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<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"> ( <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 )</span><br/>

<a NAME="E1:7"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
 not  not <font color="Maroon" title="c1">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default">7 <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>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  ...  &amp;  not <font color="Maroon" title="c1">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default">7 <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> 7
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="number03.html#T11" onmouseover="rs('number03/T11')" onmouseout="rh()">NUMBER03:11</a></span>;<br/>
<a NAME="E2:7"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 8</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>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 8</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 &amp; <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> 8</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>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <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> 8</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 8</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>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 8</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D2" onmouseover="rs('nat_d/D2')" onmouseout="rh()">NAT_D:def 2</a></span>;<br/>
<a NAME="E3:7"/>
4 <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T24" onmouseover="rs('nat_d/T24')" onmouseout="rh()">NAT_D:24</a></span>;<br/>
<a NAME="E4:7"/><span class="kw">then </span>
 not  not <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default">7 <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">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8 &amp;  ...  &amp;  not <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default">7 <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">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 7</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="nat_d.html#T67" onmouseover="rs('nat_d/T67')" onmouseout="rh()">NAT_D:67</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:7"/>
( <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7"><span class="lab"><font color="Green" title="E10">A2</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/>


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