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<span class="kw">let </span><font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">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="c3">n</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  &amp; <span class="p1">(<span class="default">3 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies ( <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:123"/><span class="lab"><font color="Green" title="E99">A1</font></span>: 
<font color="Maroon" title="c3">n</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> 
 <span class="kw">and </span><br/><a NAME="E2:123"/><span class="lab"><font color="Green" title="E100">A2</font></span>: 
<span class="p1">(<span class="default">3 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">m</font></span>)</span> <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"> ( <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:123"/><span class="lab"><font color="Green" title="E101">A3</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:123"><span class="lab"><font color="Green" title="E99">A1</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E102">A4</font></span>: 3 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">k</font></span>)</span> = 
<span class="p1">(<span class="default">3 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T9" onmouseover="rs('newton/T9')" onmouseout="rh()">NEWTON:9</a></span>
<br/>.= 
<span class="p1">(<span class="default">3 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c4">k</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="ref" href="wsierp_1.html#T1" onmouseover="rs('wsierp_1/T1')" onmouseout="rh()">WSIERP_1:1</a></span>
;<br/>
<a NAME="E6:123"/><span class="lab"><font color="Green" title="E103">A5</font></span>: 
2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">3 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c3">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="#E2:123"><span class="lab"><font color="Green" title="E100">A2</font></span></a></span>;<br/>
<a NAME="E7:123"/>
<span class="p1">(<span class="default">3 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">3 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">a</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E9:123"/><span class="lab"><font color="Green" title="E104">A6</font></span>: 
<span class="p1">(<span class="default">3 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c4">k</font></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> 2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:123"><span class="lab"><font color="Green" title="E103">A5</font></span></a>, <a class="txt" href="#E4:123"><span class="lab"><font color="Green" title="E101">A3</font></span></a>, <a class="txt" href="#E5:123"><span class="lab"><font color="Green" title="E102">A4</font></span></a>, <a class="ref" href="newton03.html#T36" onmouseover="rs('newton03/T36')" onmouseout="rh()">NEWTON03:36</a>, <a class="ref" href="xprimes1.html#T2" onmouseover="rs('xprimes1/T2')" onmouseout="rh()">XPRIMES1:2</a></span>;<br/>
<a NAME="E10:123"/>
<span class="p1">(<span class="default">3 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">3 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">b</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E12:123"/><span class="lab"><font color="Green" title="E105">A7</font></span>: 
<span class="p1">(<span class="default">3 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c6">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:123"><span class="lab"><font color="Green" title="E103">A5</font></span></a>, <a class="txt" href="#E4:123"><span class="lab"><font color="Green" title="E101">A3</font></span></a>, <a class="txt" href="#E5:123"><span class="lab"><font color="Green" title="E102">A4</font></span></a>, <a class="ref" href="newton03.html#T36" onmouseover="rs('newton03/T36')" onmouseout="rh()">NEWTON03:36</a>, <a class="ref" href="xprimes1.html#T2" onmouseover="rs('xprimes1/T2')" onmouseout="rh()">XPRIMES1:2</a></span>;<br/>
<a NAME="E13:123"/>
<span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:123"><span class="lab"><font color="Green" title="E104">A6</font></span></a>, <a class="txt" href="#E12:123"><span class="lab"><font color="Green" title="E105">A7</font></span></a></span>;<br/>
<a NAME="E14:123"/><span class="kw">then </span>
<span class="p1">(<span class="default">3 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c4">k</font></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> 4
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:123"><span class="lab"><font color="Green" title="E104">A6</font></span></a>, <a class="txt" href="number08.html#E3"><span class="lab"><font color="Green" title="E3">Lm3</font></span></a>, <a class="ref" href="number08.html#T70" target="_self" onmouseover="rs('number08/T70')" onmouseout="rh()">Th70</a></span>;<br/>
<a NAME="E15:123"/><span class="kw">then </span>
3 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 1
 
;<br/>
<a NAME="E16:123"/><span class="kw">then </span>
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pepin.html#T30" onmouseover="rs('pepin/T30')" onmouseout="rh()">PEPIN:30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:123"/>
( <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:123"><span class="lab"><font color="Green" title="E101">A3</font></span></a>, <a class="txt" href="#E6:123"><span class="lab"><font color="Green" title="E103">A5</font></span></a>, <a class="txt" href="number08.html#E8"><span class="lab"><font color="Green" title="E8">Lm8</font></span></a>, <a class="txt" href="number08.html#E12"><span class="lab"><font color="Green" title="E12">Lm12</font></span></a>, <a class="ref" href="pepin.html#T30" onmouseover="rs('pepin/T30')" onmouseout="rh()">PEPIN:30</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/>


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