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

<span class="kw">let </span><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">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">  for <font color="Olive" title="b1">a</font> being   non  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> holds <br/> <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c2">k</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c1">n</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">a</font> be   non  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  <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="c2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> implies  <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">k</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">n</font>) )</span><br/>





<span class="kw">assume </span><a NAME="E1:26"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
<font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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">  <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">k</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">n</font>)</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">nk</font> = <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">k</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T21" onmouseover="rs('nat_1/T21')" onmouseout="rh()">NAT_1:21</a></span>;<br/>
<a NAME="E3:26"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
<font color="Maroon" title="c4">nk</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E26">A1</font></span></a></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>] <span class="kw">means </span>( $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>  implies  <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">k</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c3">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> $1</span>)</span>) );<br/>
<a NAME="E4:26"/><span class="lab"><font color="Green" title="E28">A3</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
;<br/>
<a NAME="E5:26"/><span class="lab"><font color="Green" title="E29">A4</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="26_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">i</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">S<sub>1</sub></font>[<font color="Maroon" title="c5">i</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:26_1"/><span class="lab"><font color="Green" title="E30">A5</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">i</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">S<sub>1</sub></font>[<font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]</span><br/>

<a NAME="E2:26_1"/><span class="lab"><font color="Green" title="E31">A6</font></span>: 
 <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c3">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c3">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">i</font></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">a</font></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"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c3">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">i</font></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilb10_1.html#T6" onmouseover="rs('hilb10_1/T6')" onmouseout="rh()">HILB10_1:6</a></span>;<br/>
<a NAME="E3:26_1"/>
 <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c3">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">i</font></span>)</span>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c3">a</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilb10_1.html#T10" onmouseover="rs('hilb10_1/T10')" onmouseout="rh()">HILB10_1:10</a></span>;<br/>
<a NAME="E4:26_1"/><span class="kw">then </span>
 <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c3">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">i</font></span>)</span>) <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="hilb10_1.html#T3" onmouseover="rs('hilb10_1/T3')" onmouseout="rh()">HILB10_1:3</a></span>;<br/>
<a NAME="E5:26_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A7</font></span>: 
<span class="p1">(<span class="default"><a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c3">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">i</font></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c3">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">i</font></span>)</span>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c3">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">i</font></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E6:26_1"/>
<span class="p1">(<span class="default"><a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c3">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">i</font></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c3">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">i</font></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_2.html#T29" onmouseover="rs('nat_2/T29')" onmouseout="rh()">NAT_2:29</a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/>
<a NAME="E7:26_1"/><span class="kw">then </span>
 <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c3">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c3">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">i</font></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_1"><span class="lab"><font color="Green" title="E31">A6</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E8:26_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A8</font></span>: 
 <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c3">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c3">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">i</font></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_1"><span class="lab"><font color="Green" title="E32">A7</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="E9:26_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A9</font></span>: 
 <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c3">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>) <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c3">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">i</font></span>)</span>)
 
<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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_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="E1:26_1_1"/>
( <font color="Maroon" title="c5">i</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="c5">i</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="26_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:26_1_1_1"/>
<font color="Maroon" title="c5">i</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">S<sub>1</sub></font>[<font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:26_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26_1"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</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="26_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:26_1_1_2"/>
<font color="Maroon" title="c5">i</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">S<sub>1</sub></font>[<font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:26_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="txt" href="#E9:26_1"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E6:26"/>
 for <font color="Olive" title="b1">n1</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n1</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E29">A4</font></span></a>);<br/></span>
<a NAME="E7:26"/><span class="kw">then </span>
 <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">k</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c3">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">nk</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E27">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:26"/>
 <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">k</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c3">a</font>,<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"> verum</span><br/>


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