<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a>  st <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/> ex <font color="Olive" title="b2">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; (  for <font color="Olive" title="b3">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b3">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b2">f</font> holds <br/>( ( <font color="Olive" title="b2">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Olive" title="b1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b3">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> ) &amp; ( <font color="Olive" title="b1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b3">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> implies <font color="Olive" title="b2">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; ( <font color="Olive" title="b2">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  not <font color="Olive" title="b1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b3">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> ) &amp; (  not <font color="Olive" title="b1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b3">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> implies <font color="Olive" title="b2">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) ) ) &amp; <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">Sum</a> <font color="Olive" title="b2">f</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font> be   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K5" title="NUMBERS:func.5">0</a>  implies  ex <font color="Olive" title="b1">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; (  for <font color="Olive" title="b2">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">f</font> holds <br/>( ( <font color="Olive" title="b1">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b2">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> ) &amp; ( <font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b2">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> implies <font color="Olive" title="b1">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; ( <font color="Olive" title="b1">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  not <font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b2">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> ) &amp; (  not <font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b2">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> implies <font color="Olive" title="b1">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) ) ) &amp; <font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">Sum</a> <font color="Olive" title="b1">f</font> ) )</span><br/>



<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span>( ( $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> $1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> ) &amp; ( <font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> $1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; ( $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  not <font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> $1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> ) &amp; (  not <font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> $1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) );<br/>
<span class="kw">set </span><font color="Maroon" title="c3">f1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</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#K7" title="NAT_D:func.7">-'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">n1</font> = <font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">n2</font> = <font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">fa</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> 1;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">fb</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>;<br/>
<span class="lab"><font color="Green" title="E51">A1</font></span>:  <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</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#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K1" title="WSIERP_1:func.1">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K1" title="WSIERP_1:func.1">Sum</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#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T75" onmouseover="rs('rvsum_1/T75')" onmouseout="rh()">RVSUM_1:75</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K1" title="WSIERP_1:func.1">Sum</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#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T80" onmouseover="rs('rvsum_1/T80')" onmouseout="rh()">RVSUM_1:80</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.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#K7" title="NAT_D:func.7">-'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T80" onmouseover="rs('rvsum_1/T80')" onmouseout="rh()">RVSUM_1:80</a></span>
;<br/>
<span class="kw">assume </span><a NAME="E2:54"/><span class="lab"><font color="Green" title="E52">A2</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#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; (  for <font color="Olive" title="b2">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">f</font> holds <br/>( ( <font color="Olive" title="b1">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b2">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> ) &amp; ( <font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b2">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> implies <font color="Olive" title="b1">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; ( <font color="Olive" title="b1">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  not <font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b2">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> ) &amp; (  not <font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b2">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> implies <font color="Olive" title="b1">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) ) ) &amp; <font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">Sum</a> <font color="Olive" title="b1">f</font> )</span><br/>

<div><span class="lab"><font color="Green" title="E53">A3</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="54_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span></a><div class="add"><a NAME="E1:54_2"/>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D4" onmouseover="rs('int_2/D4')" onmouseout="rh()">INT_2:def 4</a></span>;<br/><a NAME="E2:54_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="ref" href="nat_3.html#D7" onmouseover="rs('nat_3/D7')" onmouseout="rh()">NAT_3:def 7</a></span>;<br/><a NAME="E3:54_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A4</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</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="#E2:54"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="ref" href="nat_d.html#T7" onmouseover="rs('nat_d/T7')" onmouseout="rh()">NAT_D:7</a></span>;<br/><a NAME="E4:54_2"/><span class="lab"><font color="Green" title="E55">A5</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D4" onmouseover="rs('int_2/D4')" onmouseout="rh()">INT_2:def 4</a></span>;<br/><span class="kw">assume </span><a NAME="E5:54_2"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E6:54_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:54_2"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="ref" href="power.html#T39" onmouseover="rs('power/T39')" onmouseout="rh()">POWER:39</a></span>;<br/><a NAME="E7:54_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font> <a href="power.html#K2" title="POWER:func.2">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T41" onmouseover="rs('power/T41')" onmouseout="rh()">POWER:41</a></span>;<br/><a NAME="E8:54_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T41" onmouseover="rs('power/T41')" onmouseout="rh()">POWER:41</a></span>;<br/><a NAME="E9:54_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_2"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E10:54_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:54_2"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="ref" href="nat_4.html#T3" target="_self" onmouseover="rs('nat_4/T3')" onmouseout="rh()">Th3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:54"/><span class="lab"><font color="Green" title="E54">A6</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/> ex <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">x</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="54_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> implies  ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Olive" title="b1">x</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:54_3"/>
<font color="Maroon" title="c8">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Olive" title="b1">x</font>]</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:54_3_1"/>
( <font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> or  not <font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:54_3_1_1"/><span class="lab"><font color="Green" title="E55">A7</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Olive" title="b1">x</font>]</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c9">x</font> = 1;<br/><span class="kw">take </span>
1
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,1]</span><br/><span class="kw">thus </span><a NAME="E2:54_3_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_3_1_1"><span class="lab"><font color="Green" title="E55">A7</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="54_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:54_3_1_2"/><span class="lab"><font color="Green" title="E55">A8</font></span>: 
 not <font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Olive" title="b1">x</font>]</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c9">x</font> =  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ;<br/><span class="kw">take </span>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>, <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]</span><br/><span class="kw">thus </span><a NAME="E2:54_3_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>, <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_3_1_2"><span class="lab"><font color="Green" title="E55">A8</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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">consider </span><font color="Maroon" title="c8">f</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E6:54"/><span class="lab"><font color="Green" title="E55">A9</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E7:54"/><span class="lab"><font color="Green" title="E56">A10</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S1" onmouseover="rs('finseq_1/S1')" onmouseout="rh()">FINSEQ_1:sch 1</a>(<a class="txt" href="#E4:54"><span class="lab"><font color="Green" title="E54">A6</font></span></a>);<br/></span>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="54_4"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c8">f</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c8">f</font> implies <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:54_4"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c8">f</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:54_4"/><span class="lab"><font color="Green" title="E57">A11</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">f</font>
 <span class="kw">and </span><br/><a NAME="E4:54_4"/><span class="lab"><font color="Green" title="E58">A12</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">k</font> = <font color="Maroon" title="c10">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_4"><span class="lab"><font color="Green" title="E57">A11</font></span></a></span>;<br/><a NAME="E6:54_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:54"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="txt" href="#E7:54"><span class="lab"><font color="Green" title="E56">A10</font></span></a>, <a class="txt" href="#E3:54_4"><span class="lab"><font color="Green" title="E57">A11</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:54_4"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:54_4"><span class="lab"><font color="Green" title="E58">A12</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E9:54"/><span class="kw">then </span>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c8">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">f</font> = <font color="Maroon" title="c8">f</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" onmouseover="rs('finseq_1/D4')" onmouseout="rh()">FINSEQ_1:def 4</a></span>;<br/>
<span class="lab"><font color="Green" title="E57">A13</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</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#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</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#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</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#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="ref" href="xreal_1.html#T233" onmouseover="rs('xreal_1/T233')" onmouseout="rh()">XREAL_1:233</a></span>
;<br/>
<a NAME="E12:54"/><span class="lab"><font color="Green" title="E58">A14</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">f</font> holds <br/><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</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#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="54_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c10">ff2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">ff1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> 1;<br/>
<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">f</font> implies <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</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#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:54_6"/><span class="lab"><font color="Green" title="E59">A15</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">f</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</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#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">x1</font> = <font color="Maroon" title="c12">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  ;<br/>
<a NAME="E3:54_6"/><span class="lab"><font color="Green" title="E60">A16</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</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#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:54"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="txt" href="#E11:54"><span class="lab"><font color="Green" title="E57">A13</font></span></a>, <a class="txt" href="#E1:54_6"><span class="lab"><font color="Green" title="E59">A15</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_6_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:54_6_1"/>
( <font color="Maroon" title="c13">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> 1</span>)</span> or  ex <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b1">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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#K7" title="NAT_D:func.7">-'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> &amp; <font color="Maroon" title="c13">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">m</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_6"><span class="lab"><font color="Green" title="E60">A16</font></span></a>, <a class="ref" href="finseq_1.html#T25" onmouseover="rs('finseq_1/T25')" onmouseout="rh()">FINSEQ_1:25</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_6_1_1"><span class="kw">suppose </span></a><a NAME="E1:54_6_1_1"/><span class="lab"><font color="Green" title="E61">A17</font></span>: 
<font color="Maroon" title="c13">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> 1</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</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#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font></span><br/><div class="add"><a NAME="E2:54_6_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A18</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" onmouseover="rs('funcop_1/T13')" onmouseout="rh()">FUNCOP_1:13</a></span>;<br/><a NAME="E3:54_6_1_1"/><span class="lab"><font color="Green" title="E63">A19</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c13">x1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="54_6_1_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_6_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:54_6_1_1_1_1"/>
( <font color="Maroon" title="c13">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c13">x1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_6_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:54_6_1_1_1_1_1"/>
<font color="Maroon" title="c13">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c13">x1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font></span><br/><div class="add"><a NAME="E2:54_6_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c13">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T4" onmouseover="rs('newton/T4')" onmouseout="rh()">NEWTON:4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:54_6_1_1_1_1_1"/>
<font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c13">x1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T6" onmouseover="rs('nat_d/T6')" onmouseout="rh()">NAT_D:6</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="54_6_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:54_6_1_1_1_1_2"/>
<font color="Maroon" title="c13">x1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c13">x1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font></span><br/><div class="add"><a NAME="E2:54_6_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">x1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E3:54_6_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">x1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E4:54_6_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A20</font></span>: 
<font color="Maroon" title="c13">x1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T233" onmouseover="rs('xreal_1/T233')" onmouseout="rh()">XREAL_1:233</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c14">j</font> = <font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font>;<br/><span class="kw">set </span><font color="Maroon" title="c15">i</font> = <font color="Maroon" title="c13">x1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1;<br/><a NAME="E5:54_6_1_1_1_1_2"/>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D4" onmouseover="rs('int_2/D4')" onmouseout="rh()">INT_2:def 4</a></span>;<br/><a NAME="E6:54_6_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A21</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="ref" href="nat_3.html#D7" onmouseover="rs('nat_3/D7')" onmouseout="rh()">NAT_3:def 7</a></span>;<br/><a NAME="E7:54_6_1_1_1_1_2"/>
( <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c13">x1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c13">x1</font> &amp; <font color="Maroon" title="c13">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_6_1_1"><span class="lab"><font color="Green" title="E62">A18</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E8:54_6_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A22</font></span>: 
<font color="Maroon" title="c13">x1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:54_6_1_1_1_1_2"><span class="lab"><font color="Green" title="E64">A20</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E9:54_6_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">x1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:54_6_1_1_1_1_2"><span class="lab"><font color="Green" title="E64">A20</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E10:54_6_1_1_1_1_2"/>
<font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c13">x1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:54_6_1_1_1_1_2"><span class="lab"><font color="Green" title="E66">A22</font></span></a>, <a class="txt" href="#E6:54_6_1_1_1_1_2"><span class="lab"><font color="Green" title="E65">A21</font></span></a>, <a class="ref" href="nat_3.html#T4" onmouseover="rs('nat_3/T4')" onmouseout="rh()">NAT_3:4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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><a NAME="E4:54_6_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_6_1_1"><span class="lab"><font color="Green" title="E62">A18</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>;<br/><a NAME="E5:54_6_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</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#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_6_1_1"><span class="lab"><font color="Green" title="E61">A17</font></span></a>, <a class="ref" href="finseq_1.html#D7" onmouseover="rs('finseq_1/D7')" onmouseout="rh()">FINSEQ_1:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E6:54_6_1_1"/>
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</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#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:54"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="txt" href="#E7:54"><span class="lab"><font color="Green" title="E56">A10</font></span></a>, <a class="txt" href="#E1:54_6"><span class="lab"><font color="Green" title="E59">A15</font></span></a>, <a class="txt" href="#E3:54_6_1_1"><span class="lab"><font color="Green" title="E63">A19</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="54_6_1_2"><span class="kw">suppose </span></a><a NAME="E1:54_6_1_2"/>
 ex <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b1">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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#K7" title="NAT_D:func.7">-'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> &amp; <font color="Maroon" title="c13">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">m</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</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#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:54_6_1_2"/><span class="lab"><font color="Green" title="E61">A23</font></span>: 
<font color="Maroon" title="c14">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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#K7" title="NAT_D:func.7">-'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:54_6_1_2"/><span class="lab"><font color="Green" title="E62">A24</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c14">m</font>
 ;<br/><a NAME="E5:54_6_1_2"/>
<font color="Maroon" title="c14">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_6_1_2"><span class="lab"><font color="Green" title="E61">A23</font></span></a>, <a class="ref" href="funcop_1.html#T13" onmouseover="rs('funcop_1/T13')" onmouseout="rh()">FUNCOP_1:13</a></span>;<br/><a NAME="E6:54_6_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/><a NAME="E7:54_6_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E8:54_6_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A25</font></span>: 
<font color="Maroon" title="c13">x1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:54_6_1_2"><span class="lab"><font color="Green" title="E62">A24</font></span></a>, <a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>;<br/><a NAME="E9:54_6_1_2"/><span class="lab"><font color="Green" title="E64">A26</font></span>: 
 not <font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c13">x1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="54_6_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:54_6_1_2_1"/>
<font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c13">x1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:54_6_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A27</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c13">x1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="ref" href="nat_3.html#T30" onmouseover="rs('nat_3/T30')" onmouseout="rh()">NAT_3:30</a></span>;<br/>
<a NAME="E3:54_6_1_2_1"/>
<font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D4" onmouseover="rs('int_2/D4')" onmouseout="rh()">INT_2:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:54_6_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:54_6_1_2"><span class="lab"><font color="Green" title="E63">A25</font></span></a>, <a class="txt" href="#E2:54_6_1_2_1"><span class="lab"><font color="Green" title="E65">A27</font></span></a>, <a class="ref" href="nat_3.html#T25" onmouseover="rs('nat_3/T25')" onmouseout="rh()">NAT_3:25</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E10:54_6_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c14">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ;<br/><a NAME="E11:54_6_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</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#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_6_1_2"><span class="lab"><font color="Green" title="E61">A23</font></span></a>, <a class="txt" href="#E4:54_6_1_2"><span class="lab"><font color="Green" title="E62">A24</font></span></a>, <a class="ref" href="finseq_1.html#D7" onmouseover="rs('finseq_1/D7')" onmouseout="rh()">FINSEQ_1:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E12:54_6_1_2"/>
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</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#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:54"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="txt" href="#E7:54"><span class="lab"><font color="Green" title="E56">A10</font></span></a>, <a class="txt" href="#E1:54_6"><span class="lab"><font color="Green" title="E59">A15</font></span></a>, <a class="txt" href="#E9:54_6_1_2"><span class="lab"><font color="Green" title="E64">A26</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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">take </span>
<font color="Maroon" title="c9">f</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; (  for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">f</font> holds <br/>( ( <font color="Maroon" title="c9">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> ) &amp; ( <font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c9">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; ( <font color="Maroon" title="c9">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  not <font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> ) &amp; (  not <font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c9">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) ) ) &amp; <font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">Sum</a> <font color="Maroon" title="c9">f</font> )</span><br/>

<span class="kw">thus </span><a NAME="E13:54"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:54"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">f</font> holds <br/>( ( <font color="Maroon" title="c9">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> ) &amp; ( <font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c9">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; ( <font color="Maroon" title="c9">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  not <font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> ) &amp; (  not <font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c9">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) ) ) &amp; <font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">Sum</a> <font color="Maroon" title="c9">f</font> )</span><br/>

<span class="kw">thus </span><a NAME="E14:54"/>
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">f</font> holds <br/>( ( <font color="Maroon" title="c9">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> ) &amp; ( <font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c9">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; ( <font color="Maroon" title="c9">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  not <font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> ) &amp; (  not <font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c9">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:54"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="txt" href="#E7:54"><span class="lab"><font color="Green" title="E56">A10</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">Sum</a> <font color="Maroon" title="c9">f</font></span><br/>

<a NAME="E15:54"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</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#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:54"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="txt" href="#E11:54"><span class="lab"><font color="Green" title="E57">A13</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:54"/>
<font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">Sum</a> <font color="Maroon" title="c9">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:54"><span class="lab"><font color="Green" title="E58">A14</font></span></a>, <a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="funct_1.html#T2" onmouseover="rs('funct_1/T2')" onmouseout="rh()">FUNCT_1:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
