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<span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</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">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c3">b</font> implies  <a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c3">b</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:98"/><span class="lab"><font color="Green" title="E38">A0</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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"> (  not  <a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c3">b</font> or  <a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c3">b</font></span>)</span> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">k</font> = 2 <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c2">a</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/>
<span class="kw">assume </span><a NAME="E3:98"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
 <a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c3">b</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="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c3">b</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">l</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c2">a</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="c3">b</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> as   <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<span class="lab"><font color="Green" title="E40">A2</font></span>: <font color="Maroon" title="c5">l</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c2">a</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:98"><span class="lab"><font color="Green" title="E39">A1</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>
;<br/>
<span class="lab"><font color="Green" title="E41">A3</font></span>: 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c2">a</font></span>)</span> = 
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c4">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton05.html#D3" target="_self" onmouseover="rs('newton05/D3')" onmouseout="rh()">Def1</a></span>
<br/>.= 
2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T6" onmouseover="rs('newton/T6')" onmouseout="rh()">NEWTON:6</a></span>
;<br/>
<a NAME="E7:98"/>
2 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#D1" onmouseover="rs('abian/D1')" onmouseout="rh()">ABIAN:def 1</a></span>;<br/>
<a NAME="E8:98"/><span class="kw">then </span>
(  not 2 is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  &amp; 2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton03.html#T5" onmouseover="rs('newton03/T5')" onmouseout="rh()">NEWTON03:5</a>, <a class="txt" href="#E5:98"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E6:98"><span class="lab"><font color="Green" title="E41">A3</font></span></a></span>;<br/>
<a NAME="E9:98"/><span class="kw">then </span>
2 <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98"><span class="lab"><font color="Green" title="E38">A0</font></span></a>, <a class="ref" href="newton03.html#T59" onmouseover="rs('newton03/T59')" onmouseout="rh()">NEWTON03:59</a></span>;<br/>
<a NAME="E10:98"/><span class="kw">then </span>
2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default">2 <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pepin.html#T31" onmouseover="rs('pepin/T31')" onmouseout="rh()">PEPIN:31</a></span>;<br/>
<a NAME="E11:98"/><span class="kw">then </span>
2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98"><span class="lab"><font color="Green" title="E38">A0</font></span></a>, <a class="ref" href="newton05.html#D3" target="_self" onmouseover="rs('newton05/D3')" onmouseout="rh()">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:98"/>
 <a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c3">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98"><span class="lab"><font color="Green" title="E38">A0</font></span></a>, <a class="txt" href="#E3:98"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E6:98"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="nat_d.html#T7" onmouseover="rs('nat_d/T7')" onmouseout="rh()">NAT_D: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/>


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