</font></span></a></span>
<br/>.= 
 <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E71">A181</font></span></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1

;<br/><a NAME="E54:8_1_2_1_5_1_3"/><span class="kw">then </span>
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c23">do</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E73">A183</font></span></a>, <a class="txt" href="#E52:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E75">A185</font></span></a>, <a class="ref" href="afinsq_2.html#T69" title="AFINSQ_2:th.69">AFINSQ_2:69</a>, <a class="ref" href="xreal_1.html#T147" title="XREAL_1:th.147">XREAL_1:147</a></span>;<br/><a NAME="E55:8_1_2_1_5_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A190</font></span>: 
<span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c23">do</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T27" title="NAT_D:th.27">NAT_D:27</a></span>;<br/><a NAME="E56:8_1_2_1_5_1_3"/>
for <font color="Olive" title="b1">x</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c23">do</font> holds <br/><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/><a NAME="E57:8_1_2_1_5_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c9">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">do</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E68">A177</font></span></a>, <a class="txt" href="#E41:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E69">A178</font></span></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></span>;<br/><a NAME="E58:8_1_2_1_5_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A191</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c23">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:8_1_2_1"><span class="lab"><font color="Green" title="E32">A62</font></span></a>, <a class="ref" href="afinsq_2.html#T67" title="AFINSQ_2:th.67">AFINSQ_2:67</a></span>;<br/><a NAME="E59:8_1_2_1_5_1_3"/><span class="lab"><font color="Green" title="E78">A192</font></span>: 
<font color="Maroon" title="c26">bo</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="prepower.html#T12" title="PREPOWER:th.12">PREPOWER:12</a></span>;<br/><a NAME="E60:8_1_2_1_5_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E70">A180</font></span></a>, <a class="ref" href="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></span>;<br/><a NAME="E61:8_1_2_1_5_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p3"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c23">do</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E77">A191</font></span></a>, <a class="ref" href="nat_d.html#T19" title="NAT_D:th.19">NAT_D:19</a></span>;<br/><a NAME="E62:8_1_2_1_5_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E76">A190</font></span></a>, <a class="txt" href="#E43:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E70">A180</font></span></a>, <a class="ref" href="nat_d.html#T3" title="NAT_D:th.3">NAT_D:3</a></span>;<br/><a NAME="E63:8_1_2_1_5_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T65" title="AFINSQ_2:th.65">AFINSQ_2:65</a></span>;<br/><a NAME="E64:8_1_2_1_5_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:8_1_2_1"><span class="lab"><font color="Green" title="E30">A60</font></span></a>, <a class="txt" href="#E40:8_1_2_1"><span class="lab"><font color="Green" title="E31">A61</font></span></a>, <a class="txt" href="#E76:8_1_2_1"><span class="lab"><font color="Green" title="E43">A77</font></span></a>, <a class="txt" href="#E1:8_1_2_1_5"><span class="lab"><font color="Green" title="E44">A78</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E65:8_1_2_1_5_1_3"/>
<font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E64">A172</font></span></a>, <a class="txt" href="#E59:8_1_2_1_5_1_3"><span class="lab"><font color="Green" title="E78">A192</font></span></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_2_1_5_1_4"><span class="kw">suppose </span></a><a NAME="E1:8_1_2_1_5_1_4"/><span class="lab"><font color="Green" title="E47">A193</font></span>: 
( <font color="Maroon" title="c13">o</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c13">o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c15">o1</font> = <font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E3:8_1_2_1_5_1_4"/><span class="lab"><font color="Green" title="E48">A194</font></span>: 
<font color="Maroon" title="c15">o1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E47">A193</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:8_1_2_1_5_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A195</font></span>: 
<font color="Maroon" title="c15">o1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T149" title="XREAL_1:th.149">XREAL_1:149</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">bo1</font> = <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c15">o1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">do1</font> = <font color="Maroon" title="c7">D</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c15">o1</font> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">bo</font> = <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c13">o</font></span>)</span>;<br/><span class="kw">consider </span><font color="Maroon" title="c19">ok1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E9:8_1_2_1_5_1_4"/><span class="lab"><font color="Green" title="E50">A196</font></span>: 
<font color="Maroon" title="c19">ok1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">o1</font>
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">ok1</font> = <font color="Maroon" title="c19">ok1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">g1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">ok1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="numeral1.html#T3" target="_self" title="NUMERAL1:th.3">Th3</a></span>;<br/><a NAME="E12:8_1_2_1_5_1_4"/><span class="lab"><font color="Green" title="E51">A197</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c7">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:8_1_2_1"><span class="lab"><font color="Green" title="E14">A41</font></span></a></span>;<br/><a NAME="E13:8_1_2_1_5_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A198</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c17">do1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">o1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E49">A195</font></span></a>, <a class="ref" href="afinsq_1.html#T14" title="AFINSQ_1:th.14">AFINSQ_1:14</a></span>;<br/><a NAME="E14:8_1_2_1_5_1_4"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E15:8_1_2_1_5_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A200</font></span>: 
<font color="Maroon" title="c15">o1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/><a NAME="E16:8_1_2_1_5_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A201</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c17">do1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c21">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E52">A198</font></span></a>, <a class="txt" href="#E9:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E50">A196</font></span></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>;<br/><a NAME="E17:8_1_2_1_5_1_4"/><span class="lab"><font color="Green" title="E55">A202</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c21">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">o1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E50">A196</font></span></a>, <a class="txt" href="#E15:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E53">A200</font></span></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>;<br/><a NAME="E18:8_1_2_1_5_1_4"/><span class="lab"><font color="Green" title="E56">A203</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c17">do1</font> holds <br/><font color="Maroon" title="c17">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1_5_1_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c22">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c17">do1</font> implies <font color="Maroon" title="c17">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c23">I</font> = <font color="Maroon" title="c22">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:8_1_2_1_5_1_4_1"/><span class="lab"><font color="Green" title="E57">A204</font></span>: 
<font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c17">do1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span><br/>

<a NAME="E3:8_1_2_1_5_1_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">o1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E49">A195</font></span></a>, <a class="txt" href="#E12:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E51">A197</font></span></a>, <a class="ref" href="afinsq_1.html#T14" title="AFINSQ_1:th.14">AFINSQ_1:14</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E58">A205</font></span>: <font color="Maroon" title="c21">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">I</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">I</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E55">A202</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">I</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c3">b1</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c23">I</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#D1" title="PREPOWER:def.1">PREPOWER:def 1</a></span>
;<br/>
<a NAME="E5:8_1_2_1_5_1_4_1"/><span class="lab"><font color="Green" title="E59">A206</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c17">do1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></span>;<br/>
<a NAME="E6:8_1_2_1_5_1_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">b1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:8_1_2_1"><span class="lab"><font color="Green" title="E13">A40</font></span></a>, <a class="txt" href="#E13:8_1_2_1"><span class="lab"><font color="Green" title="E14">A41</font></span></a>, <a class="txt" href="#E2:8_1_2_1_5_1_4_1"><span class="lab"><font color="Green" title="E57">A204</font></span></a></span>;<br/>
<a NAME="E7:8_1_2_1_5_1_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A207</font></span>: 
<font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<font color="Maroon" title="c17">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font> = 
<font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1_2_1_5_1_4_1"><span class="lab"><font color="Green" title="E57">A204</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c22">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:8_1_2_1"><span class="lab"><font color="Green" title="E15">A42</font></span></a>, <a class="txt" href="#E2:8_1_2_1_5_1_4_1"><span class="lab"><font color="Green" title="E57">A204</font></span></a>, <a class="txt" href="#E5:8_1_2_1_5_1_4_1"><span class="lab"><font color="Green" title="E59">A206</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E9:8_1_2_1_5_1_4_1"/>
<font color="Maroon" title="c17">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1_2_1_5_1_4_1"><span class="lab"><font color="Green" title="E58">A205</font></span></a>, <a class="txt" href="#E7:8_1_2_1_5_1_4_1"><span class="lab"><font color="Green" title="E60">A207</font></span></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</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><a NAME="E19:8_1_2_1_5_1_4"/>
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T149" title="XREAL_1:th.149">XREAL_1:149</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">oo1</font> = <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c15">o1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E48">A194</font></span></a>, <a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E21:8_1_2_1_5_1_4"/><span class="lab"><font color="Green" title="E57">A208</font></span>: 
for <font color="Olive" title="b1">u</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">oo</font> holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">x</font>]
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c23">d9</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E23:8_1_2_1_5_1_4"/><span class="lab"><font color="Green" title="E58">A209</font></span>: 
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c23">d9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">oo</font> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">oo</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c23">d9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="stirl2_1.html#S5" title="STIRL2_1:sch.5">STIRL2_1:sch 5</a>(<a class="txt" href="#E21:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E57">A208</font></span></a>);<br/></span><div><span class="lab"><font color="Green" title="E59">A210</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1_5_1_4_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c24">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> implies <font color="Maroon" title="c23">d9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">k</font> )</span><br/><span class="kw">assume </span><a NAME="E1:8_1_2_1_5_1_4_2"/>
<font color="Maroon" title="c24">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</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" title="c23">d9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span><br/><a NAME="E2:8_1_2_1_5_1_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c24">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T36" title="AFINSQ_1:th.36">AFINSQ_1:36</a></span>;<br/><a NAME="E3:8_1_2_1_5_1_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c24">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E4:8_1_2_1_5_1_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A211</font></span>: 
<font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/><a NAME="E5:8_1_2_1_5_1_4_2"/>
<font color="Maroon" title="c14">oo</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_2_1_5"><span class="lab"><font color="Green" title="E46">A80</font></span></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></span>;<br/><a NAME="E6:8_1_2_1_5_1_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c24">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c23">d9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E58">A209</font></span></a>, <a class="txt" href="#E4:8_1_2_1_5_1_4_2"><span class="lab"><font color="Green" title="E60">A211</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c23">d9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">k</font> = 
<font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c13">o</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E58">A209</font></span></a></span>
<br/>.= 
<span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1_2_1_5_1_4_2"><span class="lab"><font color="Green" title="E60">A211</font></span></a>, <a class="ref" href="afinsq_1.html#D5" title="AFINSQ_1:def.5">AFINSQ_1:def 5</a></span>
;<br/> <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">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c15">o1</font></span>)</span>;<br/><a NAME="E25:8_1_2_1_5_1_4"/><span class="lab"><font color="Green" title="E60">A212</font></span>: 
for <font color="Olive" title="b1">u</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">oo1</font> holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">x</font>]
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c24">d91</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E27:8_1_2_1_5_1_4"/><span class="lab"><font color="Green" title="E61">A213</font></span>: 
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c24">d91</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">oo1</font> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">oo1</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c24">d91</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="stirl2_1.html#S5" title="STIRL2_1:sch.5">STIRL2_1:sch 5</a>(<a class="txt" href="#E25:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E60">A212</font></span></a>);<br/></span><span class="kw">reconsider </span><font color="Maroon" title="c25">do</font> = <font color="Maroon" title="c7">D</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c13">o</font> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E29:8_1_2_1_5_1_4"/><span class="lab"><font color="Green" title="E62">A214</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c7">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:8_1_2_1"><span class="lab"><font color="Green" title="E14">A41</font></span></a></span>;<br/><a NAME="E30:8_1_2_1_5_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A215</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c25">do</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_2_1_5"><span class="lab"><font color="Green" title="E46">A80</font></span></a>, <a class="ref" href="afinsq_1.html#T14" title="AFINSQ_1:th.14">AFINSQ_1:14</a></span>;<br/><a NAME="E31:8_1_2_1_5_1_4"/><span class="lab"><font color="Green" title="E64">A217</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c25">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c23">d9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:8_1_2_1"><span class="lab"><font color="Green" title="E14">A41</font></span></a>, <a class="txt" href="#E23:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E58">A209</font></span></a>, <a class="txt" href="#E30:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E63">A215</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1_5_1_4_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c26">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c26">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c23">d9</font> implies <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c23">d9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c26">k</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c27">K</font> = <font color="Maroon" title="c26">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">assume </span><a NAME="E2:8_1_2_1_5_1_4_3"/><span class="lab"><font color="Green" title="E65">A218</font></span>: 
<font color="Maroon" title="c26">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c23">d9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c23">d9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c26">k</font></span><br/><a NAME="E3:8_1_2_1_5_1_4_3"/><span class="kw">then </span>
<font color="Maroon" title="c26">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c23">d9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E4:8_1_2_1_5_1_4_3"/><span class="kw">then </span>
<font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c26">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E63">A215</font></span></a>, <a class="txt" href="#E31:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E64">A217</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E5:8_1_2_1_5_1_4_3"/><span class="kw">then </span>
<font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c26">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">D</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E6:8_1_2_1_5_1_4_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c26">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c26">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c26">k</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:8_1_2_1"><span class="lab"><font color="Green" title="E15">A42</font></span></a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c23">d9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">K</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c26">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c26">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E58">A209</font></span></a>, <a class="txt" href="#E2:8_1_2_1_5_1_4_3"><span class="lab"><font color="Green" title="E65">A218</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c26">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c26">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c26">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c26">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span>

;<br/><span class="kw">hence </span><a NAME="E8:8_1_2_1_5_1_4_3"/>
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c23">d9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c26">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</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><a NAME="E33:8_1_2_1_5_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A219</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c23">d9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="numeral1.html#T5" target="_self" title="NUMERAL1:th.5">Th5</a></span>;<br/><a NAME="E34:8_1_2_1_5_1_4"/>
( ( for <font color="Olive" title="b1">x</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c25">do</font> holds <br/><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c23">d9</font> holds <br/><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c25">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">d9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E58">A209</font></span></a>, <a class="txt" href="#E30:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E63">A215</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/><a NAME="E35:8_1_2_1_5_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c7">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">do</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c23">d9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E64">A217</font></span></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></span>;<br/><a NAME="E36:8_1_2_1_5_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A220</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c25">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c23">d9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:8_1_2_1"><span class="lab"><font color="Green" title="E16">A43</font></span></a>, <a class="ref" href="afinsq_2.html#T67" title="AFINSQ_2:th.67">AFINSQ_2:67</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c26">ok</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E38:8_1_2_1_5_1_4"/><span class="lab"><font color="Green" title="E67">A221</font></span>: 
<font color="Maroon" title="c26">ok</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E47">A193</font></span></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c27">ok</font> = <font color="Maroon" title="c26">ok</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c28">g</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">ok</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="numeral1.html#T3" target="_self" title="NUMERAL1:th.3">Th3</a></span>;<br/><a NAME="E41:8_1_2_1_5_1_4"/><span class="lab"><font color="Green" title="E68">A222</font></span>: 
1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a></span>;<br/><span class="lab"><font color="Green" title="E69">A223</font></span>:  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c28">g</font> = 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">ok</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T68" title="AFINSQ_2:th.68">AFINSQ_2:68</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">ok</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#T12" title="SERIES_1:th.12">SERIES_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">ok</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c27">ok</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="series_1.html#T26" title="SERIES_1:th.26">SERIES_1:26</a></span>
<br/>.= 
 <a href="binop_2.html#K13" title="BINOP_2:func.13">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="binop_2.html#K17" title="BINOP_2:func.17">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E67">A221</font></span></a></span>
<br/>.= 
 <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E68">A222</font></span></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1

;<br/><a NAME="E43:8_1_2_1_5_1_4"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E44:8_1_2_1_5_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A224</font></span>: 
<font color="Maroon" title="c13">o</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/><a NAME="E45:8_1_2_1_5_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A225</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c28">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E67">A221</font></span></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>;<br/><a NAME="E46:8_1_2_1_5_1_4"/><span class="lab"><font color="Green" title="E72">A226</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c25">do</font> holds <br/><font color="Maroon" title="c25">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1_5_1_4_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c29">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c29">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c25">do</font> implies <font color="Maroon" title="c25">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">i</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c30">I</font> = <font color="Maroon" title="c29">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:8_1_2_1_5_1_4_5"/><span class="lab"><font color="Green" title="E73">A227</font></span>: 
<font color="Maroon" title="c29">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c25">do</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">i</font></span><br/>

<a NAME="E3:8_1_2_1_5_1_4_5"/><span class="kw">then </span>
<font color="Maroon" title="c29">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_2_1_5"><span class="lab"><font color="Green" title="E46">A80</font></span></a>, <a class="txt" href="#E29:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E62">A214</font></span></a>, <a class="ref" href="afinsq_1.html#T14" title="AFINSQ_1:th.14">AFINSQ_1:14</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E74">A228</font></span>: <font color="Maroon" title="c28">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c30">I</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">I</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E71">A225</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">I</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c3">b1</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c30">I</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#D1" title="PREPOWER:def.1">PREPOWER:def 1</a></span>
;<br/>
<a NAME="E5:8_1_2_1_5_1_4_5"/><span class="lab"><font color="Green" title="E75">A229</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c25">do</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></span>;<br/>
<a NAME="E6:8_1_2_1_5_1_4_5"/><span class="kw">then </span>
<font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">b1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:8_1_2_1"><span class="lab"><font color="Green" title="E13">A40</font></span></a>, <a class="txt" href="#E13:8_1_2_1"><span class="lab"><font color="Green" title="E14">A41</font></span></a>, <a class="txt" href="#E2:8_1_2_1_5_1_4_5"><span class="lab"><font color="Green" title="E73">A227</font></span></a></span>;<br/>
<a NAME="E7:8_1_2_1_5_1_4_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A230</font></span>: 
<font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<font color="Maroon" title="c25">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">i</font> = 
<font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1_2_1_5_1_4_5"><span class="lab"><font color="Green" title="E73">A227</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">i</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:8_1_2_1"><span class="lab"><font color="Green" title="E15">A42</font></span></a>, <a class="txt" href="#E2:8_1_2_1_5_1_4_5"><span class="lab"><font color="Green" title="E73">A227</font></span></a>, <a class="txt" href="#E5:8_1_2_1_5_1_4_5"><span class="lab"><font color="Green" title="E75">A229</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E9:8_1_2_1_5_1_4_5"/>
<font color="Maroon" title="c25">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1_2_1_5_1_4_5"><span class="lab"><font color="Green" title="E74">A228</font></span></a>, <a class="txt" href="#E7:8_1_2_1_5_1_4_5"><span class="lab"><font color="Green" title="E76">A230</font></span></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E73">A231</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1_5_1_4_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c29">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c29">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c24">d91</font> implies <font color="Maroon" title="c23">d9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c29">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">d91</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">k</font> )</span><br/><span class="kw">assume </span><a NAME="E1:8_1_2_1_5_1_4_6"/><span class="lab"><font color="Green" title="E74">A232</font></span>: 
<font color="Maroon" title="c29">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c24">d91</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">d9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c29">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">d91</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">k</font></span><br/><a NAME="E2:8_1_2_1_5_1_4_6"/><span class="kw">then </span>
<font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c22">oo1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E61">A213</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E3:8_1_2_1_5_1_4_6"/><span class="kw">then </span>
<font color="Maroon" title="c29">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c22">oo1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E4:8_1_2_1_5_1_4_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A233</font></span>: 
<font color="Maroon" title="c29">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c23">d9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E58">A209</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c23">d9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c29">k</font></span>)</span> = 
<font color="Maroon" title="c23">d9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c29">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T36" title="AFINSQ_1:th.36">AFINSQ_1:36</a></span>
<br/>.= 
<font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c25">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E58">A209</font></span></a>, <a class="txt" href="#E30:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E63">A215</font></span></a>, <a class="txt" href="#E4:8_1_2_1_5_1_4_6"><span class="lab"><font color="Green" title="E75">A233</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c17">do1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c29">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E63">A215</font></span></a>, <a class="txt" href="#E13:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E52">A198</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c24">d91</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E61">A213</font></span></a>, <a class="txt" href="#E13:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E52">A198</font></span></a>, <a class="txt" href="#E1:8_1_2_1_5_1_4_6"><span class="lab"><font color="Green" title="E74">A232</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> <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c23">d9</font> = 
1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c24">d91</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E58">A209</font></span></a>, <a class="txt" href="#E27:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E61">A213</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c24">d91</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a></span>
;<br/><a NAME="E49:8_1_2_1_5_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c23">d9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c24">d91</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E59">A210</font></span></a>, <a class="txt" href="#E47:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E73">A231</font></span></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></span>;<br/><a NAME="E50:8_1_2_1_5_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A234</font></span>: 
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c23">d9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c24">d91</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T67" title="AFINSQ_2:th.67">AFINSQ_2:67</a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c13">o</font></span>)</span>;<br/><a NAME="E51:8_1_2_1_5_1_4"/><span class="lab"><font color="Green" title="E75">A235</font></span>: 
1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c29">ok</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E53:8_1_2_1_5_1_4"/><span class="lab"><font color="Green" title="E76">A236</font></span>: 
<font color="Maroon" title="c29">ok</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E47">A193</font></span></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></span>;<br/><a NAME="E54:8_1_2_1_5_1_4"/><span class="lab"><font color="Green" title="E77">A237</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c17">do1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c24">d91</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:8_1_2_1"><span class="lab"><font color="Green" title="E14">A41</font></span></a>, <a class="txt" href="#E27:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E61">A213</font></span></a>, <a class="txt" href="#E13:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E52">A198</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1_5_1_4_8"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c30">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c30">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c24">d91</font> implies <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c15">o1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c24">d91</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c30">k</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c31">K</font> = <font color="Maroon" title="c30">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">assume </span><a NAME="E2:8_1_2_1_5_1_4_8"/><span class="lab"><font color="Green" title="E78">A238</font></span>: 
<font color="Maroon" title="c30">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c24">d91</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c15">o1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c24">d91</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c30">k</font></span><br/><a NAME="E3:8_1_2_1_5_1_4_8"/><span class="kw">then </span>
<font color="Maroon" title="c30">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c24">d91</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E4:8_1_2_1_5_1_4_8"/><span class="kw">then </span>
<font color="Maroon" title="c15">o1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c30">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E52">A198</font></span></a>, <a class="txt" href="#E54:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E77">A237</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E5:8_1_2_1_5_1_4_8"/><span class="kw">then </span>
<font color="Maroon" title="c15">o1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c30">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">D</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E6:8_1_2_1_5_1_4_8"/><span class="kw">then </span>
<font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">o1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c30">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">o1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c30">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">o1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c30">k</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:8_1_2_1"><span class="lab"><font color="Green" title="E15">A42</font></span></a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c24">d91</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c31">K</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">o1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c30">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">o1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c30">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E61">A213</font></span></a>, <a class="txt" href="#E2:8_1_2_1_5_1_4_8"><span class="lab"><font color="Green" title="E78">A238</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">o1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c30">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c15">o1</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c30">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">o1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c30">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c30">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c15">o1</font></span>)</span>

;<br/><span class="kw">hence </span><a NAME="E8:8_1_2_1_5_1_4_8"/>
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c15">o1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c24">d91</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c30">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</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><a NAME="E56:8_1_2_1_5_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A239</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c15">o1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c24">d91</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="numeral1.html#T5" target="_self" title="NUMERAL1:th.5">Th5</a></span>;<br/><a NAME="E57:8_1_2_1_5_1_4"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c25">do</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c28">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E63">A215</font></span></a>, <a class="txt" href="#E38:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E67">A221</font></span></a>, <a class="txt" href="#E44:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E70">A224</font></span></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>;<br/><a NAME="E58:8_1_2_1_5_1_4"/><span class="kw">then </span>
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c25">do</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E69">A223</font></span></a>, <a class="txt" href="#E46:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E72">A226</font></span></a>, <a class="ref" href="afinsq_2.html#T69" title="AFINSQ_2:th.69">AFINSQ_2:69</a>, <a class="ref" href="xreal_1.html#T147" title="XREAL_1:th.147">XREAL_1:147</a></span>;<br/><a NAME="E59:8_1_2_1_5_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A240</font></span>: 
<span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c25">do</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T27" title="NAT_D:th.27">NAT_D:27</a></span>;<br/><a NAME="E60:8_1_2_1_5_1_4"/><span class="lab"><font color="Green" title="E80">A241</font></span>: 
1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a></span>;<br/> <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c21">g1</font> = 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">ok1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T68" title="AFINSQ_2:th.68">AFINSQ_2:68</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">ok1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#T12" title="SERIES_1:th.12">SERIES_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">ok1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c20">ok1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="series_1.html#T26" title="SERIES_1:th.26">SERIES_1:26</a></span>
<br/>.= 
 <a href="binop_2.html#K13" title="BINOP_2:func.13">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="binop_2.html#K17" title="BINOP_2:func.17">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c15">o1</font></span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E50">A196</font></span></a></span>
<br/>.= 
 <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c15">o1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E80">A241</font></span></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c15">o1</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1

;<br/><a NAME="E62:8_1_2_1_5_1_4"/><span class="kw">then </span>
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c17">do1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c15">o1</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E54">A201</font></span></a>, <a class="txt" href="#E18:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E56">A203</font></span></a>, <a class="ref" href="afinsq_2.html#T69" title="AFINSQ_2:th.69">AFINSQ_2:69</a>, <a class="ref" href="xreal_1.html#T147" title="XREAL_1:th.147">XREAL_1:147</a></span>;<br/><a NAME="E63:8_1_2_1_5_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A242</font></span>: 
<span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c17">do1</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c15">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T27" title="NAT_D:th.27">NAT_D:27</a></span>;<br/><a NAME="E64:8_1_2_1_5_1_4"/>
( ( for <font color="Olive" title="b1">x</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c17">do1</font> holds <br/><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c24">d91</font> holds <br/><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c17">do1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">d91</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E61">A213</font></span></a>, <a class="txt" href="#E13:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E52">A198</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/><a NAME="E65:8_1_2_1_5_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c7">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">do1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c24">d91</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E77">A237</font></span></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></span>;<br/><a NAME="E66:8_1_2_1_5_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A243</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c17">do1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c24">d91</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:8_1_2_1"><span class="lab"><font color="Green" title="E16">A43</font></span></a>, <a class="ref" href="afinsq_2.html#T67" title="AFINSQ_2:th.67">AFINSQ_2:67</a></span>;<br/><a NAME="E67:8_1_2_1_5_1_4"/>
<font color="Maroon" title="c16">bo1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="prepower.html#T12" title="PREPOWER:th.12">PREPOWER:12</a></span>;<br/><a NAME="E68:8_1_2_1_5_1_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c24">d91</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c15">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E78">A239</font></span></a>, <a class="ref" href="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></span>;<br/><a NAME="E69:8_1_2_1_5_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c15">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c24">d91</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c15">o1</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c17">do1</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c15">o1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E82">A243</font></span></a>, <a class="ref" href="nat_d.html#T19" title="NAT_D:th.19">NAT_D:19</a></span>;<br/><a NAME="E70:8_1_2_1_5_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A244</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c15">o1</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c15">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c24">d91</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E81">A242</font></span></a>, <a class="txt" href="#E56:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E78">A239</font></span></a>, <a class="ref" href="nat_d.html#T3" title="NAT_D:th.3">NAT_D:3</a></span>;<br/><a NAME="E71:8_1_2_1_5_1_4"/><span class="lab"><font color="Green" title="E84">A245</font></span>: 
<font color="Maroon" title="c18">bo</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="prepower.html#T12" title="PREPOWER:th.12">PREPOWER:12</a></span>;<br/><a NAME="E72:8_1_2_1_5_1_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c23">d9</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E65">A219</font></span></a>, <a class="ref" href="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></span>;<br/><a NAME="E73:8_1_2_1_5_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c23">d9</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c25">do</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E66">A220</font></span></a>, <a class="ref" href="nat_d.html#T19" title="NAT_D:th.19">NAT_D:19</a></span>;<br/><a NAME="E74:8_1_2_1_5_1_4"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c23">d9</font> &amp;  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E79">A240</font></span></a>, <a class="txt" href="#E33:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E65">A219</font></span></a>, <a class="ref" href="nat_d.html#T3" title="NAT_D:th.3">NAT_D:3</a>, <a class="ref" href="afinsq_2.html#T65" title="AFINSQ_2:th.65">AFINSQ_2:65</a></span>;<br/><a NAME="E75:8_1_2_1_5_1_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c15">o1</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c15">o1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:8_1_2_1"><span class="lab"><font color="Green" title="E14">A41</font></span></a>, <a class="txt" href="#E14:8_1_2_1"><span class="lab"><font color="Green" title="E15">A42</font></span></a>, <a class="txt" href="#E1:8_1_2_1_5"><span class="lab"><font color="Green" title="E44">A78</font></span></a>, <a class="txt" href="#E70:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E83">A244</font></span></a>, <a class="txt" href="#E50:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E74">A234</font></span></a></span>;<br/><a NAME="E76:8_1_2_1_5_1_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c15">o1</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 1</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></span>;<br/><a NAME="E77:8_1_2_1_5_1_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c15">o1</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 1</span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E78:8_1_2_1_5_1_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c15">o1</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></span>;<br/><a NAME="E79:8_1_2_1_5_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A246</font></span>: 
<font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c15">o1</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E71:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E84">A245</font></span></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c30">o1</font> = <font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E81:8_1_2_1_5_1_4"/><span class="lab"><font color="Green" title="E86">A247</font></span>: 
<font color="Maroon" title="c30">o1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E47">A193</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E82:8_1_2_1_5_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A248</font></span>: 
<font color="Maroon" title="c30">o1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T149" title="XREAL_1:th.149">XREAL_1:149</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c31">ok</font> = <font color="Maroon" title="c29">ok</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c32">g</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">ok</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="numeral1.html#T3" target="_self" title="NUMERAL1:th.3">Th3</a></span>;<br/><a NAME="E85:8_1_2_1_5_1_4"/><span class="lab"><font color="Green" title="E88">A249</font></span>: 
1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c33">bo1</font> = <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c30">o1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c34">do1</font> = <font color="Maroon" title="c9">E</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c30">o1</font> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c35">bo</font> = <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c36">ok1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E90:8_1_2_1_5_1_4"/><span class="lab"><font color="Green" title="E89">A250</font></span>: 
<font color="Maroon" title="c36">ok1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">o1</font>
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c37">ok1</font> = <font color="Maroon" title="c36">ok1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c38">g1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">ok1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="numeral1.html#T3" target="_self" title="NUMERAL1:th.3">Th3</a></span>;<br/><a NAME="E93:8_1_2_1_5_1_4"/><span class="lab"><font color="Green" title="E90">A251</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c9">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:8_1_2_1"><span class="lab"><font color="Green" title="E30">A60</font></span></a></span>;<br/><a NAME="E94:8_1_2_1_5_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A252</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c34">do1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">o1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E76:8_1_2_1"><span class="lab"><font color="Green" title="E43">A77</font></span></a>, <a class="txt" href="#E82:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E87">A248</font></span></a>, <a class="ref" href="afinsq_1.html#T14" title="AFINSQ_1:th.14">AFINSQ_1:14</a></span>;<br/><a NAME="E95:8_1_2_1_5_1_4"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E96:8_1_2_1_5_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A254</font></span>: 
<font color="Maroon" title="c30">o1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/><a NAME="E97:8_1_2_1_5_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A255</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c34">do1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c38">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E94:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E91">A252</font></span></a>, <a class="txt" href="#E90:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E89">A250</font></span></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>;<br/><a NAME="E98:8_1_2_1_5_1_4"/><span class="lab"><font color="Green" title="E94">A256</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c38">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">o1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E90:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E89">A250</font></span></a>, <a class="txt" href="#E96:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E92">A254</font></span></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>;<br/><a NAME="E99:8_1_2_1_5_1_4"/><span class="lab"><font color="Green" title="E95">A257</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c34">do1</font> holds <br/><font color="Maroon" title="c34">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c38">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1_5_1_4_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c39">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c39">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c34">do1</font> implies <font color="Maroon" title="c34">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c39">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c38">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c39">i</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c40">I</font> = <font color="Maroon" title="c39">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:8_1_2_1_5_1_4_10"/><span class="lab"><font color="Green" title="E96">A258</font></span>: 
<font color="Maroon" title="c39">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c34">do1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c34">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c39">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c38">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c39">i</font></span><br/>

<a NAME="E3:8_1_2_1_5_1_4_10"/><span class="kw">then </span>
<font color="Maroon" title="c39">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c30">o1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E76:8_1_2_1"><span class="lab"><font color="Green" title="E43">A77</font></span></a>, <a class="txt" href="#E82:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E87">A248</font></span></a>, <a class="txt" href="#E93:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E90">A251</font></span></a>, <a class="ref" href="afinsq_1.html#T14" title="AFINSQ_1:th.14">AFINSQ_1:14</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E97">A259</font></span>: <font color="Maroon" title="c38">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c40">I</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c40">I</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E98:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E94">A256</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c40">I</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c3">b1</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c40">I</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#D1" title="PREPOWER:def.1">PREPOWER:def 1</a></span>
;<br/>
<a NAME="E5:8_1_2_1_5_1_4_10"/><span class="lab"><font color="Green" title="E98">A260</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c34">do1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c9">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></span>;<br/>
<a NAME="E6:8_1_2_1_5_1_4_10"/><span class="kw">then </span>
<font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c39">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">b1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:8_1_2_1"><span class="lab"><font color="Green" title="E29">A59</font></span></a>, <a class="txt" href="#E39:8_1_2_1"><span class="lab"><font color="Green" title="E30">A60</font></span></a>, <a class="txt" href="#E2:8_1_2_1_5_1_4_10"><span class="lab"><font color="Green" title="E96">A258</font></span></a></span>;<br/>
<a NAME="E7:8_1_2_1_5_1_4_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A261</font></span>: 
<font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c39">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<font color="Maroon" title="c34">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c39">i</font> = 
<font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c39">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1_2_1_5_1_4_10"><span class="lab"><font color="Green" title="E96">A258</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c39">i</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c39">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:8_1_2_1"><span class="lab"><font color="Green" title="E31">A61</font></span></a>, <a class="txt" href="#E2:8_1_2_1_5_1_4_10"><span class="lab"><font color="Green" title="E96">A258</font></span></a>, <a class="txt" href="#E5:8_1_2_1_5_1_4_10"><span class="lab"><font color="Green" title="E98">A260</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E9:8_1_2_1_5_1_4_10"/>
<font color="Maroon" title="c34">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c39">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c38">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c39">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1_2_1_5_1_4_10"><span class="lab"><font color="Green" title="E97">A259</font></span></a>, <a class="txt" href="#E7:8_1_2_1_5_1_4_10"><span class="lab"><font color="Green" title="E99">A261</font></span></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</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> <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c38">g1</font> = 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c37">ok1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T68" title="AFINSQ_2:th.68">AFINSQ_2:68</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c37">ok1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#T12" title="SERIES_1:th.12">SERIES_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c37">ok1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c37">ok1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="series_1.html#T26" title="SERIES_1:th.26">SERIES_1:26</a></span>
<br/>.= 
 <a href="binop_2.html#K13" title="BINOP_2:func.13">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="binop_2.html#K17" title="BINOP_2:func.17">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c30">o1</font></span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E90:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E89">A250</font></span></a></span>
<br/>.= 
 <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c30">o1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E85:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E88">A249</font></span></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c30">o1</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1

;<br/><a NAME="E101:8_1_2_1_5_1_4"/><span class="kw">then </span>
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c34">do1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c30">o1</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E97:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E93">A255</font></span></a>, <a class="txt" href="#E99:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E95">A257</font></span></a>, <a class="ref" href="afinsq_2.html#T69" title="AFINSQ_2:th.69">AFINSQ_2:69</a>, <a class="ref" href="xreal_1.html#T147" title="XREAL_1:th.147">XREAL_1:147</a></span>;<br/><a NAME="E102:8_1_2_1_5_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A262</font></span>: 
<span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c34">do1</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c30">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T27" title="NAT_D:th.27">NAT_D:27</a></span>;<br/><a NAME="E103:8_1_2_1_5_1_4"/>
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T149" title="XREAL_1:th.149">XREAL_1:149</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c39">oo1</font> = <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c30">o1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E81:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E86">A247</font></span></a>, <a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E105:8_1_2_1_5_1_4"/><span class="lab"><font color="Green" title="E97">A263</font></span>: 
for <font color="Olive" title="b1">u</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">oo</font> holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">x</font>]
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c40">d9</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E107:8_1_2_1_5_1_4"/><span class="lab"><font color="Green" title="E98">A264</font></span>: 
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c40">d9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">oo</font> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">oo</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c40">d9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="stirl2_1.html#S5" title="STIRL2_1:sch.5">STIRL2_1:sch 5</a>(<a class="txt" href="#E105:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E97">A263</font></span></a>);<br/></span><div><span class="lab"><font color="Green" title="E99">A265</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1_5_1_4_12"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c41">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c41">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> implies <font color="Maroon" title="c40">d9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c41">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c41">k</font> )</span><br/><span class="kw">assume </span><a NAME="E1:8_1_2_1_5_1_4_12"/>
<font color="Maroon" title="c41">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</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" title="c40">d9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c41">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c41">k</font></span><br/><a NAME="E2:8_1_2_1_5_1_4_12"/><span class="kw">then </span>
<font color="Maroon" title="c41">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T36" title="AFINSQ_1:th.36">AFINSQ_1:36</a></span>;<br/><a NAME="E3:8_1_2_1_5_1_4_12"/><span class="kw">then </span>
<font color="Maroon" title="c41">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E4:8_1_2_1_5_1_4_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A266</font></span>: 
<font color="Maroon" title="c41">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/><a NAME="E5:8_1_2_1_5_1_4_12"/>
<font color="Maroon" title="c14">oo</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_2_1_5"><span class="lab"><font color="Green" title="E46">A80</font></span></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></span>;<br/><a NAME="E6:8_1_2_1_5_1_4_12"/><span class="kw">then </span>
<font color="Maroon" title="c41">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c40">d9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E107:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E98">A264</font></span></a>, <a class="txt" href="#E4:8_1_2_1_5_1_4_12"><span class="lab"><font color="Green" title="E100">A266</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c40">d9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c41">k</font> = 
<font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c13">o</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E107:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E98">A264</font></span></a></span>
<br/>.= 
<span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c41">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1_2_1_5_1_4_12"><span class="lab"><font color="Green" title="E100">A266</font></span></a>, <a class="ref" href="afinsq_1.html#D5" title="AFINSQ_1:def.5">AFINSQ_1:def 5</a></span>
;<br/> <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">defpred </span><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c30">o1</font></span>)</span>;<br/><a NAME="E109:8_1_2_1_5_1_4"/><span class="lab"><font color="Green" title="E100">A267</font></span>: 
for <font color="Olive" title="b1">u</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c39">oo1</font> holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">x</font>]
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c41">d91</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E111:8_1_2_1_5_1_4"/><span class="lab"><font color="Green" title="E101">A268</font></span>: 
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c41">d91</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">oo1</font> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c39">oo1</font> holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c41">d91</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="stirl2_1.html#S5" title="STIRL2_1:sch.5">STIRL2_1:sch 5</a>(<a class="txt" href="#E109:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E100">A267</font></span></a>);<br/></span><a NAME="E112:8_1_2_1_5_1_4"/><span class="lab"><font color="Green" title="E102">A269</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c9">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c34">do1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c41">d91</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:8_1_2_1"><span class="lab"><font color="Green" title="E30">A60</font></span></a>, <a class="txt" href="#E76:8_1_2_1"><span class="lab"><font color="Green" title="E43">A77</font></span></a>, <a class="txt" href="#E111:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E101">A268</font></span></a>, <a class="txt" href="#E94:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E91">A252</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1_5_1_4_13"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c42">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c42">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c41">d91</font> implies <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c30">o1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c41">d91</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c42">k</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c43">K</font> = <font color="Maroon" title="c42">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">assume </span><a NAME="E2:8_1_2_1_5_1_4_13"/><span class="lab"><font color="Green" title="E103">A270</font></span>: 
<font color="Maroon" title="c42">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c41">d91</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c30">o1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c41">d91</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c42">k</font></span><br/><a NAME="E3:8_1_2_1_5_1_4_13"/><span class="kw">then </span>
<font color="Maroon" title="c42">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c41">d91</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E4:8_1_2_1_5_1_4_13"/><span class="kw">then </span>
<font color="Maroon" title="c30">o1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c42">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c9">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E94:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E91">A252</font></span></a>, <a class="txt" href="#E112:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E102">A269</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E5:8_1_2_1_5_1_4_13"/><span class="kw">then </span>
<font color="Maroon" title="c30">o1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c42">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c9">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E6:8_1_2_1_5_1_4_13"/><span class="kw">then </span>
<font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">o1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c42">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c30">o1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c42">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c30">o1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c42">k</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:8_1_2_1"><span class="lab"><font color="Green" title="E31">A61</font></span></a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c41">d91</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c43">K</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c30">o1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c42">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c30">o1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c42">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E111:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E101">A268</font></span></a>, <a class="txt" href="#E2:8_1_2_1_5_1_4_13"><span class="lab"><font color="Green" title="E103">A270</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c30">o1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c42">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c30">o1</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c42">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c30">o1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c42">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c42">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c30">o1</font></span>)</span>

;<br/><span class="kw">hence </span><a NAME="E8:8_1_2_1_5_1_4_13"/>
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c30">o1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c41">d91</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c42">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</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><a NAME="E114:8_1_2_1_5_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A271</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c30">o1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c41">d91</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="numeral1.html#T5" target="_self" title="NUMERAL1:th.5">Th5</a></span>;<br/><a NAME="E115:8_1_2_1_5_1_4"/>
( ( for <font color="Olive" title="b1">x</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c34">do1</font> holds <br/><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c41">d91</font> holds <br/><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c34">do1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">d91</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E111:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E101">A268</font></span></a>, <a class="txt" href="#E94:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E91">A252</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/><a NAME="E116:8_1_2_1_5_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c9">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">do1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c41">d91</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E112:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E102">A269</font></span></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></span>;<br/><a NAME="E117:8_1_2_1_5_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A272</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c34">do1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c41">d91</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:8_1_2_1"><span class="lab"><font color="Green" title="E32">A62</font></span></a>, <a class="ref" href="afinsq_2.html#T67" title="AFINSQ_2:th.67">AFINSQ_2:67</a></span>;<br/><a NAME="E118:8_1_2_1_5_1_4"/>
<font color="Maroon" title="c33">bo1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="prepower.html#T12" title="PREPOWER:th.12">PREPOWER:12</a></span>;<br/><a NAME="E119:8_1_2_1_5_1_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c41">d91</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c30">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E114:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E103">A271</font></span></a>, <a class="ref" href="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></span>;<br/><a NAME="E120:8_1_2_1_5_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c30">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c41">d91</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c30">o1</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c34">do1</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c30">o1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E117:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E104">A272</font></span></a>, <a class="ref" href="nat_d.html#T19" title="NAT_D:th.19">NAT_D:19</a></span>;<br/><a NAME="E121:8_1_2_1_5_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A273</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c30">o1</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c30">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c41">d91</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E102:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E96">A262</font></span></a>, <a class="txt" href="#E114:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E103">A271</font></span></a>, <a class="ref" href="nat_d.html#T3" title="NAT_D:th.3">NAT_D:3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c42">do</font> = <font color="Maroon" title="c9">E</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c13">o</font> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E123:8_1_2_1_5_1_4"/><span class="lab"><font color="Green" title="E106">A274</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c9">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:8_1_2_1"><span class="lab"><font color="Green" title="E30">A60</font></span></a></span>;<br/><a NAME="E124:8_1_2_1_5_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A275</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c42">do</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E76:8_1_2_1"><span class="lab"><font color="Green" title="E43">A77</font></span></a>, <a class="txt" href="#E5:8_1_2_1_5"><span class="lab"><font color="Green" title="E46">A80</font></span></a>, <a class="ref" href="afinsq_1.html#T14" title="AFINSQ_1:th.14">AFINSQ_1:14</a></span>;<br/><a NAME="E125:8_1_2_1_5_1_4"/><span class="lab"><font color="Green" title="E108">A277</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c9">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c42">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c40">d9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:8_1_2_1"><span class="lab"><font color="Green" title="E30">A60</font></span></a>, <a class="txt" href="#E76:8_1_2_1"><span class="lab"><font color="Green" title="E43">A77</font></span></a>, <a class="txt" href="#E107:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E98">A264</font></span></a>, <a class="txt" href="#E124:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E107">A275</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1_5_1_4_14"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c43">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c43">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c40">d9</font> implies <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c40">d9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c43">k</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c44">K</font> = <font color="Maroon" title="c43">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">assume </span><a NAME="E2:8_1_2_1_5_1_4_14"/><span class="lab"><font color="Green" title="E109">A278</font></span>: 
<font color="Maroon" title="c43">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c40">d9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c40">d9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c43">k</font></span><br/><a NAME="E3:8_1_2_1_5_1_4_14"/><span class="kw">then </span>
<font color="Maroon" title="c43">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c40">d9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E4:8_1_2_1_5_1_4_14"/><span class="kw">then </span>
<font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c43">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c9">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E124:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E107">A275</font></span></a>, <a class="txt" href="#E125:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E108">A277</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E5:8_1_2_1_5_1_4_14"/><span class="kw">then </span>
<font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c43">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c9">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E6:8_1_2_1_5_1_4_14"/><span class="kw">then </span>
<font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c43">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c43">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c43">k</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:8_1_2_1"><span class="lab"><font color="Green" title="E31">A61</font></span></a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c40">d9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c44">K</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c43">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c43">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E107:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E98">A264</font></span></a>, <a class="txt" href="#E2:8_1_2_1_5_1_4_14"><span class="lab"><font color="Green" title="E109">A278</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c43">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c43">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c43">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c43">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span>

;<br/><span class="kw">hence </span><a NAME="E8:8_1_2_1_5_1_4_14"/>
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c40">d9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c43">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</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><a NAME="E127:8_1_2_1_5_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A279</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c40">d9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="numeral1.html#T5" target="_self" title="NUMERAL1:th.5">Th5</a></span>;<br/><a NAME="E128:8_1_2_1_5_1_4"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E129:8_1_2_1_5_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A280</font></span>: 
<font color="Maroon" title="c13">o</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/><a NAME="E130:8_1_2_1_5_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">A281</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c42">do</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c32">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E124:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E107">A275</font></span></a>, <a class="txt" href="#E53:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E76">A236</font></span></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>;<br/><a NAME="E131:8_1_2_1_5_1_4"/><span class="lab"><font color="Green" title="E112">A282</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c32">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E76">A236</font></span></a>, <a class="txt" href="#E129:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E110">A280</font></span></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>;<br/><a NAME="E132:8_1_2_1_5_1_4"/><span class="lab"><font color="Green" title="E113">A283</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c42">do</font> holds <br/><font color="Maroon" title="c42">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1_5_1_4_15"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c43">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c43">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c42">do</font> implies <font color="Maroon" title="c42">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c43">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c43">i</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c44">I</font> = <font color="Maroon" title="c43">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:8_1_2_1_5_1_4_15"/><span class="lab"><font color="Green" title="E114">A284</font></span>: 
<font color="Maroon" title="c43">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c42">do</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c42">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c43">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c43">i</font></span><br/>

<a NAME="E3:8_1_2_1_5_1_4_15"/><span class="kw">then </span>
<font color="Maroon" title="c43">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E76:8_1_2_1"><span class="lab"><font color="Green" title="E43">A77</font></span></a>, <a class="txt" href="#E5:8_1_2_1_5"><span class="lab"><font color="Green" title="E46">A80</font></span></a>, <a class="txt" href="#E123:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E106">A274</font></span></a>, <a class="ref" href="afinsq_1.html#T14" title="AFINSQ_1:th.14">AFINSQ_1:14</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E115">A285</font></span>: <font color="Maroon" title="c32">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c44">I</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c44">I</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E131:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E112">A282</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c44">I</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c3">b1</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c44">I</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#D1" title="PREPOWER:def.1">PREPOWER:def 1</a></span>
;<br/>
<a NAME="E5:8_1_2_1_5_1_4_15"/><span class="lab"><font color="Green" title="E116">A286</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c42">do</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c9">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></span>;<br/>
<a NAME="E6:8_1_2_1_5_1_4_15"/><span class="kw">then </span>
<font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c43">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">b1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:8_1_2_1"><span class="lab"><font color="Green" title="E29">A59</font></span></a>, <a class="txt" href="#E39:8_1_2_1"><span class="lab"><font color="Green" title="E30">A60</font></span></a>, <a class="txt" href="#E2:8_1_2_1_5_1_4_15"><span class="lab"><font color="Green" title="E114">A284</font></span></a></span>;<br/>
<a NAME="E7:8_1_2_1_5_1_4_15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">A287</font></span>: 
<font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c43">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<font color="Maroon" title="c42">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c43">i</font> = 
<font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c43">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1_2_1_5_1_4_15"><span class="lab"><font color="Green" title="E114">A284</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c43">i</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c43">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:8_1_2_1"><span class="lab"><font color="Green" title="E31">A61</font></span></a>, <a class="txt" href="#E2:8_1_2_1_5_1_4_15"><span class="lab"><font color="Green" title="E114">A284</font></span></a>, <a class="txt" href="#E5:8_1_2_1_5_1_4_15"><span class="lab"><font color="Green" title="E116">A286</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E9:8_1_2_1_5_1_4_15"/>
<font color="Maroon" title="c42">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c43">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c43">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1_2_1_5_1_4_15"><span class="lab"><font color="Green" title="E115">A285</font></span></a>, <a class="txt" href="#E7:8_1_2_1_5_1_4_15"><span class="lab"><font color="Green" title="E117">A287</font></span></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</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> <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c32">g</font> = 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">ok</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T68" title="AFINSQ_2:th.68">AFINSQ_2:68</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">ok</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#T12" title="SERIES_1:th.12">SERIES_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">ok</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c31">ok</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="series_1.html#T26" title="SERIES_1:th.26">SERIES_1:26</a></span>
<br/>.= 
 <a href="binop_2.html#K13" title="BINOP_2:func.13">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="binop_2.html#K17" title="BINOP_2:func.17">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E76">A236</font></span></a></span>
<br/>.= 
 <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E75">A235</font></span></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1

;<br/><a NAME="E134:8_1_2_1_5_1_4"/><span class="kw">then </span>
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c42">do</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E130:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E111">A281</font></span></a>, <a class="txt" href="#E132:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E113">A283</font></span></a>, <a class="ref" href="afinsq_2.html#T69" title="AFINSQ_2:th.69">AFINSQ_2:69</a>, <a class="ref" href="xreal_1.html#T147" title="XREAL_1:th.147">XREAL_1:147</a></span>;<br/><a NAME="E135:8_1_2_1_5_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">A288</font></span>: 
<span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c42">do</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T27" title="NAT_D:th.27">NAT_D:27</a></span>;<br/><div><span class="lab"><font color="Green" title="E115">A289</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1_5_1_4_17"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c43">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c43">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c41">d91</font> implies <font color="Maroon" title="c40">d9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c43">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">d91</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c43">k</font> )</span><br/><span class="kw">assume </span><a NAME="E1:8_1_2_1_5_1_4_17"/><span class="lab"><font color="Green" title="E116">A290</font></span>: 
<font color="Maroon" title="c43">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c41">d91</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c40">d9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c43">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">d91</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c43">k</font></span><br/><a NAME="E2:8_1_2_1_5_1_4_17"/><span class="kw">then </span>
<font color="Maroon" title="c43">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c39">oo1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E111:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E101">A268</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E3:8_1_2_1_5_1_4_17"/><span class="kw">then </span>
<font color="Maroon" title="c43">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c39">oo1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E4:8_1_2_1_5_1_4_17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">A291</font></span>: 
<font color="Maroon" title="c43">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">oo</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c40">d9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c43">k</font></span>)</span> = 
<font color="Maroon" title="c40">d9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c43">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T36" title="AFINSQ_1:th.36">AFINSQ_1:36</a></span>
<br/>.= 
<font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c42">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c43">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E107:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E98">A264</font></span></a>, <a class="txt" href="#E124:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E107">A275</font></span></a>, <a class="txt" href="#E4:8_1_2_1_5_1_4_17"><span class="lab"><font color="Green" title="E117">A291</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c34">do1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c43">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E124:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E107">A275</font></span></a>, <a class="txt" href="#E94:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E91">A252</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c41">d91</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c43">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E111:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E101">A268</font></span></a>, <a class="txt" href="#E94:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E91">A252</font></span></a>, <a class="txt" href="#E1:8_1_2_1_5_1_4_17"><span class="lab"><font color="Green" title="E116">A290</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> <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c40">d9</font> = 
1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c41">d91</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E107:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E98">A264</font></span></a>, <a class="txt" href="#E111:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E101">A268</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c41">d91</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a></span>
;<br/><a NAME="E138:8_1_2_1_5_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c40">d9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c41">d91</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E108:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E99">A265</font></span></a>, <a class="txt" href="#E136:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E115">A289</font></span></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></span>;<br/><a NAME="E139:8_1_2_1_5_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A292</font></span>: 
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c40">d9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c41">d91</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T67" title="AFINSQ_2:th.67">AFINSQ_2:67</a></span>;<br/><a NAME="E140:8_1_2_1_5_1_4"/>
( ( for <font color="Olive" title="b1">x</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c42">do</font> holds <br/><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c40">d9</font> holds <br/><font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c42">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c40">d9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E107:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E98">A264</font></span></a>, <a class="txt" href="#E124:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E107">A275</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/><a NAME="E141:8_1_2_1_5_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c9">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">do</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c40">d9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E125:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E108">A277</font></span></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></span>;<br/><a NAME="E142:8_1_2_1_5_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">A293</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c42">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c40">d9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:8_1_2_1"><span class="lab"><font color="Green" title="E32">A62</font></span></a>, <a class="ref" href="afinsq_2.html#T67" title="AFINSQ_2:th.67">AFINSQ_2:67</a></span>;<br/><a NAME="E143:8_1_2_1_5_1_4"/><span class="lab"><font color="Green" title="E118">A294</font></span>: 
<font color="Maroon" title="c35">bo</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="prepower.html#T12" title="PREPOWER:th.12">PREPOWER:12</a></span>;<br/><a NAME="E144:8_1_2_1_5_1_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c40">d9</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E127:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E109">A279</font></span></a>, <a class="ref" href="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></span>;<br/><a NAME="E145:8_1_2_1_5_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c40">d9</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c42">do</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E142:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E117">A293</font></span></a>, <a class="ref" href="nat_d.html#T19" title="NAT_D:th.19">NAT_D:19</a></span>;<br/><a NAME="E146:8_1_2_1_5_1_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c40">d9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E135:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E114">A288</font></span></a>, <a class="txt" href="#E127:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E109">A279</font></span></a>, <a class="ref" href="nat_d.html#T3" title="NAT_D:th.3">NAT_D:3</a></span>;<br/><a NAME="E147:8_1_2_1_5_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c9">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c30">o1</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c30">o1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E121:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E105">A273</font></span></a>, <a class="txt" href="#E139:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E116">A292</font></span></a>, <a class="ref" href="afinsq_2.html#T65" title="AFINSQ_2:th.65">AFINSQ_2:65</a></span>;<br/><a NAME="E148:8_1_2_1_5_1_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c30">o1</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c30">o1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:8_1_2_1"><span class="lab"><font color="Green" title="E30">A60</font></span></a>, <a class="txt" href="#E40:8_1_2_1"><span class="lab"><font color="Green" title="E31">A61</font></span></a>, <a class="txt" href="#E76:8_1_2_1"><span class="lab"><font color="Green" title="E43">A77</font></span></a>, <a class="txt" href="#E1:8_1_2_1_5"><span class="lab"><font color="Green" title="E44">A78</font></span></a></span>;<br/><a NAME="E149:8_1_2_1_5_1_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c30">o1</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 1</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></span>;<br/><a NAME="E150:8_1_2_1_5_1_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c30">o1</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 1</span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E151:8_1_2_1_5_1_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c30">o1</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></span>;<br/><span class="kw">hence </span><a NAME="E152:8_1_2_1_5_1_4"/>
<font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E79:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E85">A246</font></span></a>, <a class="txt" href="#E143:8_1_2_1_5_1_4"><span class="lab"><font color="Green" title="E118">A294</font></span></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E78:8_1_2_1"/>
<font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E76:8_1_2_1"><span class="lab"><font color="Green" title="E43">A77</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</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">assume </span><span class="kw">that </span><br/><a NAME="E3:8_1_2"/>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E4:8_1_2"/><span class="lab"><font color="Green" title="E6">A295</font></span>: 
( <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> &amp; <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</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" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font></span><br/>

<span class="kw">thus </span><a NAME="E5:8_1_2"/>
<font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1_2"><span class="lab"><font color="Green" title="E6">A295</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>
