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<span class="kw">let </span><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">p</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  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <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> <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="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">a</font> holds <br/><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="c1">n</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="Olive" title="b1">a</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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"> (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <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> <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="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">a</font> implies <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="c1">n</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>,<font color="Maroon" title="c1">n</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="c3">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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="E1:19"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <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> <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="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">a</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"> <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="c1">n</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>,<font color="Maroon" title="c1">n</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="c3">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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/>

<a NAME="E2:19"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
(  <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>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c3">a</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <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="newton.html#K1" title="NEWTON:func.1">|^</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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>, <a class="ref" href="newton.html#T4" onmouseover="rs('newton/T4')" onmouseout="rh()">NEWTON:4</a></span>;<br/>
<a NAME="E3:19"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
(  <a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c3">a</font>,<span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <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> 1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">a</font></span>)</span> &amp;  <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"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <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">1 <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"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <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="txt" href="#E2:19"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="hilb10_1.html#T6" onmouseover="rs('hilb10_1/T6')" onmouseout="rh()">HILB10_1:6</a></span>;<br/>
<a NAME="E4:19"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
 <a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c3">a</font>,<span class="p1">(<span class="default">1 <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"><a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c3">a</font>,1)</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>,1)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">a</font></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="E5:19"/><span class="lab"><font color="Green" title="E23">A5</font></span>: 
( <font color="Maroon" title="c3">a</font> <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> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; <font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T7" onmouseover="rs('int_1/T7')" onmouseout="rh()">INT_1:7</a>, <a class="ref" href="square_1.html#D1" onmouseover="rs('square_1/D1')" onmouseout="rh()">SQUARE_1:def 1</a></span>;<br/>
<a NAME="E6:19"/>
<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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><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> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilb10_1.html#T7" onmouseover="rs('hilb10_1/T7')" onmouseout="rh()">HILB10_1:7</a></span>;<br/>
<a NAME="E7:19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A6</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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <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="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> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
;<br/>
<a NAME="E8:19"/><span class="lab"><font color="Green" title="E25">A7</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <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="c3">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#D1" onmouseover="rs('square_1/D1')" onmouseout="rh()">SQUARE_1:def 1</a></span>;<br/>
<a NAME="E9:19"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <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#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T24" onmouseover="rs('xreal_1/T24')" onmouseout="rh()">XREAL_1:24</a>, <a class="ref" href="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a></span>;<br/>
<a NAME="E10:19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A8</font></span>: 
( <span class="p1">(<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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <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#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</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>, <a class="ref" href="square_1.html#D1" onmouseover="rs('square_1/D1')" onmouseout="rh()">SQUARE_1:def 1</a></span>;<br/>
<a NAME="E11:19"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <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"><font color="Maroon" title="c3">a</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:19"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/>
<a NAME="E12:19"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <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"><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> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:19"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="txt" href="#E8:19"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="ref" href="xreal_1.html#T233" onmouseover="rs('xreal_1/T233')" onmouseout="rh()">XREAL_1:233</a>, <a class="txt" href="#E5:19"><span class="lab"><font color="Green" title="E23">A5</font></span></a></span>;<br/>
<a NAME="E13:19"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:19"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E14:19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A9</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <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 class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T15" onmouseover="rs('square_1/T15')" onmouseout="rh()">SQUARE_1:15</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_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="E1:19_1"/>
( <font color="Maroon" title="c1">n</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="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or ( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; <font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T23" onmouseover="rs('nat_1/T23')" onmouseout="rh()">NAT_1:23</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1"><span class="kw">suppose </span></a><a NAME="E1:19_1_1"/>
( <font color="Maroon" title="c1">n</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="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> <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="c1">n</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>,<font color="Maroon" title="c1">n</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="c3">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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/><div class="add"><span class="kw">hence </span><a NAME="E2:19_1_1"/>
<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="c1">n</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>,<font color="Maroon" title="c1">n</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="c3">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E3:19"><span class="lab"><font color="Green" title="E21">A3</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="19_1_2"><span class="kw">suppose </span></a><a NAME="E1:19_1_2"/>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> <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="c1">n</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>,<font color="Maroon" title="c1">n</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="c3">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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/><div class="add"><span class="kw">hence </span><a NAME="E2:19_1_2"/>
<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="c1">n</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>,<font color="Maroon" title="c1">n</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="c3">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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 class="kw">by</span> <span class="lab"><a class="txt" href="#E14:19"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="ref" href="int_1.html#T7" onmouseover="rs('int_1/T7')" onmouseout="rh()">INT_1:7</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="19_1_3"><span class="kw">suppose </span></a><a NAME="E1:19_1_3"/><span class="lab"><font color="Green" title="E28">A10</font></span>: 
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; <font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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"> <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="c1">n</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>,<font color="Maroon" title="c1">n</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="c3">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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/><div class="add"><a NAME="E2:19_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</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="E3:19_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</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="#E1:19"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="newton.html#T11" onmouseover="rs('newton/T11')" onmouseout="rh()">NEWTON:11</a></span>;<br/><a NAME="E4:19_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A11</font></span>: 
<span class="p1">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">p</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#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">n</font>)</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="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a>, <a class="txt" href="#E1:19_1_3"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="ref" href="nat_1.html#T23" onmouseover="rs('nat_1/T23')" onmouseout="rh()">NAT_1:23</a></span>;<br/><a NAME="E5:19_1_3"/>
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_3"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E6:19_1_3"/><span class="kw">then </span>
(  <a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">n</font>) <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c3">a</font>,2) or  <a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c3">a</font>,2) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hilb10_1.html#T11" onmouseover="rs('hilb10_1/T11')" onmouseout="rh()">HILB10_1:11</a></span>;<br/><a NAME="E7:19_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c3">a</font>,<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"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">n</font>)</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#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c3">a</font>,<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="c3">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E4:19"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E8:19_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A12</font></span>: 
<span class="p1">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">p</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#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c3">a</font>,<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="c3">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19_1_3"><span class="lab"><font color="Green" title="E29">A11</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="E9:19_1_3"/>
<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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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"><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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E10:19_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">p</font> <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#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:19_1_3"><span class="lab"><font color="Green" title="E30">A12</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="E11:19_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">a</font>
 <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="E12:19_1_3"/><span class="kw">then </span>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T20" onmouseover="rs('xreal_1/T20')" onmouseout="rh()">XREAL_1:20</a>, <a class="ref" href="xreal_1.html#T24" onmouseover="rs('xreal_1/T24')" onmouseout="rh()">XREAL_1:24</a></span>;<br/><a NAME="E13:19_1_3"/><span class="kw">then </span>
<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>,<font color="Maroon" title="c1">n</font>)</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"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">n</font>)</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"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <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>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E14:19_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A13</font></span>: 
<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>,<font color="Maroon" title="c1">n</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="c3">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">a</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E14:19"><span class="lab"><font color="Green" title="E27">A9</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="E15:19_1_3"/>
<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>,<font color="Maroon" title="c1">n</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="c3">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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>,<font color="Maroon" title="c1">n</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="c3">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E16:19_1_3"/>
<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="c1">n</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>,<font color="Maroon" title="c1">n</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="c3">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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 class="kw">by</span> <span class="lab"><a class="txt" href="#E14:19_1_3"><span class="lab"><font color="Green" title="E31">A13</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>
