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

<span class="kw">let </span><font color="Maroon" title="c1">n</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="c1">n</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">P</font> = (<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font>;<br/>
<a NAME="E1:46"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T30" onmouseover="rs('newton/T30')" onmouseout="rh()">NEWTON:30</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">c</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">F</font> =  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Del</a> (<span class="p1">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></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">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>);<br/>
<a NAME="E2:46"/><span class="lab"><font color="Green" title="E39">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <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="newton.html#D4" onmouseover="rs('newton/D4')" onmouseout="rh()">NEWTON:def 4</a></span>;<br/>
<a NAME="E3:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A3</font></span>: 
<span class="p1">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_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">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T29" onmouseover="rs('newton/T29')" onmouseout="rh()">NEWTON:29</a></span>;<br/>
<a NAME="E4:46"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E5:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span> <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">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E6:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A5</font></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"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Del</a> (<span class="p3">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="wsierp_1.html#T20" onmouseover="rs('wsierp_1/T20')" onmouseout="rh()">WSIERP_1:20</a></span>;<br/>
<a NAME="E7:46"/><span class="lab"><font color="Green" title="E43">A6</font></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"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Del</a> (<span class="p3">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>)</span>)</span></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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="ref" href="wsierp_1.html#D1" onmouseover="rs('wsierp_1/D1')" onmouseout="rh()">WSIERP_1:def 1</a></span>;<br/>
<a NAME="E8:46"/>
 for <font color="Olive" title="b1">b</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">b</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"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Del</a> (<span class="p2">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>)</span>)</span> holds <br/><font color="Maroon" title="c1">n</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Del</a> (<span class="p2">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">b</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">b</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="c5">b</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"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Del</a> (<span class="p2">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>)</span>)</span> implies <font color="Maroon" title="c1">n</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Del</a> (<span class="p2">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:46_1"/><span class="lab"><font color="Green" title="E44">A7</font></span>: 
<font color="Maroon" title="c5">b</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"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Del</a> (<span class="p2">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>)</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="c1">n</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Del</a> (<span class="p2">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font></span><br/>

<a NAME="E2:46_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A8</font></span>: 
<font color="Maroon" title="c5">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Del</a> (<span class="p2">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E3:46_1"/><span class="lab"><font color="Green" title="E46">A9</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_1"><span class="lab"><font color="Green" title="E44">A7</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E4:46_1"/><span class="lab"><font color="Green" title="E47">A10</font></span>: 
<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> <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#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a></span>;<br/>
<a NAME="E5:46_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E7:46"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E2:46_1"><span class="lab"><font color="Green" title="E45">A8</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="E6:46_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A11</font></span>: 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Del</a> (<span class="p2">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T110" onmouseover="rs('finseq_3/T110')" onmouseout="rh()">FINSEQ_3:110</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_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="E1:46_1_1"/>
( <font color="Maroon" title="c5">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 or <font color="Maroon" title="c5">b</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">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:46"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E2:46_1"><span class="lab"><font color="Green" title="E45">A8</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:46_1_1_1"/>
<font color="Maroon" title="c5">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 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" title="c1">n</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Del</a> (<span class="p2">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font></span><br/><div class="add"><a NAME="E2:46_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T52" onmouseover="rs('int_1/T52')" onmouseout="rh()">INT_1:52</a></span>;<br/><a NAME="E3:46_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2
 ;<br/><a NAME="E4:46_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_1"><span class="lab"><font color="Green" title="E46">A9</font></span></a>, <a class="ref" href="number02.html#T41" target="_self" onmouseover="rs('number02/T41')" onmouseout="rh()">Th41</a></span>;<br/><span class="kw">hence </span><a NAME="E5:46_1_1_1"/>
<font color="Maroon" title="c1">n</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Del</a> (<span class="p2">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:46_1"><span class="lab"><font color="Green" title="E48">A11</font></span></a>, <a class="ref" href="wsierp_1.html#T1" onmouseover="rs('wsierp_1/T1')" onmouseout="rh()">WSIERP_1:1</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="46_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:46_1_1_2"/><span class="lab"><font color="Green" title="E49">A12</font></span>: 
<font color="Maroon" title="c5">b</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">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 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" title="c1">n</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Del</a> (<span class="p2">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c6">M</font> = <font color="Maroon" title="c5">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 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="#E3:46_1"><span class="lab"><font color="Green" title="E46">A9</font></span></a>, <a class="ref" href="int_1.html#T5" onmouseover="rs('int_1/T5')" onmouseout="rh()">INT_1:5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">L</font> = <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">M</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="#E2:46"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E1:46_1_1_2"><span class="lab"><font color="Green" title="E49">A12</font></span></a></span>;<br/><a NAME="E4:46_1_1_2"/>
<font color="Maroon" title="c1">n</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">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E3:46_1"><span class="lab"><font color="Green" title="E46">A9</font></span></a>, <a class="txt" href="#E4:46_1"><span class="lab"><font color="Green" title="E47">A10</font></span></a>, <a class="txt" href="#E1:46_1_1_2"><span class="lab"><font color="Green" title="E49">A12</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E5:46_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Maroon" title="c6">M</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">L</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">1 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <font color="Maroon" title="c6">M</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E1:46_1_1_2"><span class="lab"><font color="Green" title="E49">A12</font></span></a>, <a class="ref" href="newton.html#D4" onmouseover="rs('newton/D4')" onmouseout="rh()">NEWTON:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E6:46_1_1_2"/>
<font color="Maroon" title="c1">n</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Del</a> (<span class="p2">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default">(<font color="Maroon" title="c1">n</font>,1) <a href="newton.html#K5" title="NEWTON:func.5">In_Power</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E3:46_1"><span class="lab"><font color="Green" title="E46">A9</font></span></a>, <a class="txt" href="#E6:46_1"><span class="lab"><font color="Green" title="E48">A11</font></span></a>, <a class="txt" href="#E1:46_1_1_2"><span class="lab"><font color="Green" title="E49">A12</font></span></a>, <a class="ref" href="newton.html#T24" onmouseover="rs('newton/T24')" onmouseout="rh()">NEWTON:24</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">hence </span><a NAME="E9:46"/>
<font color="Maroon" title="c1">n</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E3:46"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E6:46"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="ref" href="int_4.html#T36" onmouseover="rs('int_4/T36')" onmouseout="rh()">INT_4:36</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>
