<?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">  for <font color="Olive" title="b1">D</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><br/>  for <font color="Olive" title="b2">w</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Maroon" title="c1">n</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>  holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b2">w</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <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"><a href="jordan1a.html#K1" title="JORDAN1A:func.1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Olive" title="b1">D</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">w</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span><br/><span class="kw">let </span><font color="Maroon" title="c2">D</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</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">  for <font color="Olive" title="b1">w</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Maroon" title="c1">n</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>  holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">w</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <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"><a href="jordan1a.html#K1" title="JORDAN1A:func.1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">w</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span><br/><span class="kw">let </span><font color="Maroon" title="c3">w</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</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="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <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"><a href="jordan1a.html#K1" title="JORDAN1A:func.1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">G</font> =  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c1">n</font>);<br/>
<a NAME="E1:41"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T83" onmouseover="rs('newton/T83')" onmouseout="rh()">NEWTON:83</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:41"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
<font color="Maroon" title="c1">n</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <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"><a href="jordan1a.html#K1" title="JORDAN1A:func.1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span><br/>

<a NAME="E3:41"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A3</font></span>: 
<span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pepin.html#T36" onmouseover="rs('pepin/T36')" onmouseout="rh()">PEPIN:36</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <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"><a href="jordan1a.html#K1" title="JORDAN1A:func.1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan8.html#D1" onmouseover="rs('jordan8/D1')" onmouseout="rh()">JORDAN8:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <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"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <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"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="jordan1a.html#E2"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="nat_d.html#T19" onmouseover="rs('nat_d/T19')" onmouseout="rh()">NAT_D:19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <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">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="pepin.html#T64" onmouseover="rs('pepin/T64')" onmouseout="rh()">PEPIN:64</a></span>
<br/>.= 
<font color="Maroon" title="c3">w</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="xcmplx_1.html#T98" onmouseover="rs('xcmplx_1/T98')" onmouseout="rh()">XCMPLX_1:98</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>
